n, for\nsome n ∈Z, and assume that Cn is finitely generated. The following are equivalent:\n(1) Hn(C•; ‘\nZ[G]\n±χ\n) = 0;\n(2) C• is chain homotopy equivalent over Z[ker χ] to a chain complex D• of\nprojective Z[ker χ]-modules such that Di = 0 for all i > n −1.\nTheorem A is proved by analysing the long exact sequence in cohomology in-\nduced by a special short exact sequence involving the Novikov ring (see Observa-\ntion 3.2). This short exact sequence was communicated to the author by Andrei\nJaikin-Zapirain as a tool to simplify the original proof of [Fis24b, Theorem A]. It\n\nNOVIKOV COHOMOLOGY\n3\nwas also used in [FIK25] to study the algebraic fibres of Poincaré-duality groups.\nThe most general form of Theorem A will be given in Corollary 3.11; in particular,\nthe theorem still holds if one replaces the coefficient ring Z with any ring R (we\nalways consider maps G ! Z, of course). The crux of the implication (2) ⇒(1) is\nan observation about general modules over Novikov rings: if M is a module over\n‘\nZ[G]\nχ\n(and χ ̸= 0), then either M = 0 or M is uncountable (Lemma 3.5).\nTheorem A has the following immediate consequence for groups, proving that\ntop-dimensional Novikov cohomology is a perfect obstruction for cohomological\ndimension drop. Motivated by Sikorav’s Theorem, we define the Σ∗-invariant of a\ngroup G of type FP by\nΣ∗(G) := {χ ∈H1(G; R) ∖{0} : Hcd(G)(G; ‘\nZ[G]\nχ\n) = 0}.\nCorollary B. Let G be a group of type FP. If χ: G ! Z is an integral character,\nthen cd(ker χ) = cd(G) −1 if and only if ±χ ∈Σ∗(G).\nIt is well-known that vanishing Novikov homology is an open property for groups\nof finite type; this applies just as well to Novikov cohomology, so we obtain the\nfollowing corollary (see Corollary 4.8), which is an analogue of Corollary 1.2 for\ncohomological dimension drop.\nCorollary C. If G is a group of type FP, then cohomological dimension drop is\nan open property among integral characters. More precisely, the set\n{χ: G −! Q : cd(ker χ) = cd(G) −1}\nis an open subset of H1(G; Q). In particular, if cd(G) = 2, then the set of integral\ncharacters with free kernel is open in H1(G; Q).\nFrom a slightly stronger version of Theorem A, we obtain a result on the L2-\ninvariants of RFRS groups which generalises [Fis24b, Theorem A]. Let U(G) be the\nalgebra of operators affiliated to the group von Neumann algebra of G. The L2-\nBetti numbers of G are defined by b(2)\ni (G) = dimU(G) Hi(G; U(G)), where dimU(G)\ndenotes the von Neumann dimension (see [Lüc02] for details). The weak dimension\nof an R-module M is the supremal integer n for which there exists an R-module N\nsuch that TorR\nn (M, N) ̸= 0.\nTheorem D. Let G be a RFRS group of type FP(Q). The following are equivalent:\n(1) U(G) is of weak dimension at most one as a Q[G]-module;\n(2) for every subgroup H ⩽G, we have b(2)\ni (H) = 0 for all i > 1;\n(3) G is an iterated (cyclic or finite) extension of a free group.\nTheorem D will be used to prove that RFRS groups admitting an Abelian hier-\narchy are iterated (cyclic or finite) extensions of free groups, extending a result of\nHagen and Wise [HW10] (see Corollary 5.11). The weak dimension one property\nis a crucial step in Jaikin-Zapirain and Linton’s proof that group algebras of one-\nrelator groups are coherent [JZL25]. In a similar vein, a conjecture of Wise predicts\nthat groups of cohomological dimension at most two are coherent if and only if they\nhave vanishing second L2-Betti number. If G is RFRS and cd(G) ⩽2, then an easy\nconsequence of [Fis24b, Theorem A] is that the vanishing of the second L2-Betti\nnumber is equivalent to U(G) being of weak dimension at most one. Motivated by\nall this, we make the following prediction for coherence in the class of finite type\nRFRS groups.\n\n4\nSAM P. FISHER\nConjecture 1.4. If G is a RFRS group of type FP, then the following are equiva-\nlent:\n(1) G is coherent;\n(2) Q[G] is coherent;\n(3) U(G) is of weak dimension one as a Q[G]-module.\nThe method of proving Theorem A also yields a new proof of Ranicki’s Criterion\n[Ran95] for finite domination of cyclic covers (see Theorem 3.6), as well as a new\nproof of the implication\nH1(G; ‘\nZ[G]\n±χ\n) = 0\n=⇒\nker χ is finitely generated\n(see Proposition 3.8). This is the more subtle direction of Sikorav’s Theorem, and\nis usually the one used in practice (such as in Kielak’s fibring theorem). The new\nproofs are more conceptual, as there is no longer the need to deal with individual\nNovikov cycles. One instead works with long exact sequences of homology mod-\nules and other standard results from homological algebra, and uses the vanishing of\nNovikov homology to show that the homology of the kernel commutes with direct\nproducts. By Brown’s homological characterisation of finite domination [Bro75],\nthis yields finite generation of the kernel. Thus, the method links two well-known\ncriteria for finite domination of chain complexes: Brown’s and Ranicki’s. Another\nsatisfying aspect of this proof is that it unifies the points of view on Novikov co-\nhomology controlling cohomological dimension and Novikov homology controlling\nfiniteness properties of the cyclic covers; these phenomena follow from dual argu-\nments. A drawback, however, is that it is crucial to work with integral characters\nand the vanishing of Novikov homology associated to both χ and −χ simultaneously.\nWe do not obtain an new proof of the fact that χ ∈Σ(G) when H1(G; ‘\nZ[G]\nχ\n) = 0\nfor general characters χ: G ! R.\nFrom the Lyndon–Hochschild–Serre spectral sequence, it is easy to see that the\ncohomological dimension of ker(χ: G ! Z) is at least cd(G) −1, so vanishing of\nNovikov cohomology in low codimensions cannot imply further cohomological di-\nmension drop of the kernel (we will actually see that vanishing of low-codimensional\nNovikov cohomology implies the functors Hi(ker χ; −) commute with direct limits\nin the same low codimensions, which can be seen as a type of finiteness property).\nHowever, it still makes sense to ask whether Novikov cohomology can control fur-\nther cohomological drop of the kernel of a non-integral character. We define the\nhigher Σ∗-invariants of a group G of finite type by\nΣ∗\nm(G) := {χ ∈H1(G; R) ∖{0} : Hi(G; ‘\nZ[G]\nχ\n) = 0 for all i > cd(G) −m}\nfor m ⩾0. Hence, Σ∗\n0(G) = H1(G; R) ∖{0} and Σ∗\n1(G) = Σ∗(G). We obtain the\nfollowing sufficient condition for a map to a free Abelian group to have kernel of\nthe minimal theoretically allowed cohomological dimension.\nTheorem E. Let G be a group of type FP. Let N P G be a normal subgroup such\nthat G/N ∼= Zd. If S(G, N) ⊆Σ∗\nd(G), then cd(N) = n −d.\nGiven the similarity with Theorem 1.1(2), this raises the obvious question of\nwhether the converse statement holds (Corollary B is the important special case\nd = 1); it does not, see Example 4.17. Theorem E has the following corollary for\nPoincaré-duality groups.\n\nNOVIKOV COHOMOLOGY\n5\nCorollary F. Let G be a Poincaré-duality group of dimension n and let χ: G ! Zd.\nIf ker χ is of type FPd−1, then cd(G) = n −d.\nThe condition FP0 is vacuous, so in the case where d = 1, the corollary says that\nall kernels of maps onto Z drop in cohomological dimension. This is a very special\ncase of Strebel’s theorem, which states that all infinite-index subgroups of PDn-\ngroups have cohomological dimension at most n −1. There are various situations\nwhere cohomological dimension is additive under group extensions.\nOne of the\nmost general results is Fel’dman’s theorem [Fel71, Theorem 2.4], which states that\nif 1 ! N ! G ! Q ! 1 is an extension of groups and k is a field such that N is\nof type FP(k) and cdk(Q) < ∞, then cdk(N) + cdk(Q) = cdk(G). An interesting\nfeature of Corollary F is that the finiteness property required on the kernel is much\nweaker than in Fel’dman’s theorem. We will also see in Corollary 4.18 that the\nresult holds over arbitrary rings, whereas it is important in Fel’dman’s theorem\nthat k be a field.\n1.1. Structure of the paper. Section 2 introduces the necessary general homo-\nlogical algebra preliminaries for the article. In Section 3, we will prove Theorem 1.1\nin its full generality. This means working with a general chain complex of finitely\ngenerated R-modules, and arbitrary Laurent series instead of Novikov rings. We will\nalso give a new proof of Ranicki’s Criterion and of the fact that vanishing Novikov\nhomology detects finiteness properties. In Section 4, we define the Σ∗-invariants\nof a group over a general ring, and prove Corollary B and Corollary 1.2, as well\nas Theorem E and Corollary F. In Section 5, we use Kielak’s connection between\nNovikov and L2-cohomology to prove Theorem D. Finally, in Section 6 we look at\nthe example of one-relator groups to show that the top-dimensional Novikov homol-\nogy does not seem to be a useful invariant, in contrast to top-dimensional Novikov\ncohomology. We will see that torsion-free one-relator groups have vanishing second\nNovikov homology with respect to all characters, though this is far from the case\nfor the second Novikov cohomology.\n1.2. Acknowledgments. The author is grateful to Andrei Jaikin-Zapirain, Dawid\nKielak, Ian Leary, and Boris Okun for useful comments. The author also thanks\nICMAT, Andrei, Javier, Marco, and Sara, for their hospitality.\n2. Homological algebra preliminaries\nConvention 2.1. All rings associative and unital, with 1 ̸= 0.\nUnless specified\notherwise, R always denotes an arbitrary ring and G always denotes an arbitrary\ngroup.\nLet R be a ring. A chain complex · · · ! Ci+1 ! Ci ! · · · of R-modules will\nusually be denoted by C• and its boundary maps by ∂i : Ci ! Ci−1 (or just by\n∂when the dimension is understood). Suppose that C• is a chain complex of left\nR-modules. If M is a right R-module, then ith homology of C• with coefficients in\nM is ith homology of the chain complex M ⊗R C• and is denoted by Hi(C•; M).\nSimilarly, if M is a left R-module, then the ith cohomology of C• with coefficients in\nM is the ith cohomology of the cochain complex HomR(C•, M) and is denoted by\nHi(C•; M). There are analogous definitions when C• consists of right R-modules.\nIf C• and D• are chain complexes, then a chain map f• : C• ! D• is a sequence\nof morphisms fi : Ci ! Di such that fi−1∂i = ∂ifi for all i.\nTwo chain maps\n\n6\nSAM P. FISHER\nf•, g• : C• ! D• are said to be chain homotopic if there is a sequence of morphisms\nhi : Ci ! Di+1 such that fi −gi = hi−1∂i + ∂i+1hi for all i. A chain map f• : C• !\nD• is a chain homotopy equivalence if there is chain map g• : D• ! C• such that\nf•g• and g•f• are each chain homotopic to the identity maps. In this case, we say\nthat C• and D• are chain homotopic or of the same chain homotopy type.\nSuppose that C• is a chain complex of projective R-modules. We say that C• is\nfinitely dominated or of finite type if C• is chain homotopic to a chain complex of\nfinitely generated projective R-modules. If Ci = 0 for all i < 0 and n is an integer,\nwe say that C• is of finite n-type if C• is chain homotopic to a chain complex D•\nof projective modules such that Di is finitely generated for i ⩽n and Di = 0 for all\ni < 0. A group G is said to be of type FPn(R) (resp. FP∞(R), resp. FP(R)) if the\ntrivial R[G]-module R admits a projective resolution which is finitely generated in\ndegrees at most n (resp. finitely generated in all degrees, resp. of finite length and\nfinitely generated in all degrees).\nBrown gives the following characterisation of finite domination in terms of the\n(co)homology functors Hi(C•; −) and Hi(C•; −).\nTheorem 2.2 ([Bro75, Theorem 2]). Let C• be a chain complex of projective R-\nmodules such that Ci = 0 for all i < 0. The following are equivalent:\n(1) C• is of finite n-type;\n(2) for all directed systems of R-modules {Mj}j∈J, the natural map\nlim\n−! Hi(C•; Mj) −! Hi(C•; lim\n−! Mj)\nis an isomorphism for i < n and a monomorphism for i = n;\n(3) for all directed systems {Mj}j∈J of R-modules with lim\n−! Mj = 0, we have\nlim\n−! Hi(C•; Mj) = 0 for i ⩽n;\n(4) for any family {Mj}j∈J of R-modules, the natural map\nHi(C•;\nY\nMj) −!\nY\nHi(C•; Mj)\nis an isomorphism for i < n and an epimorphism for i = n;\n(5) for any index set J, the natural map Hi(C•; Q\nJ R) ! Q\nJ Hi(C•; R) is an\nisomorphism for i < n and an epimorphism for i = n.\nIn particular, C• is finitely dominated if and only if Hi(C•; −) preserves direct\nlimits if and only if Hi(C•; −) preserves direct products.\nDefinition 2.3 (Cohomological dimension). The cohomological dimension cd(C•)\nof a chain complex C• of R-modules is the supremal integer n such that there exists\nan R-module M with Hn(C•; M) ̸= 0.\nWe will always use this definition of cohomological dimension of a chain complex,\nbut it is nice to know that it also has the following characterisation when C• consists\nof projective modules.\nLemma 2.4. If C• is a chain complex of projective R-modules, then C• is chain\nhomotopy equivalent to a complex D• such that Di = 0 for all i > n if and only if\ncd(C•) ⩽n.\nProof. If C• is chain homotopy equivalent to D• as in the statement, then it is\nclear that Hi(C•; M) = 0 for all R-modules M and all i > n, since chain homotopy\nequivalences induce isomorphisms on homology. Hence cd(C•) ⩽n.\n\nNOVIKOV COHOMOLOGY\n7\nConversely, suppose that cd(C•) ⩽n. Let M = im(∂n+1) ⩽Cn. By assumption,\nthe cocycle z : Cn+1\n∂−! M is zero in Hn+1(C•; M), so there is a cochain c: Cn ! M\nsuch that c∂= z. This implies that M is a direct summand of Cn; write Cn =\nM ⊕P. Then C• splits as the direct sum of the chain complexes\n{· · · ! Cn+2 ! Cn+1 ! M ! 0 ! · · · } ⊕{· · · ! 0 ! P ! Cn−1 ! Cn−2 ! · · · }\nof projectives modules, where M and P both lie in degree n. We claim that the\ncomplex on the left (call it E•) is null-homotopic, which will prove the lemma.\nIt is an easy exercise to check that Hi(E•; L) = 0 for all integers i and all R-\nmodules L. Let N = im(∂n+2) and consider the short exact sequence\n0 −! N −! Cn+1 −! M −! 0.\nBy assumption, the cocycle w: Cn+2\n∂−! N is zero in cohomology, so there is some\nd: Cn+1 ! N such that w = d∂. Hence, the short exact sequence above is split, so\nE• = {· · · ! Cn+2 ! N ! 0 ! · · · } ⊕{· · · ! 0 ! M ! M ! 0 ! · · · }.\nAgain, the complex {· · · ! Cn+2 ! N ! 0 ! · · · } has vanishing cohomology\nwith all coefficients, so we can split off {· · · ! 0 ! N ! N ! 0 ! · · · } as a\ndirect summand, whose complement has vanishing cohomology with all coefficients.\nContinuing in this way show shows that E• is a direct sum of complexes of the form\n{0 ! L ! L ! 0} and is therefore null-homotopic.\n□\nThe Eilenberg–Watts theorem will be useful to us in induction arguments, in-\ncluding in the proof of Lemma 2.6 just below. Morally, it says that tensoring is the\nonly right-exact additive functor that preserves direct sums.\nTheorem 2.5 ([Wat60]). Let R and S be rings, and let F : ModR ! ModS be an\nadditive functor. If F is right-exact and preserves arbitrary direct sums, then there\nis a natural isomorphism\nF(−) ∼= −⊗R F(R),\nwhere R is viewed as a right module over itself via multiplication, and F(R) is\nviewed as a left R-module via r · x := F(mr)(x) (and mr is the left-multiplication\nby r map).\nLemma 2.6. Let R be a ring, let C• be a chain complex of projective left R-modules\nand let m and n be integers.\n(1) Suppose Ci = 0 for all i < 0.\nIf Hi(C•; R) = 0 for all i ⩽n, then\nHi(C•; M) = 0 for all i ⩽n and all R-modules M.\n(2) Suppose Ci = 0 for all i > n, that Hi(C•; −) commutes with direct limits\nfor all i ⩾m, and that Hi(C•; R) = 0 for all i ⩾m. Then Hi(C•; M) = 0\nfor all i ⩾m and all R-modules M.\nProof. For (1), see [Bro75, Lemma 2]. For (2), we will assume that n ⩾m, since\notherwise the statement is vacuous. We will prove that Hi(C•; M) = 0 for all R-\nmodules M by reverse induction on i, starting with i = n + 1 and terminating with\ni = m. There is nothing to prove in the base case, since Cn+1 = 0 and therefore\nHn+1(C•; M) trivially vanishes for all modules M.\nSuppose Hi(C•; M) = 0 for all R-modules M and for some i > m. Let\n0 −! M0 −! M1 −! M2 −! 0\n\n8\nSAM P. FISHER\nbe a short exact sequence of modules. The associated long exact sequence in coho-\nmology contains the portion\nHi−1(C•; M1) −! Hi−1(C•; M2) −! Hi(C•; M0) = 0,\nso Hi−1(C•; −) is right exact. The functor Hi−1(C•; −) also preserves direct limits\nby assumption, so the Eilenberg–Watts theorem implies that Hi−1(C•; −) is natu-\nrally isomorphic to the functor Hi−1(C•; R)⊗R −, which is zero by assumption.\n□\n3. Twisted Laurent series and finite domination\nLet R be a ring and let α: R ! R be a ring automorphism. The α-twisted\nLaurent polynomial ring in one variable is denoted by Rα[t−1, t]. This is the Laurent\npolynomial ring over R, where we impose the rule rtn = tnαn(r) for all r ∈R and\nall n ∈Z.\nLet Rα[t−1, tK denote the ring of Laurent series in the variable t, i.e. the ring\nof expressions of the form P∞\nn=k rntn with rn ∈R for all n ⩾k, where k is\nsome integer.\nMultiplication is defined on Rα[t−1, tK by formally extending the\nmultiplication on Rα[t−1, t]. Similarly, let RαJt−1, t] be the ring of formal series of\nthe form Pk\nn=−∞rntn for some k ∈Z, i.e. the ring of Laurent series in the variable\nt−1.\nExample 3.1 (The Novikov ring). The main example of the construction above\nwill come from groups and maps to Z. Let χ: G ! R be an homomorphism with\nkernel N and let R be a ring. The Novikov ring associated to this data is\n‘\nR[G]\nχ\n=\nnX\nrgg : |{g : rg ̸= 0} ∩χ−1(]−∞, t])| < ∞for all t ∈R\no\n.\nNow suppose the image of χ is infinite cyclic and let t ∈G map to a generator of\nχ(G). Denoting by α the automorphism of R[N] induced by the conjugation action\nof t, we have ‘\nR[G]\nχ ∼= R[N]α[t−1, tK.\nIn view of Example 3.1, we will often call the (co)homology of a complex C• of\nRα[t−1, t]-modules with coefficients in Rα[t−1, tK the Novikov cohomology of C•.\nGiven a right R-module M, let\nMαJt−1, tK =\nY\ni∈Z\nMti.\nThe elements of MαJt−1, tK are thought of as bi-infinite series P∞\ni=−∞miti, where\nmi ∈M for each i ∈Z.\nWe endow MαJt−1, tK with the structure of a right\nRα[t−1, t]-module by defining\n \n∞\nX\ni=−∞\nmiti\n!\n· rtn =\n∞\nX\ni=−∞\n(miα−i(r))ti+n.\nWe also define Mα[t−1, tK to be the submodule of MαJt−1, tK consisting of series of\nthe form P∞\ni=k miti for some k ∈Z.\nLet Mα[t−1, tK be the Rα[t−1, t]-submodule of series of the form P∞\nn=k mntn\nfor some k ∈Z. Similarly, let MαJt−1, t] be the submodule of series of the form\nPk\nn=−∞mntn for some k ∈Z. Crucially, Mα[t−1, tK is a right Rα[t−1, tK-module,\nand MαJt−1, t] is a right RαJt−1, t]-module.\nFinally, let Mα[t−1, t] be the right\nRα[t−1, t]-module of finitely supported sums P\ni miti.\n\nNOVIKOV COHOMOLOGY\n9\nObservation 3.2. There is a short exact sequence\n0\nMα[t−1, t]\nMαJt−1, t] ⊕Mα[t−1, tK\nMαJt−1, tK\n0\nx\n(x, x)\n(x, y)\nx −y\nof right Rα[t−1, t]-modules.\nLet ι: R ! Rα[t−1, t] be the natural inclusion. If M is an Rα[t−1, t]-module,\ndenote the restricted R-module by ι!M.\nLemma 3.3 (Shapiro’s Lemma). Let C• be a chain complex of Rα[t−1, t]-modules.\nFor any right R-module M, we have\n(1) Hi(C•; Mα[t−1, t]) ∼= Hi(ι!C•; M) when C• is a complex of left modules, and\n(2) Hi(C•; MαJt−1, tK) ∼= Hi(ι!C•; M) when C• is a complex of right modules\nfor all i ∈Z.\nProof. The isomorphism in (1) is an immediate consequence of the isomorphism\nMα[t−1, t] ∼= M ⊗R Rα[t−1, t]. For (2), there is an isomorphism\nΦ: HomRα[t−1,t](L, MαJt−1, tK) ! HomR(ι!L, M)\nfor any right Rα[t−1, t]-module L; it is given by Φ(φ)(l) = φ(l)0, where l ∈L is\narbitrary. It’s inverse Ψ is given by Ψ(ψ)(l) = P∞\ni=−∞ψ(lti)t−i. We leave it to\nthe reader to verify that these isomorphisms are well-defined and that they are\nfunctorial in the first entry, i.e. that given a map L0 ! L1 of right Rα[t−1, t]-\nmodules, the diagram\nHomRα[t−1,t](L1, MαJt−1, tK)\nHomRα[t−1,t](L0, MαJt−1, tK)\nHomR(ι!L1, M)\nHomR(ι!L0, M)\ncommutes.\nThe claimed isomorphism is then an immediate consequence of the\nisomorphism Φ.\n□\nThe power of Observation 3.2 is that it relates the (co)homology of C• with that\nof ι!C• via the following immediate consequence of Shapiro’s Lemma.\nCorollary 3.4. Let C• be a chain complex of projective Rα[t−1, t]-modules. For\nany right R-module M, the short exact sequence of Observation 3.2 induces the\nfollowing long exact sequences in homology and cohomology:\n· · ·\nHi(C•; MαJt−1, t] ⊕Mα[t−1, tK)\nHi(C•; MαJt−1, tK)\nHi−1(ι!C•; M)\nHi−1(C•; MαJt−1, t] ⊕Mα[t−1, tK)\n· · ·\n· · ·\nHi(C•; MαJt−1, t] ⊕Mα[t−1, tK)\nHi(ι!C•; M)\nHi+1(C•; Mα[t−1, t])\nHi+1(C•; MαJt−1, t] ⊕Mα[t−1, tK)\n· · ·\nBefore establishing some of the main results, we need the following key lemma.\n\n10\nSAM P. FISHER\nLemma 3.5. Any non-zero Rα[t−1, tK-module is uncountable.\nProof. Let S ⊂Rα[t−1, K be the set of non-zero expressions of the form P∞\ni=0 εiti,\nwhere εi ∈{0, 1} for each i ⩾0. Observe that if s0 ̸= s1 are distinct elements of\nS, then s1 −s0 is a unit of Rα[t−1, tK. This is because s1 −s0 = (1 −p)t−k for\nsome k ∈Z and some element p of the form p = P∞\ni=1 δiti. Hence (s1 −s0)−1 =\ntk(1 + p + p2 + · · · ).\nLet M be a non-zero module over Rα[t−1, tK and let m ∈M ∖{0}. We claim\nthat the map S ! M sending s to m·s is injective. Suppose m·s0 = m·s1 for some\ns0, s1 ∈S. If s0 ̸= s1, then m = m(s1−s0)(s1−s0)−1 = 0, which is a contradiction.\nHence, the map is injective, and since S is uncountable, M is too.\n□\nWe are now ready to give a proof of Ranicki’s Criterion based on the long exact\nsequences of Corollary 3.4. The result differs from Ranicki’s in two ways: first, it\nis in terms of vanishing Novikov cohomology (instead of homology) and second, it\ndoes not require that the chain complex be bounded above or below. Note also that\nRanicki’s original proof was for chain complexes of free modules and with trivial\ntwisting α.\nTheorem 3.6 (Ranicki’s Criterion, cohomology version). Let R be a countable ring\nand let C• be a chain complex of finitely generated projective Rα[t−1, t]-modules. For\nany integer n ⩾0, the following are equivalent:\n(1) ι!C• is finitely dominated;\n(2) Hi(C•; RαJt−1, t]) = Hi(C•; Rα[t−1, tK) = 0 for all i.\nProof. Assume that (1) holds. The long exact sequence in cohomology of Corol-\nlary 3.4 yields exact sequences\nHi(C•; Rα[t−1, t]) −! Hi(C•; RαJt−1, t] ⊕Rα[t−1, tK) −! Hi(ι!C•; R)\nfor all i ⩾0. Since C• is a chain complex of finitely generated Rα[t−1, t]-modules\nand Rα[t−1, t] is countable, it follows that Hi(C•; Rα[t−1, t]) is countable. Moreover,\nsince ι!C• is chain homotopy equivalent to a chain complex that is finitely generated\nin degrees at most n, we also have that Hi(ι!C•; R) is countable. Since the sequence\nabove is exact, we conclude that Hi(C•; RαJt−1, t] ⊕Rα[t−1, tK) is countable, and\ntherefore vanishes by Lemma 3.5.\nNow we assume that (2) holds. Let M be an arbitrary right R-module. The long\nexact sequence in cohomology of Corollary 3.4 yields isomorphisms\nHi(ι!C•; M) ∼= Hi+1(C•; Mα[t−1, t])\nfor all i ⩾0, by Lemma 2.6. Let {Mj}j∈J be any directed system of R-modules\nsuch that lim\n−! Mj = 0. Then\nlim\n−!\n\u0000(Mj)α[t−1, t]\n\u0001 ∼= lim\n−!\n\u0000(Mj)α ⊗R Rα[t−1, t]\n\u0001\n= 0\nbecause tensor products commute with direct limits.\nSince C• is a complex of\nfinitely generated projective modules, we have\nlim\n−! Hi(ι!C•; Mj) ∼= lim\n−! Hi+1(C•; (Mj)α[t−1, t])\n∼= Hi+1(C•; lim\n−!\n\u0000(Mj)α[t−1, t]\n\u0001\n) = 0.\nBy Theorem 2.2, ι!C• is finitely dominated.\n□\n\nNOVIKOV COHOMOLOGY\n11\nRemark 3.7. Let S be an Rα[t−1, t]-algebra. If C• is a bounded chain complex\n(meaning Ci = 0 for |i| sufficiently large), then Hi(C•; S) = 0 for all i if and only if\nHi(C•; S) = 0 for all i (this is an easy consequence of [Bro75, Lemma 2]). Hence,\nthe proof of Theorem 3.6 gives a short new proof of Ranicki’s Criterion.\nWe decided to give the proof of Theorem 3.6 in the case that R is countable,\nsince it is more straightforward and this covers most cases of interest (such as\nwhen R = k[N], where k is a countable ring and N = ker(G ! Z) for some\ncountable group G). As explained below, the case of a general ring R follows from\nthe countable case. Note that only the proof of the implication (1) ⇒(2) used the\ncountability assumption.\nProof (of Theorem 3.6, for a general ring R). We first set up some general nota-\ntion: if F is a free S-module for some ring S and with a fixed basis B, then let F|S′\ndenote the free S′-submodule L\nb∈B S′b ⩽F.\nBy assumption, there is a chain complex D• of finitely generated projective R-\nmodules chain homotopy equivalent to ι!C•. We may assume that C• and D• are\ncomplexes of finitely generated free modules, at the potential cost of them being\nnonzero in infinitely many degrees. Fix bases for all the modules in C• and D•.\nSince Rα[t−1, t] is free as an R-module with basis given by the powers of t, this\nchoice induces a basis for ι!C• as a free R-module. Note that all the bases are\ncountable.\nWe consider the following module maps:\n• the boundary maps of the chain complexes ι!C• and D•;\n• a homotopy equivalence h• : D• ! ι!C• with its chain homotopy inverse\ng• : ι!C• ! D•;\n• a chain map s• : D• ! D•+1 witnessing the fact that g•h• is chain homo-\ntopic to the identity (i.e. a map satisfying idDi −gihi = si−1∂i + si+1∂i)\nas well as a chain map r• : ι!C• ! ι!C•+1 witnessing the fact that h•g• is\nchain homotopic to the identity.\nLet E be the union of the R-bases of all the modules ι!Ci and Di. Let R′ ⩽R\nbe a countable subring such that f(e) ∈M|R′, where e ∈E is arbitrary and f is\nany one of the maps above whose domain contains e and whose codomain is M.\nThen ι!C•|R′ and D•|R′ are well defined chain complexes of free R′-modules, and\nthe restrictions of the maps above witness the fact that they are chain homotopic.\nWe will now prove that Hi(C•; Rα[t−1, tK) = 0 for all i ⩽n. Let\nz : Ci −! Rα[t−1, tK\nbe a cocycle for some i ⩽n, and suppose that {e1, . . . , em} is the fixed finite\ngenerating set of Ci as an Rα[t−1, t]-module. By potentially enlarging the subring\nR′ from the previous paragraph, we may assume it is countable and contains all the\ncoefficients of each element z(ej). We may moreover assume that R′ is α-invariant\n(and countable) by replacing it with the subring generated by S\ni∈Z αi(R′). Then\nz restricts to a cochain\nz′ : Ci|R′ −! R′\nα[t−1, t].\nBy Theorem 3.6, we have Hi(C•|R′; R′\nα[t−1, tK) = 0, so there exists a cochain\nc′ : Ci−1|R′ ! R′\nα[t−1, tK such that z′ = c′∂. But c′ defines an R-linear map\nc: Ci−1 −! Rα[t−1, tK\n\n12\nSAM P. FISHER\nby setting c(e) = c′(e) on all basis elements e. We then also have z = c∂, proving\nthat Hi(C•; Rα[t−1, tK) = 0, as desired.\n□\nWe will need the next two propositions later on. The first is a refinement of\nRanicki’s Criterion, which is often called Sikorav’s Theorem, and has already been\nobtained in this form by Hillman and Kochloukova [HK07, Theorem 5]. We will\ngive a new proof the direction (2) ⇒(1). Together with Hillman and Kochloukova’s\nproof of the implication (1) ⇒(2), this gives a proof of Sikorav’s Theorem that does\nnot involve analysing individual Novikov cycles in the complex of Novikov chains,\nand instead uses standard methods from homological algebra (such as mapping\ncones of chain complexes and long exact sequences in homology).\nProposition 3.8 (Sikorav’s Theorem). Let C• be a chain complex of projective\nRα[t−1, t]-modules such that Ci is finitely generated for all i ⩽n and Ci = 0 for all\ni < 0. The following are equivalent:\n(1) ι!C• is of finite n-type;\n(2) Hi(C•; RαJt−1, t]) = Hi(C•; Rα[t−1, tK) = 0 for all i ⩽n.\nProof. Assume that (2) holds. There is a finitely generated direct summand C′\nn+1\nof Cn+1 such that the modified chain complex\nC′\n• = {0 −! C′\nn+1 −! Cn −! · · · −! C0 −! 0}\nstill has Hn(C′\n•; RαJt−1, t]) = Hn(C′\n•; Rα[t−1, tK) = 0. This is because RαJt−1, t] is\na ring, and therefore the module Zi(RαJt−1, t]⊗Rα[t−1,t] C•) of cycles degrees i ⩽n\nis a direct summand of RαJt−1, t] ⊗Rα[t−1,v] Ci for all i ⩽n. A similar remark\napplies to Rα[t−1, tK.\nNow,\nHi(C′\n•; MαJt−1, t]) = Hi(C′\n•; Mα[t−1, tK) = 0\nfor any right R-module M by Lemma 2.6 and all i ⩽n. The long exact sequence\nin homology of Corollary 3.4 yields isomorphisms\nHi+1(C′\n•; MαJt−1, tK) ∼= Hi(ι!C•; M)\nfor all i < n. Note that Q RαJt−1, tK ∼= (Q R)α Jt−1, tK, where the product is taken\nover an arbitrary index set. Since Ci+1 is finitely generated for i < n, the homology\nof C• commutes with products, and therefore\nHi(ι!C•;\nY\nR) = Hi(ι!C′\n•;\nY\nR) ∼= Hi+1(C′\n•; (\nY\nR)αJt−1, tK)\n∼= Hi+1(C′\n•;\nY\nRαJt−1, tK)\n∼=\nY\nHi+1(C′\n•; RαJt−1, tK)\n∼=\nY\nHi(ι!C′\n•; R) =\nY\nHi(ι!C•; R)\nfor i < n.\nWe have a commuting diagram\nHn+1(C′\n•; Q RαJt−1, tK)\nHn(ι!C′\n•; Q R)\n0\nQ Hn+1(C′\n•; RαJt−1, tK)\nQ Hn(ι!C′\n•; R)\n0\n∼\n=\n\nNOVIKOV COHOMOLOGY\n13\nwith exact rows. Hence, Hn(ι!C′\n•; Q R) ! Q Hn(ι!C′\n•; R) is an epimorphism. But\nthere is another commuting diagram\nHn(ι!C′\n•; Q R)\nHn(ι!C•; Q R)\n0\nQ Hn(ι!C′\n•; R)\nQ Hn(ι!C•; Q R)\n0\nwith exact rows. Hence Q Hn(ι!C′\n•; R) ! Hn(ι!C•; Q R) is an epimorphism, and\nhence (1) follows from Theorem 2.2.\n□\nThe following result is the dual of the previous theorem and is proved similarly.\nIt will be used in Theorem 4.14.\nTheorem 3.9. Let C• be a chain complex of projective Rα[t−1, t]-modules with\nCi = 0 for all i > n. Suppose that the functors Hi(C•; −) commute with direct\nlimits of Rα[t−1, t]-modules for all i ⩾m. If\nHi(C•; RαJt−1, t]) = Hi(C•; Rα[t−1, tK) = 0\nfor all i ⩾m, then Hi(ι!C•; −) commutes with direct limits for all i ⩾m. If n ⩾m,\nthen ι!C• is of cohomological dimension at most n −1.\nRemark 3.10. It is unclear to us whether the converse holds, i.e. whether Hi(ι!C•; −)\ncommuting with direct limits for i ⩾m implies vanishing Novikov cohomology in\ndegrees at least m. It holds if m = n (see Corollary 3.11), and if we additionally as-\nsume that C• is a chain complex of finitely generated projectives, then the converse\nalso holds at m = n −1.\nProof (of Theorem 3.9). The proof that Hi(ι!C•; −) commutes with direct limits\nis similar to the proof of Proposition 3.8, where it was shown that Hi(ι!C•; −)\ncommutes with direct products in a certain range.\nWe leave the details to the\nreader.\nBy Lemma 2.6,\nHi(C•; MαJt−1, t]) = Hi(C•; Mα[t−1, tK) = 0\nfor all i ⩾m and any right R-module M. The long exact sequence in cohomology\nof Corollary 3.4 then yields Hn(ι!C•; M) = 0. Since M was arbitrary, this shows\nthat ι!C• has cohomological dimension at most n −1.\n□\nAs a corollary, we find that the top-dimensional Novikov cohomology of C•\ncompletely controls the cohomological dimension of ι!C•, proving Theorem A from\nthe introduction in a more general form.\nCorollary 3.11. Let C• be a chain complex of projective Rα[t−1, t]-modules of\ncohomological dimension n and such that Cn is finitely generated. The following\nare equivalent:\n(1) ι!C is of cohomological dimension n −1;\n(2) Hn(C•; RαJt−1, t]) = Hn(C•; Rα[t−1, tK) = 0.\nProof. If ι!C• is of cohomological dimension n −1, then the long exact sequence in\ncohomology of Corollary 3.4 yields the exact sequence\nHn(C•; Rα[t−1, t]) −! Hn(C•; RαJt−1, t] ⊕Rα[t−1, tK) −! 0.\n\n14\nSAM P. FISHER\nIf R is a countable ring, then Hn(C•; Rα[t−1, t]) is countable (since Cn) is finitely\ngenerated. If not, we can use the fact that Cn is finitely generated to reduce to the\ncountable case, much in the same way as in the proof of Theorem 3.6 in the general\ncase. We sketch the argument below.\nWe may assume that the modules Ci are all free.\nThe fact that ι!C• is of\ncohomological dimension n −1 means that there is a map ρ: ι!Cn−1 ! ι!Cn such\nthat ρ∂= idι!Cn. Since Cn is finitely generated, there is a finitely generated direct\nsummand A of Cn−1 containing ∂(Cn). Writing Cn−1 = A ⊕B, we can take ρ to\nbe zero on B. Representing ∂and ρ as matrices over R, they contain countably\nmany entries, and there is an α-invariant countable subring R′ of R containing all\nof these entries. By the argument above,\nHn(C•; R′\nαJt−1, t]) = Hn(C•; R′\nα[t−1, tK) = 0.\nWe conclude that Hn(C•; RαJt−1, t]) = Hn(C•; Rα[t−1, tK) = 0 in the same way as\nin the proof of the general case of Theorem 3.6.\n□\n4. The Σ∗-invariant\nLet G be a group and let R be a ring such that cdR(G) = n. If the trivial\nR[G]-module R admits a projective resolution P• ! R ! 0 such that Pi is finitely\ngenerated for all i > n −m, then we will say that G is of type FP∗\nm(R). In other\nwords, the trivial module admits a resolution that is finitely generated in the top\nm degrees. Note that the finiteness property FP∗\nm(R) only makes sense for groups\nwith finite cohomological dimension over R, and that if a group is of type FP(R)\nthen it is of type FP∗\nm(R) for all m.\nDefinition 4.1. Let G be of type FP∗\nm(R). The mth Σ∗-invariant of G (over R)\nis\nΣ∗\nm(G; R) = {χ ∈H1(G, R) ∖{0} : Hi(G; ‘\nR[G]\nχ\n) = 0 for all i > n −m}.\nIn particular, Σ∗\n1(G; R) is the set of characters with respect to which the top-\ndimensional Novikov homology of G vanishes.\nIt follows immediately from the\ndefinition that Σ∗\nm+1(G; R) ⊆Σ∗\nm(G; R) (whenever G is of type FP∗\nm+1(R)).\nProposition 4.2. Let G be a finitely generated group of type FP∗\nm(R) for some\nring R and integer m. Then Σ∗\nm(G; R) is open in H1(G; R).\nThe proposition will follow quickly from Lemma 4.5 below, which will be well\nknown to experts. The proof given here relies on chain contractions, and the follow-\ning simple fact: if χ: G ! R is positive on the support of x ∈R[G], then there is\nan open neighbourhood U of χ such that 1 −x is invertible in ‘\nR[G]\nψ\nfor all ψ ∈U.\nWe first give a couple of definitions.\nDefinition 4.3 (Truncation). Let χ: G ! R be a character. The truncation of an\nelement P\ng∈G rgg ∈‘\nR[G]\nχ\nat height t ∈R is the element P\ng∈G rgg ∈R[G], where\nrg = rg if χ(g) < t, and rg = 0 otherwise.\nDenote by RG the set of all formal series of elements of G with coefficients in\nR (which can also be thought of as the set of all functions G ! R). For every\ncharacter χ: G ! R, there is a natural inclusion ‘\nR[G]\nχ\n⊆RG.\n\nNOVIKOV COHOMOLOGY\n15\nDefinition 4.4 (Definability over Novikov rings). Let χ, ψ: G ! R be two char-\nacters. Say that x ∈‘\nR[G]\nχ\nis definable over ‘\nR[G]\nψ\nif x ∈‘\nR[G]\nχ\n∩‘\nR[G]\nψ\nwhen\nviewing both Novikov rings as subsets of RG.\nDefinitions 4.3 and 4.4 extend immediately to matrices over Novikov rings. Let\nA ∈Mat( ‘\nR[G]\nχ\n). The truncation of A at height t ∈R is obtained by truncating all\nentries of A at height t, and we say that A is definable over ‘\nR[G]\nψ\nif all its entries\nare definable over ‘\nR[G]\nψ\n.\nLemma 4.5. Let C• be a chain complex of finitely generated projective R[G]-\nmodules such that Ci = 0 for i < 0 and Ci. For any integer m, the set of non-zero\ncharacters χ such that Hi(C•; ‘\nR[G]\nχ\n) = 0 for all i ⩽m is open in H1(G; R) ∖{0}.\nProof. By taking direct sums, we may assume that the modules Ci are finitely\ngenerated and free. Let χ be a character such that Hi(C•; ‘\nR[G]\nχ\n) = 0 for all i ⩽m.\nFixing bases for the modules Ci induces representations of maps between modules\n‘\nR[G]\nχ\n⊗R[G] Ci as matrices over ‘\nR[G]\nχ\n.\nBy induction on i, we will show that ‘\nR[G]\nχ\n⊗R[G] C• admits a partial chain\ncontraction of length i whose maps are defined over ‘\nR[G]\nψ\nfor all ψ in some open\nneighbourhood U ⊆H1(G; R) ∖{0} of χ (recall that a partial chain contraction of\nlength l is a sequence of maps\nsi : ‘\nR[G]\nχ\n⊗R[G] Ci −! ‘\nR[G]\nχ\n⊗R[G] Ci+1\nsuch that id = si−1∂i + ∂i+1si for all i ⩽l).\nSet s−1 = 0, which is defined over every Novikov ring, so there is nothing to prove\nin the base case. Suppose that m ⩾i > −1 and that we have found maps sj for all\nj < i defining a length i −1 partial chain contraction of ‘\nR[G]\nχ\n⊗R[G] C•, which is\ndefined over ‘\nR[G]\nψ\nfor all ψ in some open neighbourhood U of χ. By the Novikov\nacyclicity assumption, there is a map σi : ‘\nR[G]\nχ\n⊗R[G] Ci ! ‘\nR[G]\nχ\n⊗R[G] Ci+1\nextending the partial chain contraction to one of length i. We will now modify σi\nso that it is defined over an open set of characters.\nLet σi be a truncation of σi at a height sufficiently large so that every entry of\nthe matrix M = ∂i+1(σi −σi) has positive support under χ. We then have\nM = id −σi−1∂i −∂i+1σi,\nso M is defined over ‘\nR[G]\nψ\nfor ψ in some open neighbourhood V of χ. Moreover,\nsince M has only finitely many non-zero entries, we may shrink V so that the\nentries of M are positively supported with respect to all ψ ∈V . It follows that\nid −M is invertible with inverse P∞\ni=0 M i defined over ‘\nR[G]\nψ\nfor all ψ ∈V . Let\nsi = σi(id −M)−1. One easily checks that this extends the partial chain contrac-\ntion which is defined in the open neighbourhood U ∩V of χ. This completes the\ninduction.\nThe existence of a length m partial chain contraction defined over ‘\nR[G]\nψ\nfor all\nψ in an open neighbourhood of χ implies that Hi(C•; ‘\nR[G]\nψ\n) = 0 for all i ⩽m and\nall ψ in said neighbourhood.\n□\n\n16\nSAM P. FISHER\nRemark 4.6. The proof above would be shorter if definability over the Novikov ring\nwas an open condition on characters. This is not the case. Indeed, it is easy to\nconstruct examples of elements in ‘\nR[G]\nχ\n(with G = Z2, for instance) that are not\ndefined over any ‘\nR[G]\nψ\nwith ψ ̸= χ.\nProof (of Proposition 4.2). Let P• ! R ! 0 be a length n resolution witnessing\nthe FP∗\nm(R) property for G. Let P ′\nn−m be a finitely generated free factor of Pn−m\ncontaining the image of Pn−m+1 under the boundary map. The complex\n0 −! Pn −! · · · −! Pn−m+1 −! P ′\nn−m\nhas vanishing cohomology (with any coefficients) in degrees above n−m if and only\nif P• does. Hence, we will work with this modified complex, and continue to denote\nit by P•. The cohomology Hi(P•; ‘\nR[G]\nχ\n) in degrees i > n −m is the cohomology of\nthe cochain complex HomR[G](Pi, ‘\nR[G]\nχ\n) ∼= HomR[G](Pi, R[G]) ⊗R[G] ‘\nR[G]\nχ\n. Since\nPi is finitely generated, HomR[G](Pi, R[G]) is projective and finitely generated as\nan R[G]-module. The result then immediately follows from Lemma 4.5.\n□\nAs we saw in Example 3.1, the Novikov ring associated to an integral character\nχ: G ! Z is a special case of the construction Rα[t−1, tK. Thus, the results of\nthe previous section readily apply to give theorems about group cohomology. The\nfirst such result is an immediate consequence of Corollary 3.11, and establishes the\nconverse to [Fis24b, Theorem 3.5] in the case of integral characters.\nTheorem 4.7. Let G be a group and let R be a ring such that cdR(G) < ∞and\nG is of type FP∗\n1(R). Let χ: G ! Z be an integral character. Then cdR(ker χ) =\ncdR(G) −1 if and only if ±χ ∈Σ∗\n1(G).\nCorollary C follows at once from Theorem 4.7 and Proposition 4.2.\nCorollary 4.8. Let G be a group of type FP(R) for some ring R.\nThe set of\ncharacters\n{χ: G −! Q : cdR(ker χ) = cdR(G) −1}\nis open in H1(G; Q).\nThe remainder of the section will be devoted to proving Theorem E. We first\nneed a result about the finiteness properties of kernels of maps to poly-Z groups.\nNotation 4.9. Recall that a group P is poly-Z if it admits a subnormal series\n{1} = Pd ⩽Pd−1 ⩽· · · ⩽P1 ⩽P0 = P\nsuch that Pi/Pi+1 ∼= Z for all 0 ⩽i < d. The integer d is the length of the poly-Z\ngroup P; it coincides with its cohomological dimension, and therefore is uniquely\ndetermined.\nWe will always assume that our poly-Z groups come with a fixed\nsubnormal series as above.\nFor an epimorphism χ: G ! P with P poly-Z, there is a distinguished sequence\nof integral characters\nχi : Gi −! Gi/Gi+1 ∼= Z\nwhere Gi := χ−1(Pi). Note that if P is of length d, then Gd = ker χ.\n\nNOVIKOV COHOMOLOGY\n17\nThe following two results are easy consequences of Proposition 3.8 and Theo-\nrem 3.9. We will need them when studying We use the following notation: if M is\nan R[G]-module and N ⩽G is a pair of groups, then the R[N]-module obtained by\nrestricting M will be denoted by resG\nN M. We follow Notation 4.9 in the statement\nof the next proposition.\nProposition 4.10. Let χ: G ! P be an epimorphism, where P is a poly-Z group\nof length d. Let C• be a chain complex of projective R[G]-modules of finite n-type.\nThe following are equivalent:\n(1) resG\nGd C• is of finite n-type;\n(2) Hi(resG\nGj C•; ’\nR[Gj]\n±χj) = 0 for all 0 ⩽j < d.\nProof. We prove the result by induction on d. If d = 1, then the statement of the\nproposition is identical to that of Proposition 3.8.\nSuppose (1) holds. Then resG\nG1 C• is of finite n-type. Indeed, there is a convergent\nspectral sequence\nHp(G1/Gd; Hq(resG\nGd C•; M)) =⇒Hp+q(resG\nG1 C•; M)\nfor any R[G1]-module M. This is just the Lyndon–Hochschild–Serre spectral se-\nquence, except one replaces a resolution of the trivial R[G]-module R by the chain\ncomplex resG\nGd C•; the proof of its existence is identical (see [Bro94, Chapter VII]).\nBy Theorem 2.2, lim\n−! Hq(resG\nGd C•; Mj) = 0 for all q ⩽n and all directed systems\n{Mj} with vanishing direct limit. Moreover, Hp(G1/Gd; −) preserves direct limits\nin all degrees, since G1/Gd is a poly-Z group and therefore of type FP.\nThus,\nlim\n−! Hp(resG\nG1 C•; Mj) = 0 for all p ⩽n and all directed systems {Mj} with van-\nishing direct limit. Again by Theorem 2.2, resG\nG1 C• has the appropriate finiteness\nconditions. By Proposition 3.8, we conclude that\nHj(resG\nG1 C•; ‘\nR[G]\n±χ0) = 0\nfor all j ⩽n. By induction, we also have\nHj(resG\nGi+1 C•; ’\nR[Gi]\n±χi) ∼= Hj(resG1\nGi+1 C•; ’\nR[Gi]\n±χi) = 0\nfor all j ⩽n and all 0 < i < d.\nIf (2) holds, then resG\nG1 C• is of finite n-type by Proposition 3.8. By induction,\nresG1\nGd C• is of finite n-type.\n□\nProposition 4.11. Let χ: G ! P be an epimorphism, where P is a poly-Z group\nof length d. Let C• be a chain complex of projective R[G]-modules such that Ci = 0\nfor all i > n and such that Hi(G; −) commutes with direct limits of R[G]-modules\nfor all i > n −d. If\nHi(resG\nGj C•; ’\nR[Gj]\n±χj) = 0\nfor all i > n −d and all 0 ⩽j < n, then resG\nGd C• is of cohomological dimension at\nmost n −d.\nProof. We prove the result by induction on d. If d = 1, then it follows immediately\nfrom Corollary 3.11. Suppose that d > 1. Again by Corollary 3.11, resG\nG1 C• is\nchain homotopy equivalent to a chain complex of projective modules D• which is\nof cohomological dimension at most n −1 and such that Hi(D•; −) commutes with\ndirect limits for all i > n −d. The claim now follows by induction.\n□\n\n18\nSAM P. FISHER\nFor the proof of Theorem E, we will need the following result of Farber, Geoghe-\ngan, and Schütz, which gives a version of Sikorav’s Theorem for chain complexes.\nTheorem 4.12 ([FGS10, Theorem 8]). Let C• be a chain complex of projective\nR[G]-modules such Ci is finitely generated for i ⩽n and zero for i < 0. If N P G\nis such that G/N is Abelian, then the following are equivalent:\n(1) resG\nN C• is of finite n-type;\n(2) for all χ ∈S(G, N), we have Hi(C•; ‘\nR[G]\nχ\n) = 0 for all i ⩽n.\nRemark 4.13. In [FGS10], the result is stated for R = Z. Just as in the case of\nSikorav’s Theorem for groups, the proof goes through to the case of a general unital\nassociative ring R without modification.\nWe are now ready to prove Theorem E for chain complexes of projectives. Recall\nthat if N ⩽G is a pair of groups and M is a left R[N]-module, then the corre-\nsponding induced module is IndG\nN M = R[G] ⊗R[N] M, and is a left R[G]-module.\nTheorem 4.14. Let C• be a chain complex of finitely generated projective R[G]-\nmodules such that Ci = 0 for all i > n, and let χ: G ! R be a character with free\nAbelian image of rank d. If\nHi(C•; ‘\nR[G]\nφ\n) = 0\nfor all φ ∈S(G, ker χ) and all i > n −d, then resG\nker χ C• has cohomological dimen-\nsion at most n −d.\nProof. Since the modules Ci are finitely generated,\nHomR[G](C•, ‘\nR[G]\nφ\n) ∼= HomR[G](C•, R[G]) ⊗R[G] ‘\nR[G]\nφ\n.\nThe Novikov acyclicity assumption and a Theorem 4.12 imply that\nresG\nker χ HomR[G](C•, R[G])\nis chain homotopy equivalent to a chain complex D• of projective R[ker χ]-modules\nsuch that Di is finitely generated for all i > n −d.\nFor the remainder of the proof, we follow Notation 4.9. Let P = χ(G) ∼= Zd and\nfix a series of subgroups\n{1} = Pd ⩽Pd−1 ⩽· · · ⩽P1 ⩽P0 = P\nsuch that Pi/Pi+1 ∼= Z for all 0 ⩽i < d. Let the maps χi : Gi = χ−1(Pi) ! Pi/Pi+1\nbe the associated integral characters. By Proposition 4.10, for all 0 ⩽j < d, the\ncochain complexes\nresG\nGj HomR[G](C•, R[G]) ⊗R[Gj] ’\nR[Gj]\n±χj\n∼= HomR[G](C•, R[G]) ⊗R[G] IndG\nGj ’\nR[Gj]\n±χj\n∼= HomR[G](C•, IndG\nGj ’\nR[Gj]\n±χj)\nare acyclic in degrees greater than n −d, so we have Hi(C•; IndG\nGj ’\nR[Gj]\n±χj) = 0\nfor all i > n −d and 0 ⩽j < d. The result will follow quickly from the following\nclaim.\n\nNOVIKOV COHOMOLOGY\n19\nClaim 4.15. Let χ: G ! P be an epimorphism, where P is a poly-Z group of\nlength d. Let F• be a chain complex of projective R[G]-modules such that Fi = 0\nfor all i > n and such that Hi(F•; −) commutes with direct limits for all i > n −d.\nIf\nHi(F•; IndG\nGj ’\nR[Gj]\n±χj) = 0\nfor all i > n −d and 0 ⩽j < d, then Hi(resG\nGj F•; ’\nR[Gj]\n±χj) = 0 for all i > n −d\nand 0 ⩽j < d.\nProof. We will prove this by induction on d. If d = 1, then there is nothing to\nshow, so we assume that d > 1. Fix some j < d. Let M = IndG1\nGj ’\nR[Gj]\n±χj, and let\nt ∈G map to the generator of G/G1 such that χ0(t) = 1. As in Observation 3.2,\nthere is a short exact sequence\n0 −! IndG\nG1 M −! MαJt−1, t] ⊕Mα[t−1, tK −! coIndG\nG1 M −! 0,\nwhere α is induced by the conjugation action of t. We know\nHi(F•; IndG\nG1 M) = Hi(F•; IndG\nGj ’\nR[Gj]\n±χj) = 0\nfor all i > n −d. We also know that\nHi(F•; MαJt−1, t]) = Hi(F•; Mα[t−1, tK) = 0\nby Lemma 2.6, since Mα[t−1, tK is a right ‘\nR[G]\nχ0 = R[G1]α[t−1, tK-module and\nHi(F•; ‘\nR[G]\nχ0) = 0\n(and similarly for MαJt−1, t]). It then follows that\nHi(F•; coIndG\nG1 M) = Hi(resG\nG1 F•; IndG1\nGj ’\nR[Gj]\n±χj) = 0.\nfor all i > n−d. Since Hi(resG\nG1 F•; −) commutes with direct limits for all i > n−d\nby Proposition 4.11, the claim holds by induction.\n⋄\nBy the claim, Hi(resG\nGj C•; ’\nR[Gj]\n±χj) = 0 for all i > n −d and all 0 ⩽j < d. By\nProposition 4.11, resG\nker χ C• is of cohomological dimension at most n −d.\n□\nTheorem E follows immediately by taking C• to be a projective resolution of G\nwitnessing the FP∗\nd(R) property.\nCorollary 4.16. Let G be a group of type FP∗\nd(R) with cdR(G) = n, and suppose\nthat N P G is such that G/N ∼= Zd. If S(G, N) ⊆Σ∗\nd(G; R), then cdR(N) = n −d.\nWhile Corollary 4.16 has a converse for d = 1, the converse fails in general, as\nthe following example shows. The author thanks Grigori Avramidi for telling him\nthat many closed hyperbolic 3-manifolds are free-by-Z2.\nExample 4.17. Let M be a closed fibred hyperbolic 3-manifold with b1(M) ⩾2,\nand let χ: π1(M) ! Z be an algebraic fibration induced by a fibration of M over S1\nwith fibre a hyperbolic surface Sg. By openness of the Σ-invariant and the fact that\nb1(M) ⩾2, we may perturb χ to obtain a new algebraic fibration χ′ : π1(M) ! Z\nnot equal to ±χ. The kernel of the map f : π1(M) ! Z2 given by α 7! (χ(α), χ′(α))\nis an infinite-index subgroup of ker χ ∼= π1(Sg) and therefore is non-finitely gener-\nated free. Hence, π1(M) is free-by-Z2. Let F = ker f. By Theorem 1.1, there is\n\n20\nSAM P. FISHER\na character χ ∈S(π1(M), F) such that H1(π1(M); Ÿ\u0000Z[π1(M)]\nχ\n) ̸= 0 (because F is\nnot finitely generated), so H2(π1(M); Ÿ\u0000Z[π1(M)]\nχ\n) ̸= 0 by Poincaré-duality. But the\nconverse of Corollary 4.16 would imply that Hi(π1(M); Ÿ\u0000Z[π1(M)]\nχ\n) = 0 for i = 2, 3,\nsince cd(F) = 1, so it cannot hold.\nWe conclude the section by proving Corollary F. A group G is an n-dimensional\nPoincaré-duality group over R (or, more briefly, a PDn\nR-group) if it satisfies\nHi(G; M) ∼= Hn−i(G; M)\nfor every R[G]-module M. Homology commutes with direct limits, which implies\nthat the cohomology of a Poincaré-duality group commutes with direct limits.\nHence, by Theorem 2.2, PDn\nR-groups are of type FP(R).\nCorollary 4.18. Let G be a PDn\nR-group and let χ: G ! Zd be an epimorphism. If\nker χ is of type FPd−1(R), then cdR(G) = n −d.\nProof. By the higher version of Theorem 1.1 (see [BR88], or [Fis24a, Theorem 5.3]\nfor the case of an arbitrary ring R), S(G, N) ⊆Σd−1(G; R), where N = ker χ. By\nPoincaré-duality, S(G, N) ⊆Σ∗\nd(G; R). By Corollary 4.16, cdR(G) = n −d.\n□\n5. An application to RFRS groups\nDefinition 5.1. A group G is said to be residually finite rationally solvable, or\nRFRS, if\n(1) there is a chain of finite-index normal subgroups G = G0 ⩾G1 ⩾G2 ⩾. . .\nsuch that T\ni⩾0 Gi = {1}, and\n(2) ker(Gi ! Q ⊗Z Gi/[Gi, Gi]) is a subgroup of Gi+1 for all i ⩾0.\nA chain satisfying these properties will be called a witnessing chain for the RFRS\ngroup.\nDefinition 5.2. It is easy to see that finitely generated RFRS groups are residually\n(torsion-free solvable), and therefore satisfy the strong Atiyah conjecture by [Sch02].\nBy a result of Linnell [Lin93], this implies that the division closure of Q[G] in the\nalgebra of affiliated operators U(G) is a division ring, which we call DQ[G]. The nth\nL2-Betti numbers of G is\nb(2)\nn (G) := dimDQ[G] Hn(G; DQ[G]),\nwhere dimDQ[G] M the rank of a (necessarily free of unique rank) DQ[G]-module M.\nFor more details, we refer the reader to [Lüc02, Chapter 10].\nIn fact, for RFRS groups G, Jaikin-Zapirain shows that k[G] embeds into a\ndivision ring Dk[G] for any field k, which shares many properties with DQ[G] [JZ21,\nCorollary 1.3]. We may use this division ring to define the L2-Betti numbers of a\nRFRS group over a field k by\nb(2)\nn (G; k) := dimDk[G] Hn(G; Dk[G]).\nIn particular, b(2)\nn (G; Q) = b(2)\nn (G).\nFor us, the important common feature of all the division rings Dk[G] (as the\nground field k varies) is that they can be expressed as directed unions of rings\nwhich are closely related to the Novikov rings (over k) of G and its finite-index\nsubgroups. For k = Q, this is the crux of Kielak’s fibring theorem [Kie20]. This\n\nNOVIKOV COHOMOLOGY\n21\nwas extended by Jaikin-Zapirain to all fields k in the appendix of [JZ21]. We give\na rough sketch of how this works, emphasising the details that will be necessary for\nus.\nFix a finitely generated RFRS group G and a witnessing chain (Gi)i⩾0.\nIn\n[Kie20, Definition 4.3], Kielak introduces the notion of a rich set of characters in\nH1(Gi; R) ∖{0}, a notion which depends the chosen witnessing chain (Gi)i⩾0. The\ndefinition of a rich set is somewhat technical, but we will only need the following\nproperties:\n• rich sets are open;\n• rich sets are nonempty;\n• the intersection of rich sets is rich [Kie20, Lemma 4.4].\nRemark 5.3. In [Kie20, Definition 4.3], it is not required that rich sets be open.\nThis is however simply a misprint, and indeed it is important that rich sets be open\nfor the proofs of [Kie20, Lemma 4.4] and [Kie20, Lemma 4.13].\nIf U ⊆H1(Gi; R) is any set, then Kielak defines [Kie20, Definition 3.15] an asso-\nciated subring Dk[Gi],U ⊆Dk[G] (see also [JZ21, Section 5.2]). Again the definition\nof Dk[Gi],U is somewhat technical, but we will really only use some basic properties\n(references are provided for the interested reader):\n• k[Gi] is contained in Dk[Gi],U for every choice of U (follows directly from\n[Kie20, Definition 3.15]);\n• if U ⊆V ⊆H1(Gi; R), then Dk[Gi],V ⊆Dk[Gi],U (follows directly from\n[Kie20, Definition 3.15]);\n• for any choice of U, the conjugation action of G on k[Gi] extends to an\naction on Dk[Gi],U, and we may thus form the twisted group ring\nDk[Gi],U ∗G/Gi ∼= Dk[Gi],U ⊗k[Gi] k[G],\nand there is a natural embedding Dk[Gi],U ∗G/Gi ⊆Dk[G] (see [JZ21,\nProposition 5.3]);\n• for every χ ∈U, there is a map Dk[Gi],U ! ’\nk[Gi]\nχ\n, or in other words, ’\nk[Gi]\nχ\nis a Dk[Gi],U-algebra (see [Kie20, Lemma 3.13]).\nWe continue with the notation above: G is a RFRS group with witnessing chain\n(Gi)i⩾0. Let n be a non-negative integer and let U = (Un, Un+1, . . . ), where Ui ⩽\nH1(Gi; R) is a rich set for each i ⩾n. We call such a U a sequence of rich sets.\nDefine\nDk[G],U =\n\\\ni⩾n\nDk[Gi],Ui ∗G/Gi.\nThe main technical theorem of Kielak is that Dk[G] is covered by the rings Dk[G],U.\nTheorem 5.4 ([Kie20, Theorem 4.13]). Dk[G] = S\nU Dk[G],U, where the union is\ntaken over all sequences of rich sets U.\nWe only need the following additional fact.\nLemma 5.5. The rings {Dk[G],U}U form a directed system under inclusion.\nProof. Let m and n be non-negative integers, and let U = (Um, Un+1, . . . ) and\nV = (Vn, Vn+1, . . . ) be sequences of rich sets. Let M = max{m, n}. Then Dk[G],U\n\n22\nSAM P. FISHER\nand Dk[G],V are each contained in Dk[G],U∩V, where U ∩V := (UM ∩VM, UM+1 ∩\nVM+1, . . . ). By the facts listed above, U ∩V is a sequence of rich sets, and\nDk[G],U ⊆Dk[G],U∩V ⊇Dk[G],V.\nThis proves that the system of rings Dk[G],U is directed.\n□\nLemma 5.6. Let G be a group with cdk(G) = n for some field k. Let S = lim\n−! Sj,\nwhere {Sj}j∈J is a directed system of k[G]-algebras. Suppose that Hi(G; −) com-\nmutes with direct limits of k[G]-modules for all i ⩾m. If Hi(G; S) = 0 for all\ni ⩾m, then there is some j ∈J such that Hi(G; Sj) = 0 for all i ⩾m.\nProof. Suppose that, for a fixed l ⩾m, there exists j ∈J with Hi(G; Sj) = 0 for\nall i > l (we trivially know this holds for l = n and all j ∈J). Then Hl(G; −) is\na right exact functor on the category of Sj-modules. By Theorem 2.5, there is a\nnatural isomorphism\nHl(G; −) ∼= Hl(G; Sj) ⊗Sj −\nas functors on the category of Sj-modules.\nThe functor Hl(G; −) commutes with direct products of Sj-modules, and there-\nfore so does the functor Hl(G; Sj) ⊗Sj −.\nIt is a well-known fact (and an easy\nexercise) that Hl(G; Sj) must be finitely generated as an Sj-module. Let z1, . . . , zs\nbe a finite generating set for Hl(G; Sj). Since\nHl(G; S) ∼= Hl(G; Sj) ⊗Sj S ∼= lim\n−!\nj′⩾j\nHl(G; Sj) ⊗Sj Sj′ = 0,\nwe see that there is some j′ ∈J such that zi ⊗1 = 0 in Hl(G; Sj) ⊗Sj Sj′ for each\ni = 1, . . . , s. But these elements also generate Hl(G; Sj) ⊗Sj Sj′, so we conclude\nthat\nHl(G; Sj′) ∼= Hl(G; Sj) ⊗Sj Sj′ = 0.\nSince Sj′ is an Sj-module, we also have Hi(G; Sj′) = 0 for all i > l by Lemma 2.6.\nThe lemma then follows by induction.\n□\nWe can now prove the main result of this section, which implies Theorem D\nby taking d = 1 and using the Stallings–Swan Theorem [Sta68, Swa69], which\ncharacterises free groups as the groups of cohomological dimension at most one.\nTheorem 5.7. Let G be a RFRS group with cdk(G) = n for some field k. Suppose\nHi(G; −) commutes with direct limits of k[G]-modules for all i > d. The following\nare equivalent:\n(1) Dk[G] is of weak dimension at most d as a k[G]-module;\n(2) for every subgroup H ⩽G, we have b(2)\ni (H; k) = 0 for all i > d;\n(3) there is a subnormal series Γl < Γl−1 < · · · < Γ0 = G, where each quotient\nΓi/Γi+1 is either finite or cyclic, and cdk(Γl) = d.\nThe proof will show that in (3), the quotients Γi/Γi+1 will alternate between\nbeing finite and infinite cyclic, and when Γi/Γi+1 ∼= Z we have cdk(Γi+1) < cdk(Γi).\nProof (of Theorem 5.7). The implication (1) ⇒(2) follows quickly from the fact\nthat\nHi(H; Dk[G]) ∼= Hi(H; Dk[H]) ⊗Dk[H] Dk[G]\nfor all i, by flatness of Dk[G] over Dk[H].\n\nNOVIKOV COHOMOLOGY\n23\nSuppose that (2) holds. By Lemmas 5.5 and 5.6, there is a sequence of rich sets\nU = (Us, Us+1, . . . ) such that Hi(G; Dk[G],U) = 0 for all i > d. But Dk[G],U ⊆\nDk[Gs],Us ∗G/Gs, and therefore\n0 = Hi(G; Dk[Gs],Us ∗G/Gs) ∼= Hi(Gs; Dk[Gs],Us)\nfor all i > d by Lemma 2.6. We will take Γ1 = Gs. Recall that ’\nk[Gs]\nχ\nis a Dk[Gs],Us-\nalgebra for every χ ∈Us. Since −Us ∩Us is also a rich (and hence open) set, there\nis an integral character χ: Gs ! Z such that\nHi(Gs; ’\nk[Gs]\n±χ\n) = 0\nfor all i > d. By Corollary 3.11, cdk(ker χ) = n −1 and Hi(ker χ; −) commutes\nwith direct limits for all i > d. We take Γ2 = ker χ, and repeat this argument with\nΓ2 instead of G to extend the chain to Γ2 > Γ3 > Γ4, where Γ2/Γ3 is finite, and\nΓ3/Γ4 is cyclic with cdk(Γ4) = n −2. Continuing in this way proves item (3).\nTo prove (3) implies (1), it suffices to prove the following claim: If\n1 −! N −! G −! Q −! 1\nis an extension with G RFRS, Q amenable, and Dk[N] of weak dimension at most\nd as a k[N]-module, then Dk[G] is of weak dimension at most d as a k[G]-module.\nThe argument is similar to that of [JZL25, Theorem 3.4]. Indeed, let M be any\nk[G]-module. Then, for all i > d,\nTork[G]\ni\n(Dk[N] ∗G/N, M) ∼= Tork[N]\ni\n(Dk[N], M) = 0\nby Shapiro’s lemma. But Dk[G] is the Ore localisation of Dk[N] ∗G/N by a result of\nTamari [Tam54], and since Ore localisation is flat, we also get Tork[G]\ni\n(Dk[G], M) = 0\nfor all i > d, as claimed.\n□\nCorollary 5.8. Let G be a RFRS group of type FP(k). Then Dk[G] is of weak\ndimension one as a k[G]-module if and only if G is an iterated extension of a free\ngroup by (cyclic or finite) groups.\nRemark 5.9. In the introduction, Corollary 5.8 was stated in terms of U(G), the\nalgebra of operators affiliated to the von Neumann algebra of G. Let k ⊆C be\na field. Since Dk[G] is the division closure of k[G] in U(G), and moreover Dk[G] is\na division ring, U(G) is faithfully flat over Dk[G]. It follows that U(G) has weak\ndimension at most one as a k[G]-module if and only if Dk[G] does.\nIt is an interesting problem to determine the difference between the invariants\nb(2)\ni (G; k) for different fields k. In the context of RFRS groups (of finite type, say),\nit is an easy consequence of the results main results of [Kie20] and [Fis24a] that\nb(2)\n1 (G; k) = 0 for one field k if and only if b(2)\n1 (G; k′) = 0 for all fields k′. It is\nopen whether the value of b(2)\n1 (G; k) depends on the field k, but for every i > 1\nthere are examples of right-angled Artin groups G (which are, in particular, RFRS)\nsuch that b(2)\ni (G; k) depends on k [AOS21]. Here we show that Dk[G] being of weak\ndimension one (or equivalently, that all subgroups of G have vanishing k-L2-Betti\nnumbers) does not depend on k.\nCorollary 5.10. Let G be a RFRS group of type FP. If Dk[G] is of weak dimension\none as a k[G]-module for some field k, then Dk′[G] is of weak dimension one as a\nk′[G]-module for all fields k′.\n\n24\nSAM P. FISHER\nWe conclude the section with applications to groups with elementary hierarchies.\nFollowing Hagen and Wise [HW10], we say that the trivial group has an elementary\nhierarchy of length 0, and that a group G has an elementary hierarchy of length n\nif it splits as a graph of groups with an elementary hierarchy of length n −1 with\ncyclic edge groups. The main theorem of [HW10] is that any finitely generated\nvirtually special group with an elementary hierarchy is virtually free-by-cyclic (a\ngroup is special if and only if it is a subgroup of a RAAG, so special groups are\nRFRS). This class of groups includes, e.g., graphs of free groups with cyclic edge\ngroups not containing unbalanced Baumslag–Solitar groups and limit groups not\ncontaining Z3-subgroups.\nWe extend the Hagen–Wise theorem in the following way: say that G has an\nAbelian hierarchy of length 0 if it is trivial and that it has an Abelian hierarchy\nof length n if it virtually splits as a graph of groups with an Abelian hierarchy of\nlength n −1 and finitely generated virtually Abelian edge groups. While all groups\nwith an elementary hierarchy of cohomological dimension at most two, groups with\nan Abelian hierarchy can be of arbitrarily high cohomological dimension (the class\ncontains all limits groups, for example).\nCorollary 5.11. Let G be a virtually RFRS group of type FP(k) for some field k\nadmitting a virtually Abelian hierarchy. Then G is an iterated extension of a free\ngroup by (cyclic or finite) groups.\nProof. It suffices to prove that every subgroup of G has vanishing k-L2-Betti num-\nbers above degree one. Every subgroup of G admits an Abelian hierarchy, so it\nsuffices to prove the claim for a fixed group Γ with an Abelian hierarchy of length\nn. If n = 0, then Γ is trivial, and therefore b(2)\ni (Γ; k) = 0 for all i > 0. Now assume\nn > 0; since k-L2-Betti numbers scale under passage to finite-index subgroups, we\nwill assume that Γ admits a graph of groups splitting (Γv, Γe), where each Γv has an\nAbelian hierarchy of length n −1 and Γe is virtually Abelian. Then Chiswell’s long\nexact sequence in the homology of a graph of groups [Chi76] and the fact that vir-\ntually Abelian groups have vanishing k-L2-Betti numbers above degree zero imply\nthat b(2)\ni (Γ; k) = 0 for all i > 1. The result now follows from Corollary 5.8.\n□\n6. Novikov cohomology versus Novikov homology\nLet G be a group of finite type and of cohomological dimension n and let\nχ: G ! Z be an integral character. If for some ring R we have Hn(G; ‘\nR[G]\nχ\n) = 0,\nthen Hn(G; ‘\nR[G]\nχ\n) = 0. A natural question is whether the converse holds. if the\nconverse fails, it still makes sense to ask whether the vanishing of top-dimensional\nNovikov homology controls some property of the kernel. A natural guess is that if\nHn(G; ‘\nR[G]\n±χ\n) = 0, then the homological dimension of ker χ over R is n −1. In\nthis brief section, we will see that these questions have negative answers.\nLemma 6.1. Let G be a torsion-free one-relator group and let R be any ring without\nzero divisors containing Z[G]. Then H2(G; R) = 0.\nProof. Since G is torsion-free, Z admits a projective resolution of the form\n0 −! Z[G]\n∂\n−−! Z[G]n −! Z[G] −! Z −! 0\n(this is a consequence of the Lyndon identity theorem [Lyn50]). We claim that the\nextended map ∂: R ! Rn is injective. Indeed, suppose ∂(r) = r∂(1) = 0. Since\n\nNOVIKOV COHOMOLOGY\n25\n∂is injective, ∂(1) is nonzero. Since R has no zero divisors, this implies r = 0, as\ndesired.\n□\nCorollary 6.2. If G is a torsion-free one-relator group and χ: G ! R is any\nhomomorphism, then H2(G; ‘\nZ[G]\nχ\n) = 0.\nProof. One-relator groups are locally indicable by a result of Brodski˘ı [Bro84] and\ntherefore the group ring Z[G] has no zero divisors by a result of Higman [Hig40].\nBy looking at terms of minimal χ-height, it is easy to show that ‘\nZ[G]\nχ\nalso has no\nzero divisors as well. The result then follows immediately from Lemma 6.1.\n□\nExample 6.3. Consider the Baumslag–Solitar group\nG = BS(2, 3) = ⟨a, t | t−1a2t = a3⟩.\nIt is well known that G is not residually finite, and therefore it cannot be vir-\ntually free-by-cyclic [Bau71]. So for any finite-index subgroup H ⩽G and any\nepimorphism χ: H ! Z, either\nH2(H; ’\nZ[H]\nχ\n) ̸= 0\nor\nH2(H; ’\nZ[H]\n−χ\n) ̸= 0\nby Corollary B. In particular, let χ: G ! Z be the map defined by χ(a) = 0 and\nχ(t) = 1. With Corollary 6.2 this gives an example of a group of finite type and a\nmap with non-vanishing top-dimensional Novikov cohomology that has vanishing\nNovikov homology. Moreover, ker χ is not of homological dimension one. Indeed,\nthere is a well defined homomorphism\n⟨x, y | x3 = y2⟩−! G,\nx 7−! a,\ny 7−! t−1at,\nwhich is injective (we leave checking injectivity as an exercise for the reader). Since\nM ∼= ⟨x, y | x3 = y2⟩is a non-free one-relator group, it is of homological dimension\ntwo, which shows that ker χ is of homological dimension two. Hence, vanishing\ntop-dimensional Novikov homology does not imply that the homological dimension\nof the kernel drops.\nReferences\n[AOS21]\nGrigori Avramidi, Boris Okun, and Kevin Schreve. Mod p and torsion homology growth\nin nonpositive curvature. Invent. Math., 226(3):711–723, 2021.\n[Bau71]\nGilbert Baumslag. Finitely generated cyclic extensions of free groups are residually\nfinite. Bull. Austral. Math. Soc., 5:87–94, 1971.\n[BG99]\nKai-Uwe Bux and Carlos Gonzalez. The Bestvina-Brady construction revisited: geo-\nmetric computation of Σ-invariants for right-angled Artin groups. J. London Math. Soc.\n(2), 60(3):793–801, 1999.\n[Bie07]\nRobert Bieri. Deficiency and the geometric invariants of a group. J. Pure Appl. Algebra,\n208(3):951–959, 2007. With an appendix by Pascal Schweitzer.\n[BNS87]\nRobert Bieri, Walter D. Neumann, and Ralph Strebel. A geometric invariant of discrete\ngroups. Invent. Math., 90(3):451–477, 1987.\n[BR88]\nRobert Bieri and Burkhardt Renz. Valuations on free resolutions and higher geometric\ninvariants of groups. Comment. Math. Helv., 63(3):464–497, 1988.\n[Bro75]\nKenneth S. Brown. Homological criteria for finiteness. Comment. Math. Helv., 50:129–\n135, 1975.\n[Bro84]\nSergei D. Brodski˘ı. Equations over groups, and groups with one defining relation.\nSibirsk. Mat. Zh., 25(2):84–103, 1984.\n[Bro94]\nKenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathemat-\nics. Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.\n\n26\nSAM P. FISHER\n[Chi76]\nI. M. Chiswell. Exact sequences associated with a graph of groups. J. Pure Appl. Al-\ngebra, 8(1):63–74, 1976.\n[Fel71]\nG. L. Fel’dman. The homological dimension of group algebras of solvable groups. Izv.\nAkad. Nauk SSSR Ser. Mat., 35:1225–1236, 1971.\n[FGS10]\nM. Farber, R. Geoghegan, and D. Schütz. Closed 1-forms in topology and geometric\ngroup theory. Uspekhi Mat. Nauk, 65(1(391)):145–176, 2010.\n[FIK25]\nSam P. Fisher, Giovanni Italiano, and Dawid Kielak. Virtual fibring of Poincaré-duality\ngroups, 2025. arXiv:2506.14666.\n[Fis24a]\nSam P. Fisher. Improved algebraic fibrings. Compos. Math., 160(9):2203–2227, 2024.\n[Fis24b]\nSam P. Fisher. On the cohomological dimension of kernels of maps to Z, 2024.\narXiv:2403.18758, (to appear in Geom. Topol.).\n[Hig40]\nGraham Higman. Units in group rings. DPhil, University of Oxford, 1940.\n[HK07]\nJ. A. Hillman and D. H. Kochloukova. Finiteness conditions and PDr-group covers of\nPDn-complexes. Math. Z., 256(1):45–56, 2007.\n[HW10]\nMark Hagen and Daniel T. Wise. Special groups with an elementary hierarchy are\nvirtually free-by-Z. Groups Geom. Dyn., 4(3):597–603, 2010.\n[JZ21]\nAndrei Jaikin-Zapirain. The universality of Hughes-free division rings. Selecta Math.\n(N.S.), 27(4):Paper No. 74, 33, 2021.\n[JZL25]\nAndrei Jaikin-Zapirain and Marco Linton. On the coherence of one-relator groups and\ntheir group algebras. Ann. of Math. (2), 201(3):909–959, 2025.\n[Kie20]\nDawid Kielak. Residually finite rationally solvable groups and virtual fibring. J. Amer.\nMath. Soc., 33(2):451–486, 2020.\n[Lat94]\nFrançois Latour. Existence de 1-formes fermées non singulières dans une classe de co-\nhomologie de de Rham. Inst. Hautes Études Sci. Publ. Math., (80):135–194 (1995),\n1994.\n[Lin93]\nPeter A. Linnell. Division rings and group von Neumann algebras. Forum Math.,\n5(6):561–576, 1993.\n[Lüc02]\nWolfgang Lück. L2-invariants:\ntheory and applications to geometry and K-theory.\nSpringer-Verlag, Berlin, 2002.\n[Lyn50]\nRoger C. Lyndon. Cohomology theory of groups with a single defining relation. Ann.\nof Math. (2), 52:650–665, 1950.\n[MMV98] John Meier, Holger Meinert, and Leonard VanWyk. Higher generation subgroup sets\nand the Σ-invariants of graph groups. Comment. Math. Helv., 73(1):22–44, 1998.\n[Ran95]\nAndrew Ranicki. Finite domination and Novikov rings. Topology, 34(3):619–632, 1995.\n[Sch02]\nThomas Schick. Erratum: “Integrality of L2-Betti numbers”. Math. Ann., 322(2):421–\n422, 2002.\n[Sik87]\nJean-Claude Sikorav. Homologie de Novikov associée à une classe de cohomologie de\ndegré un. Thèse d’État, Université Paris-Sud (Orsay), 1987.\n[Sta68]\nJohn R. Stallings. On torsion-free groups with infinitely many ends. Ann. of Math. (2),\n88:312–334, 1968.\n[Swa69]\nRichard G. Swan. Groups of cohomological dimension one. J. Algebra, 12:585–610, 1969.\n[Tam54]\nDov Tamari. A refined classification of semi-groups leading to generalised polynomial\nrings with a generalised degree concept. In Proceedings of the International Congress\nof Mathematicians, 1954, Amsterdam, volume III, pages 439–440. Erven P. Noordhoff\nN. V., Groningen; North-Holland Publishing Co., Amsterdam, 1954.\n[Wat60]\nCharles E. Watts. Intrinsic characterizations of some additive functors. Proc. Amer.\nMath. Soc., 11:5–8, 1960.\nEmail address: samuel.p.fisher@gmail.com\n"},"cleaned":{"kind":"string","value":"NOVIKOV COHOMOLOGY, FINITE DOMINATION, AND COHOMOLOGICAL DIMENSION SAM P. FISHER Abstract. We introduce the Σ∗-invariant of a group of finite type, which is defined to be the subset of non-zero characters χ ∈H1(G; R) with vanishing associated top-dimensional Novikov cohomology. We prove an analogue of Sikorav's Theorem for this invariant, namely that cd(ker χ) = cd(G)-1 if and only if ±χ ∈Σ∗(G) for integral characters χ. This implies that cohomological dimension drop is an open property among integral characters. We also study the cohomological dimension of arbitrary co-Abelian subgroups. The techniques yield a short new proof of Ranicki's criterion for finite domination of infinite cyclic covers, and in a different direction, we prove that the algebra of affiliated operators U(G) of a RFRS group G has weak dimension at most one if and only if G is an iterated (cyclic or finite) extension of a free group. 1. Introduction In [BNS87], Bieri, Neumann, and Strebel introduced the Σ-invariant of a finitely generated group G. The invariant, denoted Σ(G), is a subset of H1(G; R)∖{0}, and controls the finiteness properties of co-Abelian subgroups of G. We call elements of H1(G; R) characters; these are just homomorphisms χ: G ! R. If χ(G) ⊆Q, then we say that χ is integral (because in this case χ(G) ∼= Z). The set of all nonzero characters which vanish on a given subgroup N is denoted by S(G, N). The following celebrated result from [BNS87] establishes the fundamental properties of the Σ-invariant. Theorem 1.1 (Bieri-Neumann-Strebel, [BNS87]). Let G be a finitely generated group. The following properties hold: (1) Σ(G) is open in H1(G; R) ∖{0}; (2) if N P G is a normal subgroup such that G/N is Abelian, then N is finitely generated if and only if S(G, N) ⊆Σ(G). In particular, an integral character χ: G ! Z has finitely generated kernel if and only if ±χ ∈Σ(G). Recall that a map to Z with finitely generated kernel is called an algebraic fibration of G. Theorem 1.1 has the following well known corollary. Corollary 1.2. Algebraic fibring is an open property. More precisely, given a finitely generated group G, the set {χ: G -! Q : ker χ is finitely generated} ∖{0} is an open subset of H1(G; Q) ∖{0}. The Σ-invariant is well-understood in certain cases (e.g. for right-angled Artin groups [MMV98, BG99]), but for most classes of groups it is notoriously difficult to compute. A useful tool is provided by Sikorav's Theorem, which gives an elegant homological criterion for a character to lie in the Σ-invariant. 1 16 Oct 2025 2 SAM P. FISHER Theorem 1.3 (Sikorav, [Sik87]). If G is a finitely generated group, then χ ∈Σ(G) if and only if H1(G; ' Z[G] χ ) = 0. The object ' Z[G] χ appearing in the theorem is the Novikov ring associated to χ; it is a certain completion of the group ring Z[G] along χ, and can be thought of as the ring of infinite series of elements in G which go to infinity in the direction of χ (see Example 3.1 for a precise definition). Sikorav's Theorem plays a central role in Kielak's proof that a finitely generated RFRS group virtually algebraically fibres if and only if it has vanishing first L2-Betti number [Kie20]. His strategy was to related the L2-homology of a RFRS group G with the homologies of its finite-index subgroups with coefficients in Novikov rings associated to various characters. Sikorav's Theorem has many generalisations and variations. For instance, the kernel of an integral character χ: G ! Z being of type FPn is equivalent to the vanishing of Hi(G; ' Z[G] ±χ ) for all i ⩽n (see [BR88], as well as Schweitzer's appendix to [Bie07]). This result also holds if one replaces the coefficient ring Z by an arbitrary ring R, and type FPn by type FPn(R) (see, e.g., [Fis24a, Theorem 5.3]). A closely related result is Ranicki's criterion [Ran95, Theorem 1], which states that an infinite cyclic cover of a compact CW complex X is finitely dominated if and only if Hi(π1(X); Ÿ Z[π1(X)] ±χ ) = 0 for all i, where χ: π1(X) ! Z is the map associated to the cyclic cover (Ranicki's original result was for spaces X such that π1(X) = G × Z and χ being the projection onto the Z factor, but it holds in the generality stated here). Yet another result in this direction is Latour's theorem, which, for a manifold M of dimension at least 6, characterises the existence of closed non-singular 1-forms in a class χ ∈H1(M; R) in terms of the vanishing of Novikov homology associated to χ and of a K-theoretic invariant [Lat94]. All the results mentioned above directly or indirectly relate vanishing Novikov homology of a character χ to the finiteness properties of its kernel. In [Fis24b], the author showed that Novikov cohomology controls a different property of the kernel, namely its cohomological dimension. This was done with the goal of proving that RFRS groups of cohomological dimension two are virtually free-by-cyclic if and only if they have vanishing second L2-Betti number. The main new input was a criterion stating that if a group G of type FP has Hcd(G)(G; ' Z[G] ±χ ) = 0, then cd(ker χ) = cd(G) -1. One of the main purposes of this article is to prove a converse, thus showing that vanishing top-dimensional Novikov cohomology of integral characters completely determines the cohomological dimension of the kernel. In fact, the result holds in the generality of chain complexes of projective modules. Theorem A. Let G be a group, let χ: G ! Z be an integral character, and let C• be a chain complex of projective Z[G]-modules such that Ci = 0 for all i > n, for some n ∈Z, and assume that Cn is finitely generated. The following are equivalent: (1) Hn(C•; ' Z[G] ±χ ) = 0; (2) C• is chain homotopy equivalent over Z[ker χ] to a chain complex D• of projective Z[ker χ]-modules such that Di = 0 for all i > n -1. Theorem A is proved by analysing the long exact sequence in cohomology induced by a special short exact sequence involving the Novikov ring (see Observation 3.2). This short exact sequence was communicated to the author by Andrei Jaikin-Zapirain as a tool to simplify the original proof of [Fis24b, Theorem A]. It NOVIKOV COHOMOLOGY 3 was also used in [FIK25] to study the algebraic fibres of Poincaré-duality groups. The most general form of Theorem A will be given in Corollary 3.11; in particular, the theorem still holds if one replaces the coefficient ring Z with any ring R (we always consider maps G ! Z, of course). The crux of the implication (2) ⇒(1) is an observation about general modules over Novikov rings: if M is a module over ' Z[G] χ (and χ ̸= 0), then either M = 0 or M is uncountable (Lemma 3.5). Theorem A has the following immediate consequence for groups, proving that top-dimensional Novikov cohomology is a perfect obstruction for cohomological dimension drop. Motivated by Sikorav's Theorem, we define the Σ∗-invariant of a group G of type FP by Σ∗(G) := {χ ∈H1(G; R) ∖{0} : Hcd(G)(G; ' Z[G] χ ) = 0}. Corollary B. Let G be a group of type FP. If χ: G ! Z is an integral character, then cd(ker χ) = cd(G) -1 if and only if ±χ ∈Σ∗(G). It is well-known that vanishing Novikov homology is an open property for groups of finite type; this applies just as well to Novikov cohomology, so we obtain the following corollary (see Corollary 4.8), which is an analogue of Corollary 1.2 for cohomological dimension drop. Corollary C. If G is a group of type FP, then cohomological dimension drop is an open property among integral characters. More precisely, the set {χ: G -! Q : cd(ker χ) = cd(G) -1} is an open subset of H1(G; Q). In particular, if cd(G) = 2, then the set of integral characters with free kernel is open in H1(G; Q). From a slightly stronger version of Theorem A, we obtain a result on the L2invariants of RFRS groups which generalises [Fis24b, Theorem A]. Let U(G) be the algebra of operators affiliated to the group von Neumann algebra of G. The L2Betti numbers of G are defined by b(2) i (G) = dimU(G) Hi(G; U(G)), where dimU(G) denotes the von Neumann dimension (see [Lüc02] for details). The weak dimension of an R-module M is the supremal integer n for which there exists an R-module N such that TorR n (M, N) ̸= 0. Theorem D. Let G be a RFRS group of type FP(Q). The following are equivalent: (1) U(G) is of weak dimension at most one as a Q[G]-module; (2) for every subgroup H ⩽G, we have b(2) i (H) = 0 for all i > 1; (3) G is an iterated (cyclic or finite) extension of a free group. Theorem D will be used to prove that RFRS groups admitting an Abelian hierarchy are iterated (cyclic or finite) extensions of free groups, extending a result of Hagen and Wise [HW10] (see Corollary 5.11). The weak dimension one property is a crucial step in Jaikin-Zapirain and Linton's proof that group algebras of onerelator groups are coherent [JZL25]. In a similar vein, a conjecture of Wise predicts that groups of cohomological dimension at most two are coherent if and only if they have vanishing second L2-Betti number. If G is RFRS and cd(G) ⩽2, then an easy consequence of [Fis24b, Theorem A] is that the vanishing of the second L2-Betti number is equivalent to U(G) being of weak dimension at most one. Motivated by all this, we make the following prediction for coherence in the class of finite type RFRS groups. 4 SAM P. FISHER Conjecture 1.4. If G is a RFRS group of type FP, then the following are equivalent: (1) G is coherent; (2) Q[G] is coherent; (3) U(G) is of weak dimension one as a Q[G]-module. The method of proving Theorem A also yields a new proof of Ranicki's Criterion [Ran95] for finite domination of cyclic covers (see Theorem 3.6), as well as a new proof of the implication H1(G; ' Z[G] ±χ ) = 0 =⇒ ker χ is finitely generated (see Proposition 3.8). This is the more subtle direction of Sikorav's Theorem, and is usually the one used in practice (such as in Kielak's fibring theorem). The new proofs are more conceptual, as there is no longer the need to deal with individual Novikov cycles. One instead works with long exact sequences of homology modules and other standard results from homological algebra, and uses the vanishing of Novikov homology to show that the homology of the kernel commutes with direct products. By Brown's homological characterisation of finite domination [Bro75], this yields finite generation of the kernel. Thus, the method links two well-known criteria for finite domination of chain complexes: Brown's and Ranicki's. Another satisfying aspect of this proof is that it unifies the points of view on Novikov cohomology controlling cohomological dimension and Novikov homology controlling finiteness properties of the cyclic covers; these phenomena follow from dual arguments. A drawback, however, is that it is crucial to work with integral characters and the vanishing of Novikov homology associated to both χ and -χ simultaneously. We do not obtain an new proof of the fact that χ ∈Σ(G) when H1(G; ' Z[G] χ ) = 0 for general characters χ: G ! R. From the Lyndon-Hochschild-Serre spectral sequence, it is easy to see that the cohomological dimension of ker(χ: G ! Z) is at least cd(G) -1, so vanishing of Novikov cohomology in low codimensions cannot imply further cohomological dimension drop of the kernel (we will actually see that vanishing of low-codimensional Novikov cohomology implies the functors Hi(ker χ; -) commute with direct limits in the same low codimensions, which can be seen as a type of finiteness property). However, it still makes sense to ask whether Novikov cohomology can control further cohomological drop of the kernel of a non-integral character. We define the higher Σ∗-invariants of a group G of finite type by Σ∗ m(G) := {χ ∈H1(G; R) ∖{0} : Hi(G; ' Z[G] χ ) = 0 for all i > cd(G) -m} for m ⩾0. Hence, Σ∗ 0(G) = H1(G; R) ∖{0} and Σ∗ 1(G) = Σ∗(G). We obtain the following sufficient condition for a map to a free Abelian group to have kernel of the minimal theoretically allowed cohomological dimension. Theorem E. Let G be a group of type FP. Let N P G be a normal subgroup such that G/N ∼= Zd. If S(G, N) ⊆Σ∗ d(G), then cd(N) = n -d. Given the similarity with Theorem 1.1(2), this raises the obvious question of whether the converse statement holds (Corollary B is the important special case d = 1); it does not, see Example 4.17. Theorem E has the following corollary for Poincaré-duality groups. NOVIKOV COHOMOLOGY 5 Corollary F. Let G be a Poincaré-duality group of dimension n and let χ: G ! Zd. If ker χ is of type FPd-1, then cd(G) = n -d. The condition FP0 is vacuous, so in the case where d = 1, the corollary says that all kernels of maps onto Z drop in cohomological dimension. This is a very special case of Strebel's theorem, which states that all infinite-index subgroups of PDngroups have cohomological dimension at most n -1. There are various situations where cohomological dimension is additive under group extensions. One of the most general results is Fel'dman's theorem [Fel71, Theorem 2.4], which states that if 1 ! N ! G ! Q ! 1 is an extension of groups and k is a field such that N is of type FP(k) and cdk(Q) n if and only if cd(C•) ⩽n. Proof. If C• is chain homotopy equivalent to D• as in the statement, then it is clear that Hi(C•; M) = 0 for all R-modules M and all i > n, since chain homotopy equivalences induce isomorphisms on homology. Hence cd(C•) ⩽n. NOVIKOV COHOMOLOGY 7 Conversely, suppose that cd(C•) ⩽n. Let M = im(∂n+1) ⩽Cn. By assumption, the cocycle z : Cn+1 ∂-! M is zero in Hn+1(C•; M), so there is a cochain c: Cn ! M such that c∂= z. This implies that M is a direct summand of Cn; write Cn = M ⊕P. Then C• splits as the direct sum of the chain complexes {· · · ! Cn+2 ! Cn+1 ! M ! 0 ! · · · } ⊕{· · · ! 0 ! P ! Cn-1 ! Cn-2 ! · · · } of projectives modules, where M and P both lie in degree n. We claim that the complex on the left (call it E•) is null-homotopic, which will prove the lemma. It is an easy exercise to check that Hi(E•; L) = 0 for all integers i and all Rmodules L. Let N = im(∂n+2) and consider the short exact sequence 0 -! N -! Cn+1 -! M -! 0. By assumption, the cocycle w: Cn+2 ∂-! N is zero in cohomology, so there is some d: Cn+1 ! N such that w = d∂. Hence, the short exact sequence above is split, so E• = {· · · ! Cn+2 ! N ! 0 ! · · · } ⊕{· · · ! 0 ! M ! M ! 0 ! · · · }. Again, the complex {· · · ! Cn+2 ! N ! 0 ! · · · } has vanishing cohomology with all coefficients, so we can split off {· · · ! 0 ! N ! N ! 0 ! · · · } as a direct summand, whose complement has vanishing cohomology with all coefficients. Continuing in this way show shows that E• is a direct sum of complexes of the form {0 ! L ! L ! 0} and is therefore null-homotopic. □ The Eilenberg-Watts theorem will be useful to us in induction arguments, including in the proof of Lemma 2.6 just below. Morally, it says that tensoring is the only right-exact additive functor that preserves direct sums. Theorem 2.5 ([Wat60]). Let R and S be rings, and let F : ModR ! ModS be an additive functor. If F is right-exact and preserves arbitrary direct sums, then there is a natural isomorphism F(-) ∼= -⊗R F(R), where R is viewed as a right module over itself via multiplication, and F(R) is viewed as a left R-module via r · x := F(mr)(x) (and mr is the left-multiplication by r map). Lemma 2.6. Let R be a ring, let C• be a chain complex of projective left R-modules and let m and n be integers. (1) Suppose Ci = 0 for all i n, that Hi(C•; -) commutes with direct limits for all i ⩾m, and that Hi(C•; R) = 0 for all i ⩾m. Then Hi(C•; M) = 0 for all i ⩾m and all R-modules M. Proof. For (1), see [Bro75, Lemma 2]. For (2), we will assume that n ⩾m, since otherwise the statement is vacuous. We will prove that Hi(C•; M) = 0 for all Rmodules M by reverse induction on i, starting with i = n + 1 and terminating with i = m. There is nothing to prove in the base case, since Cn+1 = 0 and therefore Hn+1(C•; M) trivially vanishes for all modules M. Suppose Hi(C•; M) = 0 for all R-modules M and for some i > m. Let 0 -! M0 -! M1 -! M2 -! 0 8 SAM P. FISHER be a short exact sequence of modules. The associated long exact sequence in cohomology contains the portion Hi-1(C•; M1) -! Hi-1(C•; M2) -! Hi(C•; M0) = 0, so Hi-1(C•; -) is right exact. The functor Hi-1(C•; -) also preserves direct limits by assumption, so the Eilenberg-Watts theorem implies that Hi-1(C•; -) is naturally isomorphic to the functor Hi-1(C•; R)⊗R -, which is zero by assumption. □ 3. Twisted Laurent series and finite domination Let R be a ring and let α: R ! R be a ring automorphism. The α-twisted Laurent polynomial ring in one variable is denoted by Rα[t-1, t]. This is the Laurent polynomial ring over R, where we impose the rule rtn = tnαn(r) for all r ∈R and all n ∈Z. Let Rα[t-1, tK denote the ring of Laurent series in the variable t, i.e. the ring of expressions of the form P∞ n=k rntn with rn ∈R for all n ⩾k, where k is some integer. Multiplication is defined on Rα[t-1, tK by formally extending the multiplication on Rα[t-1, t]. Similarly, let RαJt-1, t] be the ring of formal series of the form Pk n=-∞rntn for some k ∈Z, i.e. the ring of Laurent series in the variable t-1. Example 3.1 (The Novikov ring). The main example of the construction above will come from groups and maps to Z. Let χ: G ! R be an homomorphism with kernel N and let R be a ring. The Novikov ring associated to this data is ' R[G] χ = nX rgg : |{g : rg ̸= 0} ∩χ-1(]-∞, t])| n. Suppose that the functors Hi(C•; -) commute with direct limits of Rα[t-1, t]-modules for all i ⩾m. If Hi(C•; RαJt-1, t]) = Hi(C•; Rα[t-1, tK) = 0 for all i ⩾m, then Hi(ι!C•; -) commutes with direct limits for all i ⩾m. If n ⩾m, then ι!C• is of cohomological dimension at most n -1. Remark 3.10. It is unclear to us whether the converse holds, i.e. whether Hi(ι!C•; -) commuting with direct limits for i ⩾m implies vanishing Novikov cohomology in degrees at least m. It holds if m = n (see Corollary 3.11), and if we additionally assume that C• is a chain complex of finitely generated projectives, then the converse also holds at m = n -1. Proof (of Theorem 3.9). The proof that Hi(ι!C•; -) commutes with direct limits is similar to the proof of Proposition 3.8, where it was shown that Hi(ι!C•; -) commutes with direct products in a certain range. We leave the details to the reader. By Lemma 2.6, Hi(C•; MαJt-1, t]) = Hi(C•; Mα[t-1, tK) = 0 for all i ⩾m and any right R-module M. The long exact sequence in cohomology of Corollary 3.4 then yields Hn(ι!C•; M) = 0. Since M was arbitrary, this shows that ι!C• has cohomological dimension at most n -1. □ As a corollary, we find that the top-dimensional Novikov cohomology of C• completely controls the cohomological dimension of ι!C•, proving Theorem A from the introduction in a more general form. Corollary 3.11. Let C• be a chain complex of projective Rα[t-1, t]-modules of cohomological dimension n and such that Cn is finitely generated. The following are equivalent: (1) ι!C is of cohomological dimension n -1; (2) Hn(C•; RαJt-1, t]) = Hn(C•; Rα[t-1, tK) = 0. Proof. If ι!C• is of cohomological dimension n -1, then the long exact sequence in cohomology of Corollary 3.4 yields the exact sequence Hn(C•; Rα[t-1, t]) -! Hn(C•; RαJt-1, t] ⊕Rα[t-1, tK) -! 0. 14 SAM P. FISHER If R is a countable ring, then Hn(C•; Rα[t-1, t]) is countable (since Cn) is finitely generated. If not, we can use the fact that Cn is finitely generated to reduce to the countable case, much in the same way as in the proof of Theorem 3.6 in the general case. We sketch the argument below. We may assume that the modules Ci are all free. The fact that ι!C• is of cohomological dimension n -1 means that there is a map ρ: ι!Cn-1 ! ι!Cn such that ρ∂= idι!Cn. Since Cn is finitely generated, there is a finitely generated direct summand A of Cn-1 containing ∂(Cn). Writing Cn-1 = A ⊕B, we can take ρ to be zero on B. Representing ∂and ρ as matrices over R, they contain countably many entries, and there is an α-invariant countable subring R′ of R containing all of these entries. By the argument above, Hn(C•; R′ αJt-1, t]) = Hn(C•; R′ α[t-1, tK) = 0. We conclude that Hn(C•; RαJt-1, t]) = Hn(C•; Rα[t-1, tK) = 0 in the same way as in the proof of the general case of Theorem 3.6. □ 4. The Σ∗-invariant Let G be a group and let R be a ring such that cdR(G) = n. If the trivial R[G]-module R admits a projective resolution P• ! R ! 0 such that Pi is finitely generated for all i > n -m, then we will say that G is of type FP∗ m(R). In other words, the trivial module admits a resolution that is finitely generated in the top m degrees. Note that the finiteness property FP∗ m(R) only makes sense for groups with finite cohomological dimension over R, and that if a group is of type FP(R) then it is of type FP∗ m(R) for all m. Definition 4.1. Let G be of type FP∗ m(R). The mth Σ∗-invariant of G (over R) is Σ∗ m(G; R) = {χ ∈H1(G, R) ∖{0} : Hi(G; ' R[G] χ ) = 0 for all i > n -m}. In particular, Σ∗ 1(G; R) is the set of characters with respect to which the topdimensional Novikov homology of G vanishes. It follows immediately from the definition that Σ∗ m+1(G; R) ⊆Σ∗ m(G; R) (whenever G is of type FP∗ m+1(R)). Proposition 4.2. Let G be a finitely generated group of type FP∗ m(R) for some ring R and integer m. Then Σ∗ m(G; R) is open in H1(G; R). The proposition will follow quickly from Lemma 4.5 below, which will be well known to experts. The proof given here relies on chain contractions, and the following simple fact: if χ: G ! R is positive on the support of x ∈R[G], then there is an open neighbourhood U of χ such that 1 -x is invertible in ' R[G] ψ for all ψ ∈U. We first give a couple of definitions. Definition 4.3 (Truncation). Let χ: G ! R be a character. The truncation of an element P g∈G rgg ∈' R[G] χ at height t ∈R is the element P g∈G rgg ∈R[G], where rg = rg if χ(g) -1 and that we have found maps sj for all j n -m is the cohomology of the cochain complex HomR[G](Pi, ' R[G] χ ) ∼= HomR[G](Pi, R[G]) ⊗R[G] ' R[G] χ . Since Pi is finitely generated, HomR[G](Pi, R[G]) is projective and finitely generated as an R[G]-module. The result then immediately follows from Lemma 4.5. □ As we saw in Example 3.1, the Novikov ring associated to an integral character χ: G ! Z is a special case of the construction Rα[t-1, tK. Thus, the results of the previous section readily apply to give theorems about group cohomology. The first such result is an immediate consequence of Corollary 3.11, and establishes the converse to [Fis24b, Theorem 3.5] in the case of integral characters. Theorem 4.7. Let G be a group and let R be a ring such that cdR(G) n and such that Hi(G; -) commutes with direct limits of R[G]-modules for all i > n -d. If Hi(resG Gj C•; ' R[Gj] ±χj) = 0 for all i > n -d and all 0 ⩽j 1. Again by Corollary 3.11, resG G1 C• is chain homotopy equivalent to a chain complex of projective modules D• which is of cohomological dimension at most n -1 and such that Hi(D•; -) commutes with direct limits for all i > n -d. The claim now follows by induction. □ 18 SAM P. FISHER For the proof of Theorem E, we will need the following result of Farber, Geoghegan, and Schütz, which gives a version of Sikorav's Theorem for chain complexes. Theorem 4.12 ([FGS10, Theorem 8]). Let C• be a chain complex of projective R[G]-modules such Ci is finitely generated for i ⩽n and zero for i n, and let χ: G ! R be a character with free Abelian image of rank d. If Hi(C•; ' R[G] φ ) = 0 for all φ ∈S(G, ker χ) and all i > n -d, then resG ker χ C• has cohomological dimension at most n -d. Proof. Since the modules Ci are finitely generated, HomR[G](C•, ' R[G] φ ) ∼= HomR[G](C•, R[G]) ⊗R[G] ' R[G] φ . The Novikov acyclicity assumption and a Theorem 4.12 imply that resG ker χ HomR[G](C•, R[G]) is chain homotopy equivalent to a chain complex D• of projective R[ker χ]-modules such that Di is finitely generated for all i > n -d. For the remainder of the proof, we follow Notation 4.9. Let P = χ(G) ∼= Zd and fix a series of subgroups {1} = Pd ⩽Pd-1 ⩽· · · ⩽P1 ⩽P0 = P such that Pi/Pi+1 ∼= Z for all 0 ⩽i n -d and 0 ⩽j n and such that Hi(F•; -) commutes with direct limits for all i > n -d. If Hi(F•; IndG Gj ' R[Gj] ±χj) = 0 for all i > n -d and 0 ⩽j n -d and 0 ⩽j 1. Fix some j n -d. We also know that Hi(F•; MαJt-1, t]) = Hi(F•; Mα[t-1, tK) = 0 by Lemma 2.6, since Mα[t-1, tK is a right ' R[G] χ0 = R[G1]α[t-1, tK-module and Hi(F•; ' R[G] χ0) = 0 (and similarly for MαJt-1, t]). It then follows that Hi(F•; coIndG G1 M) = Hi(resG G1 F•; IndG1 Gj ' R[Gj] ±χj) = 0. for all i > n-d. Since Hi(resG G1 F•; -) commutes with direct limits for all i > n-d by Proposition 4.11, the claim holds by induction. ⋄ By the claim, Hi(resG Gj C•; ' R[Gj] ±χj) = 0 for all i > n -d and all 0 ⩽j l (we trivially know this holds for l = n and all j ∈J). Then Hl(G; -) is a right exact functor on the category of Sj-modules. By Theorem 2.5, there is a natural isomorphism Hl(G; -) ∼= Hl(G; Sj) ⊗Sj - as functors on the category of Sj-modules. The functor Hl(G; -) commutes with direct products of Sj-modules, and therefore so does the functor Hl(G; Sj) ⊗Sj -. It is a well-known fact (and an easy exercise) that Hl(G; Sj) must be finitely generated as an Sj-module. Let z1, . . . , zs be a finite generating set for Hl(G; Sj). Since Hl(G; S) ∼= Hl(G; Sj) ⊗Sj S ∼= lim -! j′⩾j Hl(G; Sj) ⊗Sj Sj′ = 0, we see that there is some j′ ∈J such that zi ⊗1 = 0 in Hl(G; Sj) ⊗Sj Sj′ for each i = 1, . . . , s. But these elements also generate Hl(G; Sj) ⊗Sj Sj′, so we conclude that Hl(G; Sj′) ∼= Hl(G; Sj) ⊗Sj Sj′ = 0. Since Sj′ is an Sj-module, we also have Hi(G; Sj′) = 0 for all i > l by Lemma 2.6. The lemma then follows by induction. □ We can now prove the main result of this section, which implies Theorem D by taking d = 1 and using the Stallings-Swan Theorem [Sta68, Swa69], which characterises free groups as the groups of cohomological dimension at most one. Theorem 5.7. Let G be a RFRS group with cdk(G) = n for some field k. Suppose Hi(G; -) commutes with direct limits of k[G]-modules for all i > d. The following are equivalent: (1) Dk[G] is of weak dimension at most d as a k[G]-module; (2) for every subgroup H ⩽G, we have b(2) i (H; k) = 0 for all i > d; (3) there is a subnormal series Γl d. But Dk[G],U ⊆ Dk[Gs],Us ∗G/Gs, and therefore 0 = Hi(G; Dk[Gs],Us ∗G/Gs) ∼= Hi(Gs; Dk[Gs],Us) for all i > d by Lemma 2.6. We will take Γ1 = Gs. Recall that ' k[Gs] χ is a Dk[Gs],Usalgebra for every χ ∈Us. Since -Us ∩Us is also a rich (and hence open) set, there is an integral character χ: Gs ! Z such that Hi(Gs; ' k[Gs] ±χ ) = 0 for all i > d. By Corollary 3.11, cdk(ker χ) = n -1 and Hi(ker χ; -) commutes with direct limits for all i > d. We take Γ2 = ker χ, and repeat this argument with Γ2 instead of G to extend the chain to Γ2 > Γ3 > Γ4, where Γ2/Γ3 is finite, and Γ3/Γ4 is cyclic with cdk(Γ4) = n -2. Continuing in this way proves item (3). To prove (3) implies (1), it suffices to prove the following claim: If 1 -! N -! G -! Q -! 1 is an extension with G RFRS, Q amenable, and Dk[N] of weak dimension at most d as a k[N]-module, then Dk[G] is of weak dimension at most d as a k[G]-module. The argument is similar to that of [JZL25, Theorem 3.4]. Indeed, let M be any k[G]-module. Then, for all i > d, Tork[G] i (Dk[N] ∗G/N, M) ∼= Tork[N] i (Dk[N], M) = 0 by Shapiro's lemma. But Dk[G] is the Ore localisation of Dk[N] ∗G/N by a result of Tamari [Tam54], and since Ore localisation is flat, we also get Tork[G] i (Dk[G], M) = 0 for all i > d, as claimed. □ Corollary 5.8. Let G be a RFRS group of type FP(k). Then Dk[G] is of weak dimension one as a k[G]-module if and only if G is an iterated extension of a free group by (cyclic or finite) groups. Remark 5.9. In the introduction, Corollary 5.8 was stated in terms of U(G), the algebra of operators affiliated to the von Neumann algebra of G. Let k ⊆C be a field. Since Dk[G] is the division closure of k[G] in U(G), and moreover Dk[G] is a division ring, U(G) is faithfully flat over Dk[G]. It follows that U(G) has weak dimension at most one as a k[G]-module if and only if Dk[G] does. It is an interesting problem to determine the difference between the invariants b(2) i (G; k) for different fields k. In the context of RFRS groups (of finite type, say), it is an easy consequence of the results main results of [Kie20] and [Fis24a] that b(2) 1 (G; k) = 0 for one field k if and only if b(2) 1 (G; k′) = 0 for all fields k′. It is open whether the value of b(2) 1 (G; k) depends on the field k, but for every i > 1 there are examples of right-angled Artin groups G (which are, in particular, RFRS) such that b(2) i (G; k) depends on k [AOS21]. Here we show that Dk[G] being of weak dimension one (or equivalently, that all subgroups of G have vanishing k-L2-Betti numbers) does not depend on k. Corollary 5.10. Let G be a RFRS group of type FP. If Dk[G] is of weak dimension one as a k[G]-module for some field k, then Dk′[G] is of weak dimension one as a k′[G]-module for all fields k′. 24 SAM P. FISHER We conclude the section with applications to groups with elementary hierarchies. Following Hagen and Wise [HW10], we say that the trivial group has an elementary hierarchy of length 0, and that a group G has an elementary hierarchy of length n if it splits as a graph of groups with an elementary hierarchy of length n -1 with cyclic edge groups. The main theorem of [HW10] is that any finitely generated virtually special group with an elementary hierarchy is virtually free-by-cyclic (a group is special if and only if it is a subgroup of a RAAG, so special groups are RFRS). This class of groups includes, e.g., graphs of free groups with cyclic edge groups not containing unbalanced Baumslag-Solitar groups and limit groups not containing Z3-subgroups. We extend the Hagen-Wise theorem in the following way: say that G has an Abelian hierarchy of length 0 if it is trivial and that it has an Abelian hierarchy of length n if it virtually splits as a graph of groups with an Abelian hierarchy of length n -1 and finitely generated virtually Abelian edge groups. While all groups with an elementary hierarchy of cohomological dimension at most two, groups with an Abelian hierarchy can be of arbitrarily high cohomological dimension (the class contains all limits groups, for example). Corollary 5.11. Let G be a virtually RFRS group of type FP(k) for some field k admitting a virtually Abelian hierarchy. Then G is an iterated extension of a free group by (cyclic or finite) groups. Proof. It suffices to prove that every subgroup of G has vanishing k-L2-Betti numbers above degree one. Every subgroup of G admits an Abelian hierarchy, so it suffices to prove the claim for a fixed group Γ with an Abelian hierarchy of length n. If n = 0, then Γ is trivial, and therefore b(2) i (Γ; k) = 0 for all i > 0. Now assume n > 0; since k-L2-Betti numbers scale under passage to finite-index subgroups, we will assume that Γ admits a graph of groups splitting (Γv, Γe), where each Γv has an Abelian hierarchy of length n -1 and Γe is virtually Abelian. Then Chiswell's long exact sequence in the homology of a graph of groups [Chi76] and the fact that virtually Abelian groups have vanishing k-L2-Betti numbers above degree zero imply that b(2) i (Γ; k) = 0 for all i > 1. The result now follows from Corollary 5.8. □ 6. Novikov cohomology versus Novikov homology Let G be a group of finite type and of cohomological dimension n and let χ: G ! Z be an integral character. If for some ring R we have Hn(G; ' R[G] χ ) = 0, then Hn(G; ' R[G] χ ) = 0. A natural question is whether the converse holds. if the converse fails, it still makes sense to ask whether the vanishing of top-dimensional Novikov homology controls some property of the kernel. A natural guess is that if Hn(G; ' R[G] ±χ ) = 0, then the homological dimension of ker χ over R is n -1. In this brief section, we will see that these questions have negative answers. Lemma 6.1. Let G be a torsion-free one-relator group and let R be any ring without zero divisors containing Z[G]. Then H2(G; R) = 0. Proof. Since G is torsion-free, Z admits a projective resolution of the form 0 -! Z[G] ∂ --! Z[G]n -! Z[G] -! Z -! 0 (this is a consequence of the Lyndon identity theorem [Lyn50]). We claim that the extended map ∂: R ! Rn is injective. Indeed, suppose ∂(r) = r∂(1) = 0. Since NOVIKOV COHOMOLOGY 25 ∂is injective, ∂(1) is nonzero. Since R has no zero divisors, this implies r = 0, as desired. □ Corollary 6.2. If G is a torsion-free one-relator group and χ: G ! R is any homomorphism, then H2(G; ' Z[G] χ ) = 0. Proof. One-relator groups are locally indicable by a result of Brodski ̆ı [Bro84] and therefore the group ring Z[G] has no zero divisors by a result of Higman [Hig40]. By looking at terms of minimal χ-height, it is easy to show that ' Z[G] χ also has no zero divisors as well. The result then follows immediately from Lemma 6.1. □ Example 6.3. Consider the Baumslag-Solitar group G = BS(2, 3) = ⟨a, t | t-1a2t = a3⟩. It is well known that G is not residually finite, and therefore it cannot be virtually free-by-cyclic [Bau71]. So for any finite-index subgroup H ⩽G and any epimorphism χ: H ! Z, either H2(H; ' Z[H] χ ) ̸= 0 or H2(H; ' Z[H] -χ ) ̸= 0 by Corollary B. In particular, let χ: G ! Z be the map defined by χ(a) = 0 and χ(t) = 1. With Corollary 6.2 this gives an example of a group of finite type and a map with non-vanishing top-dimensional Novikov cohomology that has vanishing Novikov homology. Moreover, ker χ is not of homological dimension one. Indeed, there is a well defined homomorphism ⟨x, y | x3 = y2⟩-! G, x 7-! a, y 7-! t-1at, which is injective (we leave checking injectivity as an exercise for the reader). Since M ∼= ⟨x, y | x3 = y2⟩is a non-free one-relator group, it is of homological dimension two, which shows that ker χ is of homological dimension two. Hence, vanishing top-dimensional Novikov homology does not imply that the homological dimension of the kernel drops. References [AOS21] Grigori Avramidi, Boris Okun, and Kevin Schreve. Mod p and torsion homology growth in nonpositive curvature. Invent. Math., 226(3):711-723, 2021. [Bau71] Gilbert Baumslag. Finitely generated cyclic extensions of free groups are residually finite. Bull. Austral. Math. Soc., 5:87-94, 1971. [BG99] Kai-Uwe Bux and Carlos Gonzalez. The Bestvina-Brady construction revisited: geometric computation of Σ-invariants for right-angled Artin groups. J. London Math. Soc. (2), 60(3):793-801, 1999. [Bie07] Robert Bieri. Deficiency and the geometric invariants of a group. J. Pure Appl. Algebra, 208(3):951-959, 2007. With an appendix by Pascal Schweitzer. [BNS87] Robert Bieri, Walter D. Neumann, and Ralph Strebel. A geometric invariant of discrete groups. Invent. Math., 90(3):451-477, 1987. [BR88] Robert Bieri and Burkhardt Renz. Valuations on free resolutions and higher geometric invariants of groups. Comment. Math. Helv., 63(3):464-497, 1988. [Bro75] Kenneth S. Brown. Homological criteria for finiteness. Comment. Math. Helv., 50:129135, 1975. [Bro84] Sergei D. Brodski ̆ı. Equations over groups, and groups with one defining relation. Sibirsk. Mat. Zh., 25(2):84-103, 1984. [Bro94] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. 26 SAM P. FISHER [Chi76] I. M. Chiswell. Exact sequences associated with a graph of groups. J. Pure Appl. Algebra, 8(1):63-74, 1976. [Fel71] G. L. Fel'dman. The homological dimension of group algebras of solvable groups. Izv. Akad. Nauk SSSR Ser. Mat., 35:1225-1236, 1971. [FGS10] M. Farber, R. Geoghegan, and D. Schütz. Closed 1-forms in topology and geometric group theory. Uspekhi Mat. Nauk, 65(1(391)):145-176, 2010. [FIK25] Sam P. Fisher, Giovanni Italiano, and Dawid Kielak. Virtual fibring of Poincaré-duality groups, 2025. . [Fis24a] Sam P. Fisher. Improved algebraic fibrings. Compos. Math., 160(9):2203-2227, 2024. [Fis24b] Sam P. Fisher. On the cohomological dimension of kernels of maps to Z, 2024. , (to appear in Geom. Topol.). [Hig40] Graham Higman. Units in group rings. DPhil, 1940. [HK07] J. A. Hillman and D. H. Kochloukova. Finiteness conditions and PDr-group covers of PDn-complexes. Math. Z., 256(1):45-56, 2007. [HW10] Mark Hagen and Daniel T. Wise. Special groups with an elementary hierarchy are virtually free-by-Z. Groups Geom. Dyn., 4(3):597-603, 2010. [JZ21] Andrei Jaikin-Zapirain. 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Higher generation subgroup sets and the Σ-invariants of graph groups. Comment. Math. Helv., 73(1):22-44, 1998. [Ran95] Andrew Ranicki. Finite domination and Novikov rings. Topology, 34(3):619-632, 1995. [Sch02] Thomas Schick. Erratum: \"Integrality of L2-Betti numbers\". Math. Ann., 322(2):421422, 2002. [Sik87] Jean-Claude Sikorav. Homologie de Novikov associée à une classe de cohomologie de degré un. Thèse d'État, Université Paris-Sud (Orsay), 1987. [Sta68] John R. Stallings. On torsion-free groups with infinitely many ends. Ann. of Math. (2), 88:312-334, 1968. [Swa69] Richard G. Swan. Groups of cohomological dimension one. J. Algebra, 12:585-610, 1969. [Tam54] Dov Tamari. A refined classification of semi-groups leading to generalised polynomial rings with a generalised degree concept. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, volume III, pages 439-440. Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1954. [Wat60] Charles E. Watts. Intrinsic characterizations of some additive functors. Proc. Amer. Math. Soc., 11:5-8, 1960. Email address:"}}},{"rowIdx":187,"cells":{"id":{"kind":"string","value":"2510.14798"},"raw":{"kind":"string","value":"Balls and Bins and the Infinite Process with Random Deletions\nPetra Berenbrink∗\nTom Friedetzky†\nPeter Kling‡\nLars Nagel§\nAbstract\nWe consider an infinite balls-into-bins process with deletions where in each discrete step t\na coin is tossed as to whether, with probability β(t) ∈(0, 1), a new ball is allocated using the\nGreedy[2] strategy (which places the ball in the lower loaded of two bins sampled uniformly at\nrandom) or, with remaining probability 1 −β(t), a ball is deleted from a non-empty bin chosen\nuniformly at random. Let n be the number of bins and m(t) the total load at time t. We\nare interested in bounding the discrepancy xmax(t) −m(t)/n (current maximum load relative\nto current average) and the overload xmax(t) −mmax(t)/n (current maximum load relative to\nhighest average observed so far).\nWe prove that at an arbitrarily chosen time t the total number of balls above the average is\nO(n) and that the discrepancy is O(log(n)). For the discrepancy, we provide a matching lower\nbound. Furthermore we prove that at an arbitrarily chosen time t the overload is log log(n) +\nO(1). For “good” insertion probability sequences (in which the average load of time intervals\nwith polynomial length increases in expectation) we show that even the discrepancy is bounded\nby log log(n) + O(1).\nOne of our main analytical tools is a layered induction, as per [ABKU99]. Since our model\nallows for rather more general scenarios than what was previously considered, the formal analysis\nrequires some extra ingredients as well, in particular a detailed potential analysis. Furthermore,\nwe simplify the setup by applying probabilistic couplings to obtain certain “recovery” properties,\nwhich eliminate much of the need for intricate and careful conditioning elsewhere in the analysis.\n1\nIntroduction.\nBalls-into-bins processes have been formalized in the mathematical community since at least the\nearly 18th century [dM18]. Their objective is, e.g., to allocate at random a set of balls into a number\nof bins such that the maximum number of balls per bin is minimized. These processes have been\nvery well studied, with many results being largely folklore. However, even seemingly minor modifi-\ncations of the protocol or the model tend to result in vastly different and often far more complex\nanalyses as well as, occasionally, very surprising results. Apart from the undoubted usefulness as\nprobabilistic paradigms, balls-into-bins processes have many applications, often under the bonnet as\nprimitive black-box operations. Among others, this includes areas like (the obvious) load balancing\nand allocation of jobs to servers [CL97], routing in networks [CMMadH+98], hashing [CRSW13],\ndictionary data structures [DW05], design of file systems [LCB13], peer-to-peer networks [SMK+01],\nand many more.\nThe most simple balls-into-bins process, which allocates m balls independently and uniformly\nat random to n bins, is very well understood. For example, if m = n then the maximum number of\n∗University of Hamburg, Germany (petra.berenbrink@uni-hamburg.de).\n†Durham University, U.K. (tom.friedetzky@durham.ac.uk).\n‡Darmstadt University of Applied Sciences, Germany (peter.kling@h-da.de).\n§Loughborough University, U.K. (L.Nagel@lboro.ac.uk).\n1\narXiv:2510.14798v1 [cs.DC] 16 Oct 2025\n\nballs (load) in any bin is, w.h.p.1, log(n)/ log log(n) · (1 + o(1)) [RS98]. In [ABKU99] the authors\nintroduced the Greedy[d] process. Here, balls are allocated sequentially; each ball chooses d ∈N\nbins uniformly at random and is allocated to a least loaded of them.\nGreedy[1] is identical\nto the simple process described above. But if d ≥2, the maximum load is, w.h.p., reduced to\nlog log(n)/ log(d) + O(1) [ABKU99] for m = n. This exponential decrease in the maximum load is\noften referred to as the power of two choices. Greedy[2] was later analyzed for m ≥n (the heavily\nloaded case), which led to the surprising result that, in contrast to Greedy[1], the difference\nbetween the maximum and average load is independent of m [BCSV06] (namely, w.h.p.\nit is\nlog log(n)/ log(d) + O(1)).\nBalls-into-Bins with Deletions.\nWe study Greedy[2] in the heavily loaded case where balls are\nnot only inserted but also deleted. Specifically, we ask whether allocation via Greedy[2] can over-\ncome the negative effect of random deletions (effectively “holes” allocated according to Greedy[1],\nwhich without insertions would cause a rift of depth p m\nn log n for large m).\nOne can differentiate balls-into-bins with deletions depending on whether, each time a ball is\ninserted, the ball gets d fresh choices (insertion/deletion model) or whether the ball might be a\npreviously deleted ball that is reinserted with its original d choices (reinsertion/deletion model).\nRecently, the authors of [BK22] showed that, somewhat surprisingly, even in the insertion/deletion\nmodel, an oblivious adversary2 can make sure that Greedy[2] does not overcome the negative effect\nof deletions: already for n = 4 bins, it can cause a maximum load of m/4+Ω(√m) with probability\nΩ(1). Intuitively, the lower bound uses the fact that if there is ever a bin i that contains far fewer\n(say k) balls than other bins, Greedy[2] is biased toward i for the next k throws. After filling the\nsystem to m balls with at least k insertions, the adversary exploits the insertion bias to perform\nk deletions that are now biased towards bin i. Adding replacement balls for the just deleted balls,\none can show that enough of these land on bins ̸= i to cause the desired overload.\nResults in a Nutshell.\nWe consider the insertion/deletion model for n ∈N bins with random\nbut dynamic deletions, starting from an initially empty system. Each time step t, with probability\nβ(t) a new ball with fresh 2 choices is inserted by Greedy[2]. Otherwise, with probability 1−β(t),\na ball is deleted from a bin chosen uniformly at random among all non-empty bins. (All our results\nalso hold for the protocol that does not delete from a random bin, but a ball chosen uniformly\nat random among all balls, see Appendix A.) Our main result can be seen as complementing this\nunfortunate lower bound of [BK22] for Greedy[2] with a positive result: In a model with random\ndeletions (but still dynamic, that is, resulting in a fluctuating average load), Greedy[2]’s allocation\novercomes the negative deletion effect as long as insertions dominate deletions ever so slightly in\nthe recent past of a given time step t. This highlights a fundamental difference between our model\nand the one of [BK22]. Our result suggests that in many practical applications, where deletions\n(if at all) occur non-adversarially, Greedy[2] easily overcomes random Greedy[1]-type deletions.\nThis holds true even if the system utilization (ratio between insertions and deletions) fluctuates\nsignificantly. See Section 1.1 for a full and formal description of our results.\n1We say an event E happens with high probability (w.h.p.) if Pr [ E ] ≥1 −n−Ω(1).\n2The adversary knows only the d choices and insertion time of each ball. Each time step it can either (a) insert\na new ball with d fresh random bin choices or (b) delete the r-th most recently inserted ball (r chosen freely). In\nparticular, it is oblivious to the exact load situation, since it cannot see to which of the d choices the allocation\nstrategy (like Greedy[d]) assigned a given ball.\n2\n\n1.1\nOur Contribution.\nThe system state at any time can be modeled by a load vector x = (x1, . . . , xn) ∈Nn\n0, where xi\nis the load of the i-th fullest bin. We use m(x) := ∥x∥1 to denote the total load of x and define,\nfor clarity, xmax := x1 and xmin := xn as the maximum and minimum load of x, respectively.\nWe say x is perfectly balanced if xmin, xmax ∈{ ⌊m(x)/n⌋, ⌈m(x)/n⌉}. The discrepancy disc(x) :=\nxmax−m(x)/n of x is the difference between the current maximum and average load. In some cases,\nwe also use the stronger absolute discrepancy adisc(x) := max { xmax −m(x)/n, m(x)/n −xmin }\nof x (the largest difference of any load to the current average load). With this, our process can\nbe described as a random sequence\n\u0000x(t)\n\u0001\nt∈N0 of load vectors with total load m(t) := m\n\u0000x(t)\n\u0001\n,\ndiscrepancy disc(t) := disc\n\u0000x(t)\n\u0001\n, and absolute discrepancy adisc(t) := adisc\n\u0000x(t)\n\u0001\nat time t. We\nuse mmax(t) := max { m(0), . . . , m(t) } to denote the maximum total load up to time t. The overload\novld(t) := xmax(t) −mmax(t)/n at time t is the difference between the current maximum and the\nhighest average load up to time t. The height of a ball is its position in the bin modelled as a stack\nof balls counting from bottom to top.\nWe assume a dynamic insertion probability β(t) that may change from step to step to any value\nin [0, 1]. In other words, insertions and deletions are governed by an infinite insertion probability\nsequence\n\u0000β(t)\n\u0001\nt∈N giving the (independent) probabilities of an insertion/deletion in each time step.\nWe say value x ∈(0, 1) is bounded away from 0 (from 1) if x ≥0 + Ω(1) (x ≤1 −Ω(1)). While\nballs-into-bins processes typically aim to bound the discrepancy, this cannot work in general when\ndeletions are involved. For example, the insertion probability sequence could contain a stretch that\ndeletes m(t)/2 balls after some time t. Translating a folklore example (also mentioned in [BK22])\nto our setting, deletions can be regarded as random “holes” allocated via Greedy[1]. This would\ncause a discrepancy of order ˜Θ(\np\nm(t)/n), a bound trivially achieved by allocations via Greedy[1],\neven holds if the system starts out perfectly balanced.\nThus, in this paper we aim to bound the overload.\nWe characterize which (possibly local)\nproperties an insertion probability sequence requires for Greedy[2] to achieve a small discrepancy.\nAlso, in an infinite process it is quite impossible to always have a small discrepancy (or even\noverload). Thus, our main results (given below) all hold with high probability at an arbitrary (also\nsuper-exponentially large) point t in time:\n(a) Assume all insertion probabilities up to time t are bounded away from 0. Then the discrepancy\nat time t is O(log(n)) (Theorem 2.1) and the number of balls above average at time t is O(n)\n(Theorem 2.2). The bound on the discrepancy is tight if the insertion probabilities are bounded\naway from 1/2 from above in the previous O(n log(n)) time steps (see Appendix E).\n(b) Assume we are in a well (e.g., perfectly) balanced situation at time t. Then we can maintain\nan overload of log log(n) + O(1) for poly(n) steps as long as the insertion probabilities during\nthat time are bounded away from 0 and 1. If, additionally, the insertion probabilities ensure an\nexpected load increase for polynomially long subintervals (see definition of c-good time intervals\nin Section 3), we can maintain a discrepancy (instead of only overload) of log log(n) + O(1)\n(see Theorem 3.1 and Lemma 3.4 in Section 3 for the full, formal statements).\n(c) Our final result shows that we achieve an overload of log log(n) + O(1) at an arbitrary time\nt ∈N0 as long as the insertion probability sequence till that time is bounded away from 0\nand 1 (Theorem 4.1). Similar to the previous result, if the insertion probabilities ensure an\nexpected (possibly quite small) load increase for long subintervals during the previous n4 time\nsteps, we even get a discrepancy of log log(n) + O(1) at time t.\nTo the best of our knowledge, these are the first results for the discrete, heavily-loaded inser-\ntion/deletion setting that provide characterizations for when Greedy[2] achieves strong guarantees\n3\n\non the discrepancy instead of only on the overload.\nMain Technical Contributions and Proof Idea.\nTo prove that the discrepancy is O(log(n))\n(see (a) above) we use a potential function argument similar to [PTW15, TW14].\nTo reduce\nthe discrepancy to double-logarithmic (see (c) above) we use the excellent recovery properties of\nGreedy[2]: In [Czu00], Czumaj studied the recovery of Greedy[2] for a process that alternates\nbetween ball insertions and deletions, maintaining the total load invariant. He showed that the\nprocess is able to recover, i.e., reach a steady state distribution, in polynomial time. However, this\ndoes not reveal any properties (e.g., the discrepancy) of the steady state.\nIn contrast to [Czu00] we cannot have a unique steady state as the total load varies over time.\nHowever, we can show via a path coupling argument that when we start two copies of our process in\nany two configurations of logarithmic discrepancy with the same number of balls – which we have at\nany time via (a) – it causes both configurations to converge to each other quickly (in o(n4) steps).\nThis can be interpreted as a “typical” state to which all configurations with the same number of\nballs converge. Again, as in [Czu00] this does not yet reveal any further information about such a\ntypical state.\nThis is where (b) comes into play, which uses a layered induction to show that, starting from a\ncertain class of configurations (with discrepancy at most log log(n)+c), the process remains in such\na well-balanced configuration for a long time, Ω(n4) steps. That is, we maintain well-balancedness\nfor longer than it takes to reach the typical state, implying that such typical states lie in this class\nof well-balanced configurations. Note that this deviates significantly from previous usages of layered\ninduction in this area: For example, in [TW14] the authors used layered induction to reduce the\ndiscrepancy from O(log n) to O(log log n). Our layered induction is “merely” used to establish that\na typical state (reached quickly due to the good recovery) has a low discrepancy. Apart from being\nflexible enough to deal with a rather general deletion model, our approach also allows for stronger\nprobability guarantees (of order 1 −1/poly(n) instead of 1 −1/polylog(n) in [TW14]).\nIn our layered induction we have to cope with two new problems in comparison with previous\nwork: The total load m(t) changes over time, and we may not assume it to be linear in n. The\nclassical layered induction proof of [ABKU99] heavily relies on the base case, which by means of a\nsimple pigeon-hole argument states that there are at most n/c bins with c balls. This holds because\nin their setting the total load cannot exceed n. In our case, however, the total load is unbounded. We\ndeal with this first problem by performing our layered induction only on the highest log log(n)+O(1)\nmany load levels. This leads us to the second challenge, since the changing total load m(t) also\nimplies that the (absolute) height of the considered levels changes over time. To deal with this,\nwe carefully formulate our layered induction relative to the maximum average and use a flexible\nrandom walk argument to prove strong bounds on the number of balls above a certain level over\nthe maximum average. This yields a bound on the overload measure, from which we derive the\ndiscrepancy bound – under the assumption that there was no large, recent drop of the total load.\nNote that in the case where such a large drop occurred, our lower bound mentioned in (a) shows\nthat we cannot hope for a good discrepancy. To the best of our knowledge, this is the first layered\ninduction for the case m ≫n as well as for a process with a changing load.\n1.2\nRelated Work.\nThere is a vast amount of literature on balls-into-bins games in a multitude of settings. For an\noverview see [Wie17] and for results with one random choice per ball see [BCN+15, BFK+18, LS23].\nAfter a very brief overview of results with multiple choices we focus on processes with deletions.\n4\n\nMultiple Choice.\nThe Greedy[d] process was introduced and analyzed in [ABKU99], although\na similar result in a different setting had been shown in [KLMadH92]. In [BCSV06] the authors\ngeneralize the results to the heavily loaded case m ≫n. [TW14] presents a simpler analysis for\nthe heavily loaded case, albeit showing a slightly worse upper bound on the maximum load. The\nauthor of [V¨03] considers the always-go-left process Left[d] where bins are divided into d clusters\nof n/d bins each. Every ball randomly chooses one bin per clusters and is allocated to a least\nloaded among the chosen bins; ties are broken in favor of the leftmost cluster. This results in a\nmaximum load of log log(n)/(d·ϕd) with ϕd ≤2 (ϕd ∈(1, 2) is the generalized golden ratio), w.h.p.,\na surprising improvement over [ABKU99]. For more recent results see [LSS22, LSS23, LSS24]. The\npotential function analysis for Part (a) is inspired by methods introduced in [PTW15] where the\nauthors consider a very interesting variant of Greedy[d], which they call (1 + β)-process. Here\neach ball is allocated using Greedy[1] with probability 1 −β, and using Greedy[2] with the\nremaining probability β. The authors showed that the difference between maximum and average\nload is Θ(log(n)/β) for each β < 1. Hence, as long as the number of allocations with Greedy[2] are\nsufficiently large, the load is rebalanced and the “damage” caused by Greedy[1] is undone. In this\npaper we use the same potential function and generalize the analysis to deletions (see Section 2).\nProcesses with Deletion, m ≤n.\nIn [ABKU99] the authors analyze the Greedy[d] allocation\nstrategy in an insertion/deletion model where, after n initial insertions, random deletions and in-\nsertions happen alternately. They show that for a polynomial number of steps the maximum load\nis, w.h.p., log log(n)/log(d) + O(1). The authors of [CFM+98] provide an alternative proof for the\nsame bound that can handle arbitrary insertion/deletion sequences, as long as the total system load\nis at most n. Under the same restriction they provide bounds for the reinsertion/deletion model\nwith adversarial deletions, showing that the maximum load is, w.h.p., O(log log(n)).\nProcesses with Deletions, m > n.\nThe infinite insertion/deletion model with adversarial dele-\ntions for the moderately loaded case is considered by [V¨03] for the Left[d] process where the\nmaximum system load is always bounded by h · n. He proves that the maximum (bin) load is\nln ln(n)/(d · ln(ϕd)) + O(h) w.h.p. (again, ϕd ∈(1, 2) is the generalized golden ratio). Note that for\nh = Ω(log(n)), this is not better than the bound achieved by the trivial Greedy[1] allocation.\nThe authors of [BCF+23] consider the moderately loaded case but for the reinsertion/deletion\nmodel.\nThey introduce the Iceberg[d] process, combining Greedy[1] and Left[d].\nSlightly\nsimplified, the Iceberg[d] process allocates a ball via Greedy[1] if the target bin contains at most\nh + ˜O(\n√\nh) balls that were allocated by Greedy[1]. Otherwise, either Left[d] is used (if there are\ncurrently fewer than n/d balls allocated by Left[d]) or the ball is allocated to bin 1. This process\nachieves a maximum load of h + log log(n)/(d log ϕd)) + ˜O(\n√\nh). Note that this process must know\nh and keep track of the number of balls placed with specific strategies, making it difficult to use in\ndistributed settings. In contrast, Greedy[d] places balls solely based on the selected bins’ loads.\nThe focus of [Czu00] is the analysis of the recovery time (the time it takes a process to move from\nan arbitrary to a “typical” state) in the heavily loaded case (m arbitrary). The author considers a\nsetting starting with m arbitrarily placed balls followed by alternating deletions (either a random\nball or from a random bin) and insertions. The recovery time under Greedy[d] allocation is proven\nto be polynomial in m and n. While parts of our analysis (Section 4) are inspired by this result,\nnote that in our case the system load m fluctuates over time and that we do not only show fast\nrecovery but actually characterize what the “typical” state looks like.\nAs mentioned at the beginning of the introduction, [BK22] give lower bounds for Greedy[2] in\nthe heavily loaded, adversarial insertion/deletion model. They also provide a more general lower\n5\n\nbound for any oblivious allocation strategy (like Greedy[2] or Left[2]) in the reinsertion/deletion\nmodel (with the system load capped at m): an oblivious adversary can force a maximum load of\nm/4 + mΩ(1) for n = 4 bins within the first poly(m) steps. On the positive side, they introduce\nthe allocation strategy ModulatedGreedy for the insertion/deletion model, which achieves an\noverload of O(log(m)). To overcome their lower bound for Greedy[2], ModulatedGreedy de-\ncides probabilistically between two bin candidates, based on their current load. The algorithm must\nknow global guarantees on the load situation (in particular, the maximum number of balls that will\never be present and the current maximum and minimum load).\nQueuing Systems.\nA prominent example of a queuing system is the supermarket model consid-\nered in, e.g., [VDK96, BCFV00, Mit01, LN05, LM05]. It can be seen as a variant of balls-into-bins\nprocess in continuous time. Here, balls typically arrive in a Poisson stream of rate λn and are\nallocated according to Greedy[d]. Bins process (delete) balls in first-in first-out order, with a\nprocessing time that is exponentially distributed with mean 1. These systems and their results can\noften be readily mapped to the balls-into-bins settings yielding results similar to those discussed\nabove. However, the focus in queuing systems lies typically on properties like the balls’ service times\n(the average time a ball spends in the system), how long it takes to reach an equilibrium state, or\nhow the system behaves in the equilibrium. Note that the (expected) unit processing time together\nwith λ < 1 means that we are dealing here with the lightly/moderately loaded case (m ≤poly(n))\nin the equilibrium. Later results (e.g., [BLP10, MBvLW16, BFL18]) considered settings where d\nand/or λ might depend on n or where the balls’ service time adhere to more general distributions.\nFurther Related Work in Load Balancing.\nThere are also results such as [ABS98, BFK+18]\nthat consider parallel load balancing settings, where balls are not sequentially allocated but thrown\nin batches of size Θ(n) and where, similar to the supermarket model, bins have a processing time\n(in these cases simply 1). One is interested in which batch sizes and allocation strategies yield small\nwaiting times. [ABS98] shows for Greedy[2] that, if the batches are not too large, the maximum\nwaiting time is, w.h.p., at most ln ln(n)/ ln(d) + O(1). For a batch size of λn, [BFK+18] prove a\nmaximum waiting time of O(\n1\n1−λ · log\nn\n1−λ) for Greedy[1] and O(log\nn\n1−λ) for Greedy[2].\nAnother line of work considers a mixture of load balancing and averaging processes on general\ngraphs (in contrast to the balls-into-bins settings, that can be thought of as load balancing on a\nclique of n nodes). Two recent results are [ANS22, BHH+23]. The major difficulty here typically\nstems from dealing with the given graph’s structure, which changes the considered processes and\nanalysis techniques considerably compared to our setting.\n2\nMax Load & Above-average Ball Count via Potential Functions.\nIn this section we extend the analysis of [PTW15] to allow for ball deletions (with variable deletion\nprobabilities). The following Theorem 2.1 uses the potential function Γα(x(t)) from [PTW15]. We\nshow that, at an arbitrary point of time, the expected potential is linear in n. The potential is\ndefined as Γα(x(t)) = Φα(x(t)) + Ψα(x(t)), where\nΦα(x(t)) =\nn\nX\ni=1\neα·(xi(t)−m(t)/n),\nand\nΨα(x(t)) =\nn\nX\ni=1\ne−α·(xi(t)−m(t)/n).\n(1)\nThe potential Φα(x(t)) quantifies the impact of bins with loads above the average while Ψα(x(t))\ndoes the same for the bins below the average. Although bounds on the maximum load are derived\n6\n\nonly from Φα(x(t)), both parts are needed to ensure a potential drop in every load situation where\nthe potential is large. Theorem 2.1, whose proof can be found in Appendix B, demonstrates that the\npotential function is robust enough to tolerate deletions if the β(t) are lower-bounded by a constant\nβ > 0. It states that the absolute discrepancy at any time t is, w.h.p., at most O(log n) higher than\nit was at the start. In particular, if the process starts with an absolute discrepancy of O(log n), it\nwill almost certainly be O(log n) at time t.\nTheorem 2.1. Consider an insertion probability sequence\n\u0000β(t)\n\u0001\nt∈N bounded away from 0 by a\nconstant β > 0.\nIf α ≤β/16, then for any t ∈N0 we have E[Γα(x(t))] = O(n) · eα·adisc(0).\nMoreover, for any a ≥0 we have Pr\n\u0002\nadisc(t) −adisc(0) ≥a+2\nα · ln n\n\u0003\n≤n−a.\nIn particular, if we start in a perfectly balanced situation with discrepancy 0 (e.g., empty system),\nthe expected potential at an arbitrary time t is only linear. Equipped with the above, we can prove\na strong upper bound on the number of balls that lie above the average at a given time t:\nTheorem 2.2. Fix an arbitrary step t. Let β ∈(0, 1] be a constant. Assume β(τ) ≥β for all\n1 ≤τ ≤t and α ≤β/16. For any fixed non-negative ℓand any fixed µ ∈(0, 1), there is a γ = γ(µ)\nso that, with probability at least 1 −n−(3+ℓ), the number of balls above height ⌈m(t)\nn ⌉+ γ is at most\nµn.\nThe proof is based on Lemma C.1 in Appendix C, an adaptation of a result in [LS22, LS21],\nwhich gives us a high-probability guarantee for the potential Γα(x(t)) being linear. In fact, the\npurpose of the adaptation of [LS22, LS21]’s results is two-fold: (i) to incorporate our deletions and\ntheir respective probability bounds, and (ii) to improve the high-probability guarantee from the\noriginal 1 −1/n3 (which was sufficient for the purposes of [LS22, LS21]). Note that the probability\nstated in Theorem 2.2 is inherited from Lemma C.1; apart from that, the theorem’s actual proof is\nan entirely deterministic counting argument. The proof details may be found in Appendix C.\n3\nMaintaining a Well-balanced State.\nBefore we state this section’s main result (Theorem 3.1), we introduce the notion of c-good time\nintervals I for a given insertion probability sequence. Intuitively, such an interval guarantees us\nthat the total system load increases in expectation in any sufficiently long subinterval of I.\nDefinition 3.1 (c-good interval). Given a sequence β(t) (t ∈N) of insertion probabilities, we call\na discrete time interval I c-good if for each subinterval (t1, t2] ⊆I with t2 −t1 ≥c · n we have\n\u0000Pt2\nt=t1+1 β(t)\n\u0001\n/(t2 −t1) ≥1\n2(1 + ϵ) for a constant ϵ > 0.\nTheorem 3.1 states that, w.h.p., our process, when started in a perfectly balanced configuration,\nremains well-balanced for at least n4 steps.3 Here, well-balanced (this is formalized later in Defini-\ntion 3.2 and Lemma 3.4) is measured relative to the maximum average observed so far for general\ninsertion probability sequences and relative to the current average if the considered time window is\nO(1)-good.\nTheorem 3.1. Consider an insertion probability sequence\n\u0000β(t)\n\u0001\nt∈N bounded away from 0 and 1.\nLet x(0) be a perfectly balanced configuration. Then, w.h.p., xmax(t) ≤mmax(t)/n+log log n+O(1)\nfor any t ≤n4. Moreover, if there is a constant c ∈N such that the time interval [0, n4] is c-good,\nthen, w.h.p., xmax(t) ≤m(t)/n + log log n + O(1) for any t ≤n4.\n3In fact, our proof shows that starting from any well-balanced configuration, the situation remains well-balanced\nfor polynomial many steps.\n7\n\nTo prove Theorem 3.1, we conduct a layered induction.\nA major difficulty stems from the\nfact that we have to deal with a permanently changing average load. For this an important in-\ngredient is Theorem 2.2, which gives us a high probability bound on the number of balls above\nthe current average.\nThis enables us to derive a base case for our layered induction, indepen-\ndent of the changing average load. The layered induction works roughly as follows: For each level\nℓ∈{ 0, 1, . . . , log log(n) + Θ(1) } we consider the number of balls at height greater than average\n+ℓ. For each level we define a critical threshold αℓthat decreases doubly exponentially with ℓ.\nMoreover, each level has a safe threshold αℓ/2. We show that, as long as all levels remain below\ntheir critical threshold, any level that is above its safe threshold has a strong drift back towards the\nsafe threshold. A standard result on random walks then shows that it is unlikely that during the\npoly(n) many considered time steps, any of the log log(n) + Θ(1) many levels (that all start below\ntheir respective safe threshold) crosses the critical interval [αℓ/2, αℓ).\nFormalizing Well-balanced Configurations.\nIn the following, let ˆβ ∈(0, 1) be a constant\nupper bound on the insertion probability of our process, such that β(t) ≤ˆβ for all considered time\nsteps t. For a height h ∈N0 and a configuration x define mh(x) as the number of balls that have\nheight at least h in configuration x. For our process starting in an initial configuration x(0), we\nuse the shorthand mh(t) := mh(x(t)). In our analysis, it will be helpful to measure the number of\nballs at or above some level ℓ, which, in turn, is above a given base height h (which might vary over\ntime). To this end, we introduce the notation m(h)\nℓ(x) := mh+ℓ(x) and m(h)\nℓ(t) := mh+ℓ(t).\nWe consider ℓ∗+ 2 = log log(n) + Θ(1) many levels ℓ∈{ 0, 1, . . . , ℓ∗+ 1 } with their critical\nthresholds αℓdefined as follows:\nαℓ:=\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1−ˆβ\n128ˆβ · n\nif ℓ= 0,\n32ˆβ\n1−ˆβ · α2\nℓ−1/n\nif αℓ−1 >\nr\n3(1−ˆβ)\n2ˆβ\n· n log(n),\n12 log(n)\nif ℓ= ℓ∗:= min\n\u001a\nℓ\n\f\f\f\f αℓ−1 ≤\nr\n3(1−ˆβ)\n2ˆβ\n· n log(n)\n\u001b\n, and\n24\nif ℓ= ℓ∗+ 1.\n(2)\nNote that the recursive definition ensures ℓ∗= log log(n) + Θ(1). Moreover, it implies the following\nrelations between the critical thresholds αℓand αℓ−1:\nObservation 3.1. For any ℓ∈{ 1, 2, . . . , ℓ∗+ 1 } and large enough n we have\n8ˆβ\n1 −ˆβ\n· α2\nℓ−1\nn\n≤αℓ≤1\n4 · αℓ−1.\n(3)\nUsing the critical threshold αℓ, we define the valid interval of level ℓ∈{ 0, 1, . . . , ℓ∗+ 1 } as\nVℓ:= [0, αℓ). Similarly, we define the level’s safe interval as Sℓ:= [0, αℓ/2) and its critical interval\nas Cℓ:= [αℓ/2, αℓ).\nDefinition 3.2. For a base height h ∈N0 and level ℓ∈{ 0, 1, . . . , ℓ∗+ 1 } we say level ℓof con-\nfiguration x is h-valid if m(h)\nℓ(x) ∈Vℓ, h-safe if m(h)\nℓ(x) ∈Sℓ, and h-critical if m(h)\nℓ(x) ∈Cℓ.\nConfiguration x is h-valid/ -safe/ -critical if all levels 0 to ℓ∗+ 1 are h-valid/-safe/-critical.\nIn proofs, we sometimes omit the base height if it is clear from the context.\n8\n\nAnalysis.\nWe start with a simple lemma that provides the probabilities for the increase and\ndecrease of the number of balls at or above a given height h.\nLemma 3.1. For any configuration x and any height h ∈N we have\n(a) Pr [ mh(t + 1) > mh(t) | x(t) = x ] = β(t) ·\n\u0000mh−1(t) −mh(t)\n\u00012/n2 and\n(b) Pr [ mh(t + 1) < mh(t) | x(t) = x ] ≥\n\u00001 −β(t)\n\u0001\n·\n\u0000mh(t) −mh+1(t)\n\u0001\n/n.\nProof.\n(a) For mh(t) to increase during the next step, a ball must be inserted by Greedy[2]\n(probability β(t)) and both of the choices of Greedy[2] must fall into a bin that has load\nat least h −1. The number of such bins is exactly mh−1(t) −mh(t), yielding the probability\n\u0000mh−1(t) −mh(t)\n\u00012/n2.\n(b) For mh(t) to decrease during the next step, a ball must be deleted (probability 1 −β(t)), and\nthe chosen bin must have load at least h. The number of such bins is exactly mh(t)−mh+1(t)\nand the total number of non-empty bins is at most n, yielding the probability ≥\n\u00001 −β(t)\n\u0001\n·\n\u0000mh(t) −mh+1(t)\n\u0001\n/n.\nLemma 3.2. Consider an h-valid configuration x and let x′ denote the (random) configuration\nafter one step of our process with the insertion probability β(t) ≤ˆβ < 1.\n(a) If level ℓ∈{ 1, 2, . . . , ℓ∗} of configuration x is h-critical, then\nPr\nh\nm(h)\nℓ(x′) < m(h)\nℓ(x)\ni .\nPr\nh\nm(h)\nℓ(x′) > m(h)\nℓ(x)\ni\n≥2.\n(4)\n(b) If level ℓ= ℓ∗+ 1 of configuration x is h-critical and if n is large enough, then\nPr\nh\nm(h)\nℓ(x′) < m(h)\nℓ(x)\ni .\nPr\nh\nm(h)\nℓ(x′) > m(h)\nℓ(x)\ni\n≥√n.\n(5)\nThe proof of Lemma 3.2 may be found in Appendix D.\nNext we formulate an auxiliary lemma about the probability that a random walk on integers\nreaches position b > 0 before position −a < 0 when started in 0, assuming the random walk is\nbiased towards the left by a factor r > 0. We will apply this lemma to the values m(h)\nℓ(•) (basically\ncorresponding to the random walks) to show that these values are unlikely to cross the critical\ninterval Cℓ(on which, by Lemma 3.2, m(h)\nℓ(•) is biased by a factor of at least 2 towards the safe\nleft side).\nLemma 3.3. Consider a (not necessarily i.i.d.) random walk (St)t∈N0 on Z with initial position\nS0 = 0 and position St = Pt\ni=1 Xi after t steps with step width Xi ∈{ −1, 0, +1 } at time i. Let\nF = (Ft)t∈N0 denote the random walk’s natural filtration. Assume there is r ∈R>0 \\ { 1 } such that\nfor all t ∈N, Pr [ Xt = −1 | Ft−1 ]\n\u000e\nPr [ Xt = +1 | Ft−1 ] = r. For x ∈Z define the stopping time\nτx := inf { t ∈N0 | St = x } indicating when the random walk reaches x for the first time. Then, for\nany a, b > 0, Pr [ τ+b < τ−a ] = (ra −1)/(ra+b −1).\nProof. Consider martingale M0 := 1 and Mi := rXi · Mi−1.\nDefine the stopping time τ :=\nmin { τ−a, τb }. By the optional stopping theorem, we have E[Mτ] = E[M0] = 1. This yields 1 =\nE[M0] = E[Mτ] = Pr [ τ+b < τ−a ] · rb +\n\u00001 −Pr [ τ+b < τ−a ]\n\u0001\n· r−a. Rearranging for Pr [ τ+b < τ−a ]\ngives the desired result.\n9\n\nThe next lemma is our main tool in proving Theorem 3.1. It states that if started in a safe\nconfiguration x, our process maintains safety (with respect to the base height h(t) := ⌈mmax(t)/n⌉+\nO(1)) for a polynomial time. The basic idea is to consider any time t′ in which some previously\nsafe level ℓjust became critical. Using Lemma 3.2, we then couple the random variable mℓ(•) with\na random walk that is always at least as large as mℓ(•) and is biased by a factor of exactly 2 (or\n√n, in the case of level ℓ= ℓ∗+ 1) towards the safe left side of the critical interval Cℓ. Applying\nLemma 3.3 to all possible such random walks together with a union bound then implies that, w.h.p.,\nno level crosses the critical interval and becomes invalid for poly(n) many steps.\nLemma 3.4. Let h(t) := ⌈mmax(t)/n⌉+ γ denote the base height at time t. Consider an initial\nconfiguration x(0) = x that is h(0)-safe. Fix the time T = n4 and assume n to be large enough. If\nβ(t) ≤ˆβ < 1 for all t ∈{ 0, 1, . . . , T −1 }, then\nPr [ ∀t ∈{ 0, 1, . . . , T −1 } : x(t) is h(t)-valid ] ≥1 −n−1.\n(6)\nThe proof of the lemma may be found in Appendix D. It remains to prove Theorem 3.1, which\nfollows by applying Lemma 3.4 and observing that any valid configuration has the desired maximum\nload.\nProof of Theorem 3.1. Define the base height h(t) := ⌈mmax(t)/n⌉+ γ at time t as in Lemma 3.4.\nConsider an initial configuration x = x(0) that is perfectly balanced (i.e., a configuration with\nmaximum load ⌈mmax(t)/n⌉). Note that x is trivially h(0)-safe, since (choosing γ ≥1) there are no\nballs with height ≥h(0).\nThen, by Lemma 3.4, all configurations x(t) with t ∈[0, n4) are (w.h.p.) h(t)-valid. Thus, it is\nsufficient to show that the maximum load in any h-valid configuration is h + log log n + O(1). This\nfollows immediately from the definition of the critical thresholds: Indeed, if x is h-valid, so is level\nℓ∗+ 1. Thus, there are at most αℓ∗+1 = O(1) many balls on or above level ℓ∗+ 1. In particular, the\nmaximum load in x is xmax ≤h + ℓ∗+ 1 + αℓ∗+1 = h + log log n + O(1). This proves the first part\nof the theorem.\nFor the second part, assume that the insertion probability in the interval [0, n4] is c-good for\na constant c. That is, for any discrete subinterval (t1, t2] ⊆[0, n4] of length t2 −t1 ≥c · n we\nhave (Pt2\nt=t1+1 β(t))/(t2 −t1) ≥1\n2 · (1 + ϵ) for a constant ϵ > 0. Consider the random variable\nY = Pt2\nt=t1+1 Yt where Yt = 1 with probability β(t) and Yt = 0 otherwise. Note that Y counts the\nnumber of balls added in the time interval (t1, t2]. By c-goodness we have E[Y ] ≥(1 + ϵ) · t2−t1\n2\n. A\nsimple application of Chernoff yields\nPr\n\u0014\nY < (1 + ϵ/2) · t2 −t1\n2\n\u0015\n≤e−Θ(1)·(t2−t1) = n−ω(1).\n(7)\nNote that if Y balls are added during the interval (t1, t2], at most t2 −t1 −Y balls are deleted.\nThus, using Equation (7), the total change of the system load during the interval (t1, t2] is, with\nvery high probability,\nm(t2) −m(t1) ≥Y −(t2 −t1 −Y ) = 2Y −(t2 −t1) ≥ϵ · t2 −t1\n2\n> 0.\n(8)\nThus, we can take a union bound over the poly(n) many subintervals (t1, t2] ⊆[0, n4] of length at\nleast c·n to get that, w.h.p., the average load at the end of the subinterval is at least the average load\nat its beginning. All other subintervals have length < c·n, so the average load can decrease by at most\nc during any of them. Together, we get that, w.h.p., for any t ∈[0, n4] we have m(t) ≥mmax(t) −c.\nThus, by using the theorem’s first statement we can, w.h.p., bound the maximum load at any\ntime t ∈[0, n4] by xmax(t) ≤⌈mmax(t)/n⌉+ log log n + O(1) ≤⌈m(t)/n⌉+ log log n + O(1) + c =\n⌈m(t)/n⌉+ log log n + O(1).\n10\n\n4\nQuick Recovery & Steady State Characterization.\nWith the tools from Sections 2 and 3 at hand, we are now ready to prove our main result (Theo-\nrem 4.1). It generalizes the guarantees given by Theorem 3.1 (there only for the first poly(n) many\nsteps) to an arbitrary time t ∈N0.\nTheorem 4.1. Consider an insertion probability sequence\n\u0000β(t)\n\u0001\nt∈N bounded away from 0 and 1.\nThen, w.h.p., xmax(t) ≤mmax(t)/n+log log n+O(1) for any t ∈N0. Moreover, if there is a constant\nc ∈N such that time interval [t −n4, t] is c-good, then, w.h.p., xmax(t) ≤m(t)/n + log log n + O(1).\nWe conjecture that the requirement for a constant ϵ > 0 in Definition 3.1 is an artifact of\nour analysis technique and can be reduced to o(1) or even eliminated. The same holds for our\nassumption that the insertion probability must be bounded away from 1 (note that for β = 1 the\nresult follows from [BCSV06]). We note that the first of the two results in the theorem also holds\nwhen first m ≥n balls are inserted and the process then alternates between insertions and deletions.\nAnalysis Overview.\nThe proof idea of Theorem 4.1 is as follows: For any time t ∈N0, The-\norem 2.1 implies (w.h.p.) a logarithmic absolute discrepancy at time t′ := t −n4. We then use\na path coupling argument similar to [Czu00] to show that our process causes any load situation\nwith logarithmic absolute discrepancy to recover quickly to a “typical” state. More exactly, any two\n(different) such load situations at time t′ can be coupled such that (w.h.p.) they become identical\nat time t. At this point however, we don’t know how such a “typical” looks like (and whether it\nhas small maximal load). This is where Theorem 3.1 comes into play. It tells us that if we start in\na perfectly balanced load situation at time t′, our process maintains (w.h.p.) a double-logarithmic\nmaximal load for at least n4 steps. A simple union bound over these two high probability bounds\nthen implies that the “typical” load situation at time t must also have a double-logarithmic maximal\nload.\nAnalysis.\nWe now formalize the above idea. To this end, we first introduce a measure for the\nsimilarity of two given load situations.\nDefinition 4.1. Consider two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 =\n∥y∥1. The transformation distance between x and y is ∆(x, y) := ∥x −y∥1/2.\nNote that the transformation distance obeys the identity ∆(x, y) = Pn\ni=1 max { 0, xi −yi } and\ncorresponds to the minimal number of ball movements to transform x into y (and vice versa).\nMoreover, any load vector x can be transformed into the perfectly balanced load vector by moving\nthe at most n · disc(x) ≤n · adisc(x) balls above average. As an immediate consequence, we get the\nfollowing observation.\nObservation 4.1. For two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 = ∥y∥1,\nwe have ∆(x, y) ≤2n · max { adisc(x), adisc(y) }.\nThe following lemma leverages a result from [Czu00] to show that one step of our process\ntends to reduce transformation distance between two given load vectors. In [Czu00], the author\nconsiders basically the same setting as ours, but with guaranteed alternating insertions/deletions.\nLemma 4.1 basically shows that their proof transfers directly to our more general setting with\ninsertion probability sequences.\n11\n\nLemma 4.1. Consider two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 = ∥y∥1.\nAssume ∆(x, y) = 1 and let x′ and y′ denote the load vectors after applying one step of our process\nwith insertion probability β ∈[0, 1] to x and y, respectively. There is a coupling of x′ and y′ such\nthat\n(a) E[∆(x′, y′)] ≤∆(x, y) = 1 and\n(b) Pr [ ∆(x′, y′) ̸= ∆(x, y) ] ≥(1 −β)/n.\nProof. For the random decision whether to insert or delete a ball in x and y, we use the identity\ncoupling that either inserts a ball in both processes (with probability β(t)) or deletes a ball in both\nprocesses (with probability 1−β(t)). It remains to couple the random bin choices in a deletion step\nand the two random bin choices of Greedy[2] in an insertion step.\nAssume the considered step is a deletion step.\nThen we use the first coupling described\nin [Czu00, Section 6] (who consider alternating insertions/deletions instead of our more flexi-\nble insertion probability sequences).\nIn this case, [Czu00, Claims 6.1 and 6.2] together yield\nE[∆(x′, y′) | Deletion Step] ≤∆(x, y) = 1 and Pr [ ∆(x′, y′) ̸= ∆(x, y)|Deletion Step ] ≥1/n.\nNote that this already yields the lemma’s second statement, since a deletion step occurs with prob-\nability 1 −β.\nSimilarly, for an insertion step, we use the second coupling described in [Czu00, Section 6], where\nthe exact same insertion rule is used (again, in a setting with alternating insertions/deletions). In\nthis case, the proof of [Czu00, Lemma 5.1] implies E[∆(x′, y′) | Insertion Step] ≤∆(x, y) = 1.\nTogether, we get the desired result.\nThe next lemma characterizes the expected and high probability crossing time of a simple, α-\nlazy reflecting random walk on { 0, 1, . . . , D } (i.e., a random walk that has the same probability to\nmove left/right and that stays put with probability α or if it tries to move out of bounds). We use\nthe lemma for our coupling result in Lemma 4.3.\nLemma 4.2. Let D ∈N0 and α ∈[0, 1). Consider a simple, α-lazy reflecting random walk (Wt)t∈N0\non { 0, 1, . . . , D } started in W0 = D. Let T := min { t ∈N | Wt = 0 }. Then E[T] = D·(D+1)/(1−\nα). Moreover, for any n ∈N and a > 0 we have Pr [ T ≥a · log(n) · 2E[T] ] ≤1/n−a.\nProof. Let Ei denote the expected time for the random walk to reach 0 if started at position\ni ∈{ 0, 1, . . . , D }. Clearly, E0 = 0 and ED = E[T]. Note that we have the recurrence relation\nEi = 1+Ei ·α+(Ei−1 +Ei+1)·(1−α)/2 for i ∈{ 1, 2, . . . , D −1 } and ED = 1+ED−1 ·(1−α)/2+\nED ·(1+α)/2. From this one can first deduce Ei = 2(D−i+1)/(1−α)+Ei−1 for i ∈{ 1, 2, . . . , D }\nand, then, E[T] = ED = D · (D + 1)/(1 −α).\nFor the high probability bound, we observe that Markov’s inequality implies Pr [ T ≥2E[T] ] ≤\n1/2. Thus, after at most a · log(n) repetitions of such phases of length 2E[T], the random walk has\nreached 0 with probability at least (1/2)a·log(n) = n−a.\nThe following lemma shows that it is possible to couple two instances of our process with a\nlogarithmic absolute discrepancy; they will be in the same distribution after polynomial time.\nLemma 4.3. Consider two instances of our random processes\n\u0000x(t)\n\u0001\nt∈N0 and\n\u0000y(t)\n\u0001\nt∈N0 with\n∥x(0)∥1 = ∥y(0)∥1 and adisc(x(0)), adisc(y(0)) ≤3\nα · log n. Assume both processes use the same\ninsertion probability sequence\n\u0000β(t)\n\u0001\nt∈N0 bounded away from 0 and 1.\nThen there is a coupling\nof\n\u0000x(t)\n\u0001\nt∈N0 and\n\u0000y(t)\n\u0001\nt∈N0 for whose coupling time τ := min\n\b\nt ∈N0\n\f\f ∆\n\u0000x(t), y(t)\n\u0001\n= 0\n\t\nwe\nhave, w.h.p., τ = O\n\u0000n3 · (log n)3\u0001\n.\n12\n\nProof. Consider any time t ∈N0 and define ∆(t) := ∆\n\u0000x(t), y(t)\n\u0001\n. There is a sequence x(t) =\nx(0)(t), x(1)(t), . . . , x(∆(t))(t) = y(t) of load vectors such that for all indices i ∈{ 1, 2, . . . , ∆(t) }\nwe have ∆(i)(t) := ∆\n\u0000x(i−1)(t), x(i)(t)\n\u0001\n= 1.\nBy Lemma 4.1(a), there is a coupling of each pair\n\u0000x(i−1)(t + 1), x(i)(t + 1)\n\u0001\nsuch that\nE\n\u0002\n∆(i)(t + 1)\n\u0003\n≤1. This implies a coupling of x(t + 1) and y(t + 1) with\nE\n\u0002\n∆(t + 1)\n\u0003\n≤E\n\n\n∆(t)\nX\ni=1\n∆(i)(t + 1)\n\n=\n∆(t)\nX\ni=1\nE\nh\n∆(i)(t + 1)\ni\n≤∆(t).\n(9)\nSimilarly, Lemma 4.1(b) implies\nPr [ ∆(t + 1) ̸= ∆(t) | ∆(t) > 0 ] ≥1 −β(t)\nn\n.\n(10)\nConsider the random process\n\u0000∆(t)\n\u0001\nt∈N0. Let D := n · 10\nα · log n = O(n · log n) and define the\nstopping time\nT1 := min { t ∈N0 | ∆(t) > D } .\n(11)\nLet β := sup { β(t) | t ∈N0 } < 1 and α := 1 −(1 −β)/n. Because of Equations (9) and (10),\nduring the time steps t ∈{ 0, 1, . . . , T1 −1 } the distance ∆(t) is majorized by the position W(t) of\na simple, α-lazy reflecting random walk\n\u0000W(t)\n\u0001\nt∈N0 on { 0, 1, . . . , D } starting in W(0) = D. Define\nthe stopping time\nT2 := min { t ∈N0 | W(t) = 0 } .\n(12)\nNote that if T1 > T2, the majorization implies ∆(T2) ≤W(T2) = 0 and, thus, τ ≤T2.\nBy applying Theorem 2.1 with a = 5 to the first n4 time steps and using a union bound,\nw.h.p. both\n\u0000x(t)\n\u0001\nt∈N0 and\n\u0000y(t)\n\u0001\nt∈N0 maintain an absolute discrepancy of at most adisc(0) + 7\nα ·\nlog n = 10\nα ·log n during the first n4 steps. By Observation 4.1, this implies a transformation distance\nof at most n· 10\nα ·log n = D and, thus, T1 > n4. By Lemma 4.2, w.h.p., T2 = O\n\u0000n3 ·(log n)3\u0001\n. Thus,\nby a union bound we have, w.h.p., T1 ≥T2 and T2 = O\n\u0000n3 · (log n)3\u0001\n. As argued above, together\nthese imply τ ≤T2 = O\n\u0000n3 · (log n)3\u0001\n.\nWith the coupling at hand we are ready to prove our main result.\nProof of Theorem 4.1. Note that for t ≤n4, the statement follows immediately from Theorem 3.1.\nFor larger t, Theorem 2.1 gives us, w.h.p., xmax(t −n4) −xmin(t −n4) ≤2c · log n. By this and\nby shifting each time step forward by n4 steps, it is sufficient to prove the theorem’s statement for\nt = n4 starting from an initial load vector x(0) with xmax(0) −xmin(0) ≤2c · log n.\nDefine an initial load vector y(0) with total load ∥y(0)∥1 = m(0) that is perfectly balanced.\nBy Lemma 4.3 there is a coupling of\n\u0000x(t)\n\u0001\nt∈N0 and\n\u0000y(t)\n\u0001\nt∈N0 such that, w.h.p., x(t) = y(t) for all\nt = ˜Ω(n3) (so, in particular, for t = n4). Note that Theorem 3.1 applies to\n\u0000y(t)\n\u0001\nt∈N0. This gives\nus the corresponding high probability guarantees from Theorem 3.1 for y(n4) which, by leveraging\nthe coupling, hold w.h.p. for x(n4), finishing the proof.\nAcknowledgments.\nPetra Berenbrink’s research was funded by the DFG project Distributed and Collaborative Systems\nof Agents (project number 411362735) as part of the Research Unit (Forschungsgruppe) Algorithms,\nDynamics, and Information Flow in Networks (FOR 2975).\n13\n\nReferences\n[ABKU99]\nYossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations.\nSIAM Journal on Computing, 29(1):180–200, 1999.\n[ABS98]\nMicah Adler, Petra Berenbrink, and Klaus Schröder. Analyzing an infinite parallel\njob allocation process. In In Proceedings of the European Symposium (ESA), pages\n417–428, 1998.\n[ANS22]\nDan Alistarh, Giorgi Nadiradze, and Amirmojtaba Sabour.\nDynamic averaging\nload balancing on cycles. Algorithmica, 84(4):1007–1029, 2022.\n[BCF+23]\nMichael A. 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Similarly,\nlet Y = (y(t))t∈N0 be the process in which every ball is deleted with the same probability during a\ndeletion step.\nOur goal is to show that process Y is not worse (w.r.t. discrepancy/overload) than process X. We\nformalize this using the standard notion of majorization: For a configuration x let Sk(x) = Pk\ni=1 xi\nbe the total load of the k fullest bins.\nWe say a configuration x majorizes configuration y if\nSk(y) ≤Sk(x) for all k ∈{ 1, 2, . . . , n }. Note that this immediately implies that y has a smaller\nmaximal load than x. Thus, we can formalize the desired result via the following theorem:\nTheorem A.1. Starting from an empty system, there exists a coupling between processes X and Y\nsuch that x(t) majorizes y(t) for all t ∈N0.\nCoupling.\nLet us define a suitable coupling between processes X and Y . To this end, we use the\nidentity coupling for the decision whether a ball is allocated (with probability β(t)) or deleted (with\nprobability 1 −β(t)). As a result, the total load in both processes remains the same at any time;\nwe denote it by m(t). It remains to couple the random choices for allocation and for deletion steps\nfrom configuration x(t) to configuration x(t + 1):\nAllocation Step: Here we use the identity coupling. That is, both processes use the same two\nrandom bin choices for Greedy[2].\nDeletion Step: Here we number the m(t) balls in both configurations x(t) and y(t) from left to\nright (and arbitrarily within each bin). We then draw a uniformly random number z(t) from\n[0, 1) and use it to determine the ball deletions as follows:\n(a) Y deletes the ball i ∈{ 1, . . . , m(t) } if and only if\ni −1\nm(t) ≤z(t) <\ni\nm(t).\n(13)\n(b) X deletes the ball j ∈{ 1, . . . , m(t) }, where j = Pℓ−1\na=1 xa(t) + k (i.e., the k-th ball in bin\nℓ), if and only if\nℓ−1\nˆn(t) +\nk −1\nˆn(t) · xℓ(t) ≤z(t) < ℓ−1\nˆn(t) +\nk\nˆn(t) · xℓ(t),\n(14)\nwhere ˆn(t) denotes the number of non-empty bins in configuration x(t).\nNote that our coupling of the deletions maintains the marginal distributions of X and Y . Indeed,\nprocess Y deletes ball i with probability exactly 1/m. Similarly, process X deletes ball j with\nprobability exactly 1/(ˆn(t) · xℓ(t)) (a uniformly random ball chosen from a uniformly random non-\nempty bin).\n17\n\nAnalysis.\nIn the prove of Theorem A.1, we make use of the following simple observation:\nObservation A.1. Consider the above coupling for processes X and Y during a deletion step t+1.\nAssume that y(t) is majorized by x(t). Let r and s denote the bins from which X and Y delete\nduring step t + 1, respectively. Similarly, let j and i denote the number of the balls deleted by X\nand Y during step t + 1, respectively. Then\nSs−1(y(t)) < i ≤j ≤Sr(x(t))\n(15)\nThe first inequality follows by our ball numbering scheme (from left to right), since i ≤Ss−1(y(t))\nwould imply that ball i lies in a bin < s. Similarly, the last inequality follows since j > Sr(x(t))\nwould imply that ball j lies in a bin > r. The inequality i ≤j is a simple consequence of our coupling,\nsince the probabilities for ball i being deleted in Y are constant (w.r.t. i) while the probabilities for\nball j being deleted in X are monotonously non-decreasing (w.r.t. j).\nWith this, we are ready to prove the main result of this section.\nProof of Theorem A.1. We use the coupling defined above and prove the statement via induction\nover t, with the trivial base case Sk(x(0)) = 0 = Sk(y(0)) for all k. For the inductive step we show\nthat the coupling maintains majorization when going from x(t) to x(t + 1) during time step t + 1.\nIf a ball is inserted during time step t + 1, this follows immediately from [ABKU99, Lemma 3.4].\nThus, in the rest of this proof we consider a deletion during time step t + 1.\nMoreover, as in\nObservation A.1, we use (a) r and s to denote the bins from which X and Y delete a ball in step t,\nrespectively; and (b) j and i to denote the balls deleted by X and Y , respectively.\nA plateau of a configuration x is a maximal set of consecutive bins having the same load. Note\nthat when going from x(t) to x(t+1) by deleting a ball from bin r, we have to re-sort the bins after\nthe deletion. This will swap bin r with the rightmost bin r′ of the plateau in which r lies. In other\nwords, we can think of X as deleting a ball from some bin r′ ≥r instead of bin r. Formally we get\nx(t + 1) = x(t) −er′, where ei denotes the i-th unit vector in Rn. Similarly, Y deletes a ball from\nthe rightmost bin s′ ≥s in the plateau containing bin s, yielding y(t + 1) = y(t) −es′\nSo assume x(t) majorizes y(t). That is, Sk(y(t)) ≤Sk(x(t)) for all k ∈{ 1, 2, . . . , n }. We show\nby case distinction that Sk(y(t + 1)) ≤Sk(x(t + 1)) for all k ∈{ 1, 2, . . . , n }:\nCase 1: k < min { r′, s′ } Then the first k bins remain untouched by the deletion in both processes,\nyielding\nSk(y(t + 1)) = Sk(y(t)) ≤Sk(x(t)) = Sk(x(t + 1)).\n(16)\nCase 2: k ≥max { r′, s′ } Then the first k bins both loose exactly one ball in both processes, yield-\ning\nSk(y(t + 1)) = Sk(y(t)) −1 ≤Sk(x(t)) −1 = Sk(x(t + 1)).\n(17)\nCase 3: s′ ≤k < r′ Then X and Y delete exactly zero and one ball in the first k bins, respectively,\nyielding\nSk(y(t + 1)) = Sk(y(t)) −1 ≤Sk(x(t)) −1 = Sk(x(t + 1)) −1 < Sk(x(t + 1)).\n(18)\nCase 4: r′ ≤k < s ≤s′ Then we use Observation A.1 to get\nSk(y(t + 1)) = Sk(y(t)) ≤Ss−1(y(t))\n≤Sr(x(t)) −1 ≤Sr′(x(t)) −1 = Sr′(x(t + 1)) ≤Sk(x(t + 1)).\n(19)\n18\n\nCase 5: max { r′, s } ≤k < s′ For the sake of a contradiction, let k be the minimal value in this\nrange that violates the majorization after the deletion (and normalization). That is, we have\nSk(y(t + 1)) > Sk(x(t + 1)) and Sk−1(y(t + 1)) ≤Sk−1(x(t + 1)).4 This implies yk(t + 1) >\nxk(t + 1). Now note that k lies in the plateau of y(t) between s and s′, such that for all\nq ∈{ k + 1, k + 2, s′ −1 } we have\nyq(t + 1) = yk(t + 1) > xk(t + 1) ≥xq(t + 1)\n(20)\n(the last inequality using that x(t + 1) is sorted non-increasingly) and, similarly,\nys′(t + 1) = yk(t + 1) −1 ≥xk(t + 1) ≥xq(t + 1).\n(21)\nThis implies Ss′(y(t)) = Ss′(y(t + 1)) + 1 > Ss′(x(t + 1)) + 1 = Ss′(x(t)), contradicting the\ninduction hypothesis.\nTogether, this shows that x(t + 1) majorizes y(t + 1).\nB\nOmitted Material from Proof of Theorem 2.1.\nIn this section we prove Theorem 2.1, which we restate for convenience:\nTheorem 2.1. Consider an insertion probability sequence\n\u0000β(t)\n\u0001\nt∈N bounded away from 0 by a\nconstant β > 0.\nIf α ≤β/16, then for any t ∈N0 we have E[Γα(x(t))] = O(n) · eα·adisc(0).\nMoreover, for any a ≥0 we have Pr\n\u0002\nadisc(t) −adisc(0) ≥a+2\nα · ln n\n\u0003\n≤n−a.\nWe use the potential functions from [PTW15] and show that, at an arbitrary point of time,\nthe expected potential is linear in n. Since the authors of [PTW15] considered a process without\nball deletions, we extend the potential function analysis to capture ball deletions. Recall that the\nprocess works as follows: In every time step a ball is inserted with probability β(t) using two choices.\nOtherwise, with probability 1 −β(t), a random non-empty bin is picked and a ball from that bin is\ndeleted.\nThe three potential functions defined in [PTW15] are:\nΦα(x(t)) =\nn\nX\ni=1\neα·(xi(t)−m(t)/n);\nΨα(x(t)) =\nn\nX\ni=1\ne−α·(xi(t)−m(t)/n);\nΓα(x(t)) = Φα(x(t)) + Ψα(x(t))\nwhere x(t) = (x1(t), ..., xn(t)) is the normalized load vector.\nFor convenience, we use yi(t) :=\nxi(t) −m(t)/n for the difference between the load in the ith bin and the average.\nThe remainder of the section is dedicated to prove Lemma B.8 which directly implies Theo-\nrem 2.1. For this proof, we require insertion and deletion probabilities. Given that a ball is deleted\nin round t, we denote by qi(t) the probability that it is removed from the ith bin. And given that\na ball is inserted, we denote by pi(t) the probability that it is inserted into the ith bin. If no bin is\nempty, then qi(t) = 1/n because the protocol deletes a ball from a non-empty bin chosen uniformly\nat random. If there are s empty bins, then the empty bins, which come last in the normalized array,\nhave probability qi(t) = 0, and the remaining bins have probability qi(t) = 1/(n −s). pi(t), on the\nother hand, is independent of t. pi(t) = pi =\n\u0000 i\nn\n\u00012 −\n\u0000 i−1\nn\n\u00012 because for the ith bin to be chosen\nin the normalized array, the ith bin must be among the choices next to a bin with the same or a\nsmaller index. (W.l.o.g. we assume that if bins of equal load are chosen, the bin with the larger\nindex receives the ball.) From this we can derive the following simple observation:\n4For k = s this holds by the previous case.\n19\n\nObservation B.1. P\ni≤n/4 pi ≤1/4 −ϵ and P\ni≥3n/4 pi ≥1/4 + ϵ for any ϵ ≤3\n16.\nProof. The probability for any of the first n/4 bins receiving the new ball is\nX\ni≤n/4\npi =\nX\ni≤n/4\n\u0012 i\nn\n\u00132\n−\n\u0012i −1\nn\n\u00132\n=\n\u0012n/4\nn\n\u00132\n= 1\n16.\nThe probability for any of the last n/4 bins receiving the new ball is\nX\ni>3n/4\npi =\nX\ni>3n/4\n\u0012 i\nn\n\u00132\n−\n\u0012i −1\nn\n\u00132\n=\n\u0010n\nn\n\u00112\n−\n\u00123n/4\nn\n\u00132\n= 1 −\n\u00123\n4\n\u00132\n= 7\n16\nLet ˆy(t) = (ˆy1(t), . . . , ˆyn(t)) denote the load vector above average before normalization. By\ncombining the probabilities β, pi and qi, we can express ˆyi(t + 1), i.e. the ith bin’s load above\naverage before normalization at t + 1, as a function of its previous load yi(t):\nˆyi(t + 1) =\n\n\n\n\n\n\n\n\n\n\n\nyi + 1 −1/n,\nwith probability β(t) · pi (ball is allocated to ith bin),\nyi −1/n,\nwith probability β(t) · (1 −pi) (ball is allocated to other bin),\nyi −1 + 1/n,\nwith probability (1 −β(t)) · qi (ball is deleted from ith bin),\nyi + 1/n,\nwith probability (1 −β(t)) · (1 −qi) (ball is deleted from other bin)\nLet Φi\nα(x(t)) = eα·yi(t) and Ψi\nα(x(t)) = e−α·yi(t) be the local potentials in bin i.\nAnd let\n∆ˆΦi\nα := Φi\nα(ˆx(t+1))−Φi\nα(x(t)) and ∆ˆΨi\nα := Ψi\nα(ˆx(t+1))−Ψi\nα(x(t)) be the local potential changes\nin the ith bin (before normalization). Then:\n∆ˆΦi\nα =\n\n\n\n\n\n\n\n\n\n\n\neα(yi+1−1/n) −eαyi = eαyi · (eα(1−1/n) −1),\nwith probability β(t) · pi,\neα(yi−1/n) −eαyi = eαyi · (e−α/n −1),\nwith probability β(t) · (1 −pi),\neα(yi−1+1/n) −eαyi = eαyi · (eα(−1+1/n) −1),\nwith probability (1 −β(t)) · qi,\neα(yi+1/n) −eαyi = eαyi · (eα/n −1),\nwith probability (1 −β(t)) · (1 −qi)\n∆ˆΨi\nα =\n\n\n\n\n\n\n\n\n\n\n\ne−α(yi+1−1/n) −eαyi = e−αyi · (e−α(1−1/n) −1),\nwith probability β(t) · pi,\ne−α(yi−1/n) −eαyi = e−αyi · (eα/n −1),\nwith probability β(t) · (1 −pi),\ne−α(yi−1+1/n) −eαyi = e−αyi · (e−α(−1+1/n) −1),\nwith probability (1 −β(t)) · qi,\ne−α(yi+1/n) −eαyi = e−αyi · (e−α/n −1),\nwith probability (1 −β(t)) · (1 −qi)\nWe use these local changes to prove different bounds on the overall potential changes ∆Φα =\nΦα(x(t + 1)) −Φα(x(t)) (in Lemma B.1, Lemma B.2, Corollary B.1 and Lemma B.5) and ∆Ψα =\nΨα(x(t + 1)) −Ψα(x(t)) (in Lemma B.3, Lemma B.4, Corollary B.2 and Lemma B.6).\nLemma B.1 (similar to a part of Lemma A.5 in [TW14]). If, for a constant β, β(t) ≥β for all t\nand if α ≤ϵ·β\n3 ≤1\n16, then\nE[∆Φα | x(t)] ≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n.\n20\n\nProof. We compute the expected potential difference:\nE[∆Φα | x(t)]\n=\nn\nX\ni=1\nβ(t)pi · (eα(1−1/n) −1) · eαyi + β(t)(1 −pi) · (eα(−1/n) −1) · eαyi\n+(1 −β(t))qi · (eα(−1+1/n) −1) · eαyi + (1 −β(t))(1 −qi) · (eα(1/n) −1) · eαyi\n=\nn\nX\ni=1\nβ(t)eαyi\n\u0010\npi · e−α/n · (eα −1) + e−α/n −1\n\u0011\n+(1 −β(t))eαyi\n\u0010\nqieα/n(e−α −1) + eα/n −1\n\u0011\n≤\nn\nX\ni=1\nβ(t)eαyi\n\u0010\npi(eα −1) + e−α/n −1\n\u0011\n+ (1 −β(t))eαyi\n\u0010\nqi(e−α −1) + eα/n −1\n\u0011\n=\nn\nX\ni=1\neαyi\n\u0010\nβ(t)pi(eα −1) + (1 −β(t))qi(e−α −1) + eα/n −1 + β(t)(e−α/n −eα/n)\n\u0011\n≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)pi(eα −1) + (1 −β(t))qi(e−α −1) + α\nn + 2α2\nn2 −2β(t)α\nn\n\u0013\n≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)pi(eα −1) + (1 −β(t)) 1\nn(e−α −1) + α\nn + 2α2\nn2 −2β(t)α\nn\n\u0013\n(22)\n=\nn\nX\ni=1\neαyi\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\nInequality (22) holds because, if no bin is empty, all qi = 1/n, and if bins are empty, then the\nqi are decreasing while the term (1 −β(t)) · (e−α −1) · eαyi is increasing as e−α −1 < 0.\nLemma B.2 (similar to a part of Lemma A.5 in [TW14]). If, for a constant β, β(t) ≥β for all t,\nif α ≤ϵ·β\n3 ≤1\n16 and if y3n/4(t) ≤0, then E[∆Φα | x(t)] ≤−β·α·ϵ\nn\n· Φα + α.\n21\n\nProof. We start from Lemma B.1 and continue the calculation:\nE[∆Φα | x(t)]\n≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n≤\nX\ni<3n/4\neαyi\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n+\nX\ni≥3n/4\ne0\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n≤\nX\ni<3n/4\n4Φα\n3n\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n(23)\n+β(t)\n\u0010\neα −1 −α\n4\n\u0011\n+ (1 −β(t))1\n4(e−α −1 + α) + α2\n2n\n(Obs. B.1)\n≤\n4Φα\n3n\n\u0012\nβ(t)\n\u0012\u00123\n4 −ϵ\n\u0013\n(eα −1) −3α\n4\n\u0013\n+ (1 −β(t))3\n4(e−α −1 + α) + 3α2\n2n\n\u0013\n+ α (24)\n=\nΦα\nn\n\u0012\nβ(t)\n\u0012\u0012\n1 −4\n3ϵ\n\u0013\n(eα −1) −α\n\u0013\n+ (1 −β(t))(e−α −1 + α) + 2α2\nn\n\u0013\n+ α\n=\nΦα\nn\n\u0012\n−4\n3β(t)ϵ(eα −1) + β(t) (eα −1 −α) + (1 −β(t))(e−α −1 + α) + 2α2\nn\n\u0013\n+ α\n≤\nΦα\nn\n\u0012\n−4\n3β(t)ϵα + 2\n3α2\n\u0013\n+ α\n≤\nΦα\nn (−β(t)ϵα) + α\n≤\nΦα\nn (−βϵα) + α\nIn Inequality (23), we use that Pn\ni=1 pi(eα−1)eαyi is maximal when eαyi = 4·Φα\n3·n for all i < 3n/4.\nThis is because the pi are non-decreasing and the yi are non-increasing. In Inequality (24), we use\nObservation B.1 and the fact that Pn\ni=1 pi(eα −1)eαyi is maximal if the pi are uniform. The second\nto last inequality uses α ≤β · ϵ/3 ≤β(t) · ϵ/3.\nCorollary B.1. If, for a constant β, β(t) ≥β for all t, if α ≤ϵ·β\n3 ≤\n1\n16 and if y3n/4(t) ≤0, then\nE[∆Φα | x(t)] ≤Φα · α2\nn .\nProof. We use Lemma B.1 and continue the calculation:\nE[∆Φα | x(t)]\n≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)\n\u0010\npi(eα −1) −α\nn\n\u0011\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n≤\nn\nX\ni=1\neαyi\nn\n\u0012\nβ(t)(eα −1 −α) + (1 −β(t))(e−α −1 + α) + 2α2\nn\n\u0013\n≤\nΦα · α2\nn\n22\n\nThe second inequality, for which all pi are set to 1/n, holds because the pi are non-decreasing\nand the eαyi are non-increasing. The last inequality makes use of the Taylor series.\nLemma B.3 (similar to a part of Lemma A.6 in [TW14]). If, for a constant β, β(t) ≥β for all t\nand if α ≤ϵ·β\n3 ≤1\n16, then\nE[∆Ψα | x(t)] ≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)(pi(e−α −1) + α/n) + (1 −β(t)) 1\nn(eα −1 −α) + 2α2\nn2\n\u0013\n.\nProof. We upper-bound the expected potential difference.\nE[∆Ψα | x(t)]\n=\nn\nX\ni=1\nβ(t)pi · (e−α(1−1/n) −1) · e−αyi + β(t)(1 −pi) · (e−α(−1/n) −1) · e−αyi\n+(1 −β(t))qi · (e−α(−1+1/n) −1) · e−αyi + (1 −β(t))(1 −qi) · (e−α(1/n) −1) · e−αyi\n=\nn\nX\ni=1\nβ(t)e−αyi\n\u0010\npi · eα/n · (e−α −1) + eα/n −1\n\u0011\n+(1 −β(t))e−αyi\n\u0010\nqi · e−α/n · (eα −1) + e−α/n −1\n\u0011\n≤\nn\nX\ni=1\nβ(t)e−αyi\n\u0010\npi(e−α −1) + eα/n −1\n\u0011\n+ (1 −β(t))e−αyi\n\u0010\nqi(eα −1) + e−α/n −1\n\u0011\n=\nn\nX\ni=1\ne−αyi\n\u0010\nβ(t)pi(e−α −1) + (1 −β(t))qi(eα −1) + e−α/n −1 + β(t)(eα/n −e−α/n)\n\u0011\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)pi(e−α −1) + (1 −β(t))qi(eα −1) −α\nn + 2β(t)α\nn\n+ 2α2\nn2\n\u0013\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)pi(e−α −1) + (1 −β(t)) 1\nn(eα −1) −α\nn + 2β(t)α\nn\n+ 2α2\nn2\n\u0013\n(25)\n=\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)(pi(e−α −1) + α/n) + (1 −β(t)) 1\nn (eα −1 −α) + 2α2\nn2\n\u0013\nWe use the Taylor series of the exponential function, and Inequality (25), replacing qi by 1/n,\nholds because the qi are non-increasing and the q−αyi are non-decreasing.\nLemma B.4 (similar to part of Lemma A.6 in [TW14]). If, for a constant β, β(t) ≥β for all t, if\nα ≤ϵ·β\n3 ≤1\n16 and if yn/4 ≥0, then E[∆Ψα | x(t)] ≤−β·α·ϵ\nn\n· Ψα + α.\n23\n\nProof. We use Lemma B.3 and continue the calculation:\nE[∆Ψα | x(t)]\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)(pi(e−α −1) + α/n) + (1 −β(t)) 1\nn (eα −1 −α) + 2α2\nn2\n\u0013\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)\n\u0010\npi(e−α −1) + α\nn\n\u0011\n+ (1 −β(t))α2\nn + 2α2\nn2\n\u0013\n≤\nX\ni≥n/4\ne−αyi\n\u0012\nβ(t)\n\u0010\npi(e−α −1) + α\nn\n\u0011\n+ (1 −β(t))α2\nn + 2α2\nn2\n\u0013\n+\nX\ni<n/4\ne0\n\u0012\nβ(t)\n\u0010\npi(e−α −1) + α\nn\n\u0011\n+ (1 −β(t))α2\nn + 2α2\nn2\n\u0013\n(Obs. B.1)\n≤\nX\ni≥n/4\ne−αyi\nn\n\u0012\nβ(t)\n\u0012\n1 + 4\n3ϵ\n\u0013\n(e−α −1) + β(t)α + (1 −β(t))α2 + 2α2\nn\n\u0013\n(26)\n+β(t)\n4 (e−α −1 + α) + 1 −β(t)\n4\nα2 + α2\n2n\n≤\nΨα −n/4\nn\n\u0012\nβ(t)4\n3ϵ(e−α −1) + α2 + α3\n2\n\u0013\n+ 3\n4α2\n(27)\n≤\nΨα\nn\n\u0012\nβ(t)4\n3ϵ(e−α −1) + α2\n\u0013\n+ α\n≤\nΨα\nn (−β(t)ϵα) + α\n≤\nΨα\nn (−βϵα) + α\nThe tricky part is Eq. (26) and (27). Note that (e−α −1) is negative, and pi and e−αyi are\nweakly increasing. So, to bound the expression, we can assume that the pi are uniform, and from\nP\ni≥n/4 pi ≥3/4 + ϵ it follows that then pi ≥(3/4 + ϵ)/(3/4 · n) = (1 + 4/3 · ϵ) · 1/n which implies\nEq. (26). The resulting term 1\nn\n\u0010\nβ(t)\n\u00001 + 4\n3ϵ\n\u0001\n(e−α −1) + β(t)α + (1 −β(t))(eα −1 −α) + 2α2\nn\n\u0011\nis\nnegative as well. Since P\ni<n/4 e−αyi < n/4 (because of yn/4 ≥0), it follows that P\ni≥n/4 e−αyi ≥\nΨα −n/4 and thus Eq. (27).\nThe second to last inequality uses ϵ ≥3α/β ≥3α/β(t).\nCorollary B.2. If, for a constant β, β(t) ≥β for all t, if α ≤ϵ·β\n3\n≤\n1\n16 and if yn/4 ≥0, then\nE[∆Ψα | x(t)] ≤Ψα α2\nn .\nProof. We use Lemma B.3 and continue the calculation:\nE[∆Ψα | x(t)]\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)(pi(e−α −1) + α/n) + (1 −β(t)) 1\nn(eα −1 −α) + 2α2\nn2\n\u0013\n≤\nΨα\nα2\nn\n24\n\nLemma B.5 (similar to Lemma A.7 in [TW14]). Suppose β(t) ≥β, y3n/4 > 0 and E[∆Φα | x(t)] ≥\n−α·β·ϵ\n4·n · Φα. Then either Φα < ϵ\n4 · Ψα or Γα ≤360·n\nϵ8·β4 .\nProof. In the following we use that pi ≤1−4·ϵ\nn\nfor i ≤n/3 and that P pi · eα·yi is maximized when\np is uniform. We start from Lemma B.1.\nE[∆Φα | x(t)]\n≤\nn\nX\ni=1\neαyi\n\u0012\nβ(t)(pi(eα −1) −α/n) + (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n≤\nX\ni≤n/3\neαyi\n\u0012\nβ(t)\n\u00121 −4ϵ\nn\n(eα −1) −α/n\n\u0013\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n+\nX\ni>n/3\neαyi\n\u0012\nβ(t)\n\u0012 3\n2n(eα −1) −α/n\n\u0013\n+ (1 −β(t)) 1\nn(e−α −1 + α) + 2α2\nn2\n\u0013\n≤\nΦα,≤n/3\nn\n\u0012\nβ(t) ((1 −4ϵ)(eα −1) −α) + (1 −β(t))(e−α −1 + α) + 2α2\nn\n\u0013\n+Φα,>n/3\nn\n\u0012\nβ(t)\n\u00123\n2(eα −1) −α\n\u0013\n+ (1 −β(t))(e−α −1 + α) + 2α2\nn\n\u0013\n≤\n−3 · β(t) · α · ϵ\nn\n· Φα + α\nn · Φα,>n/3\n≤\n−3 · β · α · ϵ\nn\n· Φα + α\nn · Φα,>n/3\nRecall that we assume E[∆Φα | x(t)] ≥−α·β·ϵ\n4·n · Φα.\n−α · β · ϵ\n4 · n\n· Φα\n≤\n−3 · α · β · ϵ\nn\n· Φα + α\nn · Φα,>n/3\n⇒\n2 · β · ϵ · Φα\n≤\nΦα,>n/3\nIf Φα < ϵ\n4·Ψα, we are done (see statement). So, assume Φα ≥ϵ\n4·Ψα. Since Φα,≥n/3 ≤2·n\n3 ·e3·α·B/n\nfor B = P\ni max{0, yi} = ||y||1\n2\nand since y3n/4 > 0 implies Ψα ≥n\n4 · e4·α·B/n, we get:\nn · ϵ\n16 · e4·α·B/n ≤ϵ\n4 · Ψα ≤Φα\n≤\nΦα,>n/3 ·\n1\n2 · β · ϵ ≤\nn\n3 · β · ϵ · e3·α·B/n\n=⇒\neα·B/n ≤\n6\nβ · ϵ2\nFinally, it follows that:\nΓα\n=\nΦα + Ψα ≤5\nϵ · Φα ≤5\nϵ ·\nn\n3 · β · ϵ · e3·α·B/n ≤5\nϵ ·\nn\n3 · β · ϵ ·\n\u0012\n6\nβ · ϵ2\n\u00133\n= 360 · n\nϵ8 · β4\nLemma B.6 (similar to Lemma A.8 in [TW14]). Suppose β(t) ≥β, yn/4 < 0 and E[∆Ψα | x(t)] ≥\n−α·β·ϵ\n4·n · Ψα. Then either Ψα < ϵ\n4 · Φα or Γα ≤1040000·n\nϵ8·β4\n.\n25\n\nProof. We use Lemma B.3. In Eq. (28) we use pi > 1+ϵ\nn\nfor i > 2n/3 and e−α −1 < 0.\nE[∆Ψα | x(t)]\n≤\nn\nX\ni=1\ne−αyi\n\u0012\nβ(t)\n\u0010\npi(e−α −1) + α\nn\n\u0011\n+ (1 −β(t)) 1\nn(eα −1 −α) + 2α2\nn2\n\u0013\n≤\nΨα,≤2n/3 · α\nn\n(28)\n+\nX\ni>2n/3\ne−αyi\n\u0012\nβ(t)\n\u00121 + ϵ\nn\n(e−α −1) + α\nn\n\u0013\n+ (1 −β(t)) 1\nn(eα −1 −α) + 2α2\nn2\n\u0013\n≤\nΨα,≤2n\n3 · α\nn +\nΨα,> 2n\n3\nn\n\u0012\nβ(t)ϵ(e−α −1) + α2 + α3\n2\n\u0013\n≤\nΨα,≤2n\n3 · α\nn +\nΨα,> 2n\n3\nn\n\u0000−αβ(t)ϵ + α2\u0001\n≤\nΨα,≤2n\n3 · α\nn + Ψα,> 2n\n3 · −αβ(t)ϵ\n2n\n≤\nΨα,≤2n\n3 · α\nn + Ψα,> 2n\n3 · −αβϵ\n2n\nUsing E[∆Ψα | x(t)] ≥−α·β·ϵ\n4·n · Ψα, we get:\n−α · β · ϵ\n4 · n\n· Ψα\n≤\nΨα,≤2n\n3 · α\nn −Ψα · αβϵ\n2n\nΨα\n≤\n4\nβ · ϵ · Ψα,≤2n\n3\nIf Ψα < ϵ\n4 · Φα, we would be done; so we assume Ψα ≥ϵ\n4 · Φα. Since Ψα,≤2n\n3 ≤2\n3 · n · e3·α·B/n for\nB = P\ni max{0, yi} and since yn/4 < 0 implies Φα ≥n\n4 · e4·α·B/n, we get:\nϵ · n\n16 · e4·α·B/n ≤ϵ\n4 · Φα ≤Ψα ≤\n4\nβ · ϵ · Ψα,≤2n\n3 ≤\n8\n3 · β · ϵ · n · e3·α·B/n\n⇒\neα·B/n ≤\n128\n3 · β · ϵ2\nFinally, it follows that:\nΓα\n≤\n5\nϵ · Ψα ≤5\nϵ ·\n8\n3 · β · ϵ · n · e3·α·B/n ≤\n40\n3 · β · ϵ2 · n ·\n\u0012\n128\n3 · β · ϵ2\n\u00133\n≤1040000 · n\nϵ8 · β4\nThe next Lemma B.7 utilizes all previous lemmas and corollaries to prove a bound on the\npotential change ∆Γα = Γα(x(t + 1)) −Γα(x(t)). This bound will then be used in Lemma B.8 to\nfinally prove that the expected potential E[Γα(x(t))] is linear in n.\nLemma B.7 (similar to Theorem A.9 in [TW14]). E[Γα(t+1) | x(t)] ≤(1−α·β·ϵ\n4·n )·Γα(t)+ 1040000\nϵ8·β4 .\n26\n\nProof. Case 1: yn/4 ≥0 and y3n/4 ≤0. The statement follows from Lemma B.2 and Lemma B.4:\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n≤\n−α · β · ϵ\nn\n· Φα + α −α · β · ϵ\nn\n· Ψα + α\n=\n−α · β · ϵ\nn\n· Γα + 2 · α\nCase 2: yn/4 ≥y3n/4 > 0. If E[∆Φα | x(t)] ≤−ϵ·β·α\n4·n · Φα, apply Lemma B.4:\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n≤\n−ϵ · β · α\n4 · n\n· Φα + α −Ψα · α · β · ϵ\nn\n+ α\n≤\n−α · β · ϵ\n4 · n\n· Γα + α\nIf this is not the case, apply Lemma B.5 which distinguishes two cases: If Φα < ϵ\n4 · Ψα, apply\nLemma B.4 and Corollary B.1 and use Φα < ϵ\n4 · Ψα ≤1\n16 · Ψα as well as β ≥α (which follows from\nϵ ≥2 · α/β):\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n≤\nα2\nn · Φα −Ψα · α · β · ϵ\nn\n+ α\n≤\nα · β · ϵ\nn · 4\n· Ψα −Ψα · α · β · ϵ\nn\n+ α\n=\n−Ψα · 3 · α · β · ϵ\n4 · n\n+ α\n≤\n−Γα · 3 · α · β · ϵ\n4 · n\n+ α\nIf Γα < 360·n\nϵ8·β4 , apply Corollary B.1 and Corollary B.2:\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n(29)\n≤\nα2\nn · Φα + α2\nn · Ψα\n=\nα2\nn · Γα\n≤\n−α · β · ϵ\nn\n· Γα + (1 + ϵ) · α · β\nn\n· Γα\n≤\n−α · β · ϵ\nn\n· Γα +\n360\nϵ8 · β2\n(30)\nCase 3: y3n/4 ≤yn/4 < 0. If E[∆Ψα | x(t)] ≤−α·β·ϵ\n4·n · Ψα, apply Lemma B.2:\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n≤\n−α · β · ϵ\nn\n· Φα + α −α · β · ϵ\n4 · n\n· Ψα\n≤\n−α · β · ϵ\n4 · n\n· Γα + α\n27\n\nIf this is not the case, apply Lemma B.6 which distinguishes two cases: If Ψα < ϵ\n4 · Φα, apply\nLemma B.2 and Corollary B.2 and use ϵ < 1/4:\nE[∆Γα | x(t)]\n=\nE[∆Φα | x(t)] + E[∆Ψα | x(t)]\n≤\n−α · β · ϵ\nn\n· Φα + α + α2\nn · Ψα\n≤\n−3 · α · β · ϵ\n4 · n\n· Γα + α\n(The proof for the last inequality is very similar to Case 2.)\nIf Γα < c·n\nβ4 , Inequality (29)-(30) can be reused with the constant from Lemma B.6.\nE[∆Γα | x(t)]\n≤\n−α · β · ϵ\nn\n· Γα + 1040000\nϵ8 · β2\nLemma B.8. Fix an arbitrary step t. Let ϵ ≤\n3\n16, and assume that β ∈(0, 1] is a constant probability\nand that α ≤ϵ·β\n3 . Then\nE[Γα(x(t))] ≤4160002\nα · β3 · ϵ9 · eα·adisc(0) · n.\nProof. The proof of this lemma follows from Lemma B.7 which gives us\nE[Γα(x(t + 1)) | x(t)] ≤\n\u0012\n1 −α · β · ϵ\n4 · n\n\u0013\n· Γα(x(t)) + 1040000\nϵ8 · β2 .\nTaking the expected value on both sides, we obtain\nE[Γα(x(t + 1))] ≤\n\u0012\n1 −α · β · ϵ\n4 · n\n\u0013\n· E[Γα(x(t))] + 1040000\nϵ8 · β2 .\nSolving this recursion yields\nE[Γα(x(t))] ≤\n\u0012\n1 −α · β · ϵ\n4 · n\n\u0013t\n·\n\u0012\nE[Γα(x(0))] −4160000\nα · β3 · ϵ9 · n\n\u0013\n+ 4160000\nα · β3 · ϵ9 · n\n≤E[Γα(x(0))] + 4160000\nα · β3 · ϵ9 · n\n≤2n · eα·adisc(0) + 4160000\nα · β3 · ϵ9 · n\n=\n\u0012\n2eα·adisc(0) + 4160000\nα · β3 · ϵ9\n\u0013\n· n ≤4160002\nα · β3 · ϵ9 · eα·adisc(0) · n.\nLemma B.9. Fix an arbitrary step t. Let ϵ ≤\n3\n16, and assume that β ∈(0, 1] is a constant probability\nand that α ≤ϵ·β\n3 . Then\nPr\n\u0014\nadisc(t) −adisc(0) ≥(a + 2) · ln n\nα\n\u0015\n≤n−a.\n28\n\nProof. Lemma B.8 together with Markov’s inequality yields\nPr\n\u0014\nΓα(x(t)) ≥na+1 · 4160002\nα · β3 · ϵ9 · eα·adisc(0)\n\u0015\n≤n−a.\nBy rearranging the inequality on the left-hand side, we see that the corresponding event is equivalent\nto\nln(Γα(x(t)))\nα\n−adisc(0) ≥\n(a + 1) · ln(n) + ln(4160002\nα·β3·ϵ9 )\nα\n.\nFor large n, the right-hand side of this is at most (a+2)·ln(n)\nα\n. For the left-hand side, the potential\nfunction definition yields a lower bound of adisc(t) −adisc(0). Combining the above insights, we\nget\nPr\n\u0014\nadisc(t) −adisc(0) ≥(a + 2) · ln n\nα\n\u0015\n≤Pr\n\"\nln(Γα(x(t)))\nα\n−adisc(0) ≥\n(a + 1) · ln(n) + ln(4160002\nα·β3·ϵ9 )\nα\n#\n= Pr\n\u0014\nΓα(x(t)) ≥na+1 · 4160002\nα · β3 · ϵ9 · eα·adisc(0)\n\u0015\n≤n−a.\nC\nOmitted Material from Proof of Theorem 2.2.\nWe restate Theorem 2.2 for convenience.\nTheorem 2.2. Fix an arbitrary step t. Let β ∈(0, 1] be a constant. Assume β(τ) ≥β for all\n1 ≤τ ≤t and α ≤β/16. For any fixed non-negative ℓand any fixed µ ∈(0, 1), there is a γ = γ(µ)\nso that, with probability at least 1 −n−(3+ℓ), the number of balls above height ⌈m(t)\nn ⌉+ γ is at most\nµn.\nProof. From Lemma C.1 we may assume that Γα(x(t)) = O(n) with probability at least 1−n−(3+ℓ)\nfor any fixed non-negative ℓ, where the constant factor hidden in the big-Oh depends on ℓ.\nWe assign a height hb(t) to each ball b at time t in such a way that in each bin i the Xi(t)\nballs contained in that bin at time t are numbered 1, . . . , Xi(t). Further let h′\nb(t) = hb(t) −⌈m(t)\nn ⌉\nif hb(t) ≥⌈m(t)\nn ⌉and h′\nb(t) = 0 otherwise.\nThe standard potential function Φα(t) = P\ni eαX+\ni (t) with X+\ni (t) being bin i’s load above ⌈m(t)\nn ⌉\nat time t and zero otherwise is expressed in terms of individual bins’ potentials. We will rearrange\nthis function so as to express the potential in terms of individual balls’ potentials, and then make\nsubsequent arguments based upon this new potential function.\nWe assign potential πb(t) to ball b at height hb(t) as follows:\nπb(t) =\n\n\n\n\n\n0\nif hb(t) ≤⌈m(t)\nn ⌉,\neα\nif hb(t) = ⌈m(t)\nn ⌉+ 1,\neαh′\nb(t) −eα(h′\nb(t)−1)\nif hb(t) ≥⌈m(t)\nn ⌉+ 2.\nFurthermore let Π(t) = P\nball b πb(t). Now Π(t) = Φ(t), which may be seen as follows: Consider a\nbin i with k ≥1 balls above ⌈m(t)\nn ⌉. Then those balls contribute a total potential of\nΠi(t) :=\nX\nball b: hb(t)>⌈m(t)\nn\n⌉\nπb(t) = eα +\nk\nX\nj=2\neα·j −eα·(j−1)\n= eα + (e2α −eα) + (e3α −e2α) + · · · + (e(k−1)α −e(k−2)α) + (ekα −e(k−1)α).\n29\n\nThis is a telescoping sum evaluating to Πi(t) = ekα. This however is precisely this bin’s potential\ncontribution in the potential function Φ, so that we can conclude that\nΠ(t) =\nX\nball b\nπb(t) =\nX\nbin i\nX\nball b\nin bin i\nπb(t) =\nX\nbin i\nΠi(t) =\nX\nbin i\neαX+\ni (t) = Φ(t).\nConsider a height ⌈m(t)\nn ⌉+ k for some k ≥2; the precise value will be fixed below. Assume that\nthere is a set B of at least µn balls at height ⌈m(t)\nn ⌉+ k or higher (if no such set exists the proof is\nfinished); the precise distribution of the balls in bins is irrelevant. Each of those at least µn many\nballs contributes a potential of at least\neαk −eα(k−1) = eαk ·\n\u0012\n1 −1\neα\n\u0013\n≥ϵ · eαk\nfor some constant ϵ = ϵ(α) > 0, for a total potential contribution of those balls of at least\nΠB(t) ≥ϵ · µn · eαk.\nNow ΠB(t) > c · n whenever k > 1\nα ln c\nϵµ −1, so letting d = 1\nα ln c\nϵµ proves the theorem (recall that\nthe probability is inherited from Lemma C.1 – the actual argument hee in the proof building on top\nof it is entirely deterministic).\nLemma C.1. Assume the premise of Theorem 2.2 holds. Then Γα(x(t)) = O(n) with probability\nat least 1 −n−(3+ℓ) for any fixed non-negative ℓ, where the constant factor hidden in the big-Oh\ndepends on ℓ.\nProof. The proof is a minor adaptation of Theorem 5.3 on page 18 in Los and Sauerwald’s [LS21],\nwhich is a more detailed version in arXiv of their [LS22]. All references in Appendix C to their\nlemmas etc. are with respect to the numbering in their arXiv version [LS21].\nAs mention before, the purpose of the adaptation of [LS21]’s results is two-fold: (i) to incorporate\nour deletions and their respective probability bounds, and (ii) to improve the high-probability\nguarantee from the original 1 −1/n3 (which was sufficient for the purposes of [LS22, LS21]) to\n1 −1/n3+ℓfor any arbitrary but fixed ℓ> 0.\nTo show this result we adapt Theorem 5.3 in [LS21] as follows. The ℓin the exponent of the\nstatement comes from Theorem 2.2.\nTheorem C.1 (cf.\nTheorem 5.3 in [LS21]). Consider any probability vector p that is (i) non-\ndecreasing in i, i.e., pi ≤pi+1 and (ii) for constant ϵ, P\ni≥3n/4 pi ≥1/4+ϵ and P\ni≤n/4 pi ≥1/4−ϵ.\nThen, for any t ≥0 and α2 ≤β·ϵ\n110, c = cϵ,α2 = 2 · 40 · 1283 · ϵ−7 · 4 · α−1\n2 ,\nPr\nh\n∩s∈[t,t+n log5(n)]{Γ(s)\nα2 ≤2cn}\ni\n≥1 −n−(3+ℓ).\nProof. The proof is almost identical to that of Theorem 5.3 in [LS21]. The chain of modifications\nstarts with Lemma 5.5 in [LS21] where the restriction Γ(t)\n1\n≤cn9 occurs. An application of Markov’s\ninequality turns the n9 into a probability 1/n8, which subsequent lemmas etc. further decay to 1/n3.\nThe main idea to make the result work for us is to modify the restriction on Γ(t)\n1\nto Γ(t)\n1\n≤cn9+ℓ,\nwhich accordingly will turn into a probability 1/n8+ℓ, only to subsequently be eroded to 1/n3+ℓ.\nBasically we have the exact same results with the exact same proofs as in in [LS21] but with some\nexplicit constant parameters replaced with their respective original values plus ℓ. We will need to\n30\n\nmake some equally straightforward modification of [LS21]’s results to make them compatible with\nour potential function analysis.\nWe will use two instances of the potential function Γα which only differ in α. The first one,\nΓα1, uses α1 ≤β·ϵ\n3 , the second one, Γα2, uses α2 ≤\nα1\n4·9.1. (Explanation: α1 and α2 must fulfil the\nrequirements of the Potential Function Analysis in the previous section and of Lemma C.2.) From\nLemma B.8 we can derive, for n and c large enough, i.e. c ≥4160002·n\nα·β3·ϵ9 , that Pr\n\u0002\nΓα1(x(t)) ≥c · n9+ℓ\u0003\n<\nn−(8+ℓ). The next lemma proves a stronger statement for Γα2.\nLemma C.2 (cf. Lemma 5.5 in [LS21]). For any t ≥0, if Γ(t)\nα1 ≤c·n9+ℓ, then, (i) |xi(t)| ≤9.1\nα1 ·log(n)\nfor all i ∈[n], (ii) Γ(t)\nα2 ≤n4/3, and (iii) |Γ(t+1)\nα2\n−Γ(t)\nα2| ≤n1/3.\nProof. First statement: For all i ∈[n],\nΓ(t)\nα1 = eα1·xi(t) + e−α1·xi(t) ≤c · n9+ℓ=⇒|xi(t)| ≤9.1\nα1\n· log(n).\nSecond statement:\nΓ(t)\nα2 < 2 ·\nn\nX\ni=1\nexp\n\u0012\nα2 · 9.1\nα1\n· log(n)\n\u0013\n≤2 · n · n1/4 < n4/3.\nThird statement: One of the xi(t) is increased or decreased by 1, and all xi(t) are (additionally)\ndecreased or increased by 1/n as the average load is altered by the allocation or deletion in step\nt + 1. The change affects both parts, Φα and Ψα, of potential function Γα. In total, the absolute\nℓ1-change in all xi(t) is upper-bounded by 4. Since exp(.) is convex, the hypothetical worst case is\nthat the largest exponent increases by 4 while all others remain the same:\n|Γ(t+1)\nα2\n−Γ(t)\nα2|\n≤\nexp(α2 · max{x(t)\nmax + 4, −x(t)\nmin −4})\n≤\ne4α2 · exp(α2 · 9.1\nα1\n· log(n)) ≤n1/3\nIf we define c′ = 104·106\nϵ8·β4 and ϵ′\nα = αβϵ\n4 then we obtain the following claim by applying Theorem B.7\nin the proof from [LS21].\nClaim C.1 (cf. Claim 5.7 in [LS21]). For any t ≥0, E[Γ(t+1)\nα2\n| Γ(t)\nα2, Γ(t)\nα2 ≥2c′\nϵ′α2 · n] ≤(1 −\nϵ′\nα2\n2n ) · Γ(t)\nα2.\nProof. The proof is identical to Claim 5.7 in [LS21].\nLemma C.3 (cf. Lemma 5.6 in [LS21]). For any t ≥n · log2(n), for constants c′ > 0, ϵ′\nα2 > 0\ndefined as above,\nPr\n\u0014\n∪s∈[t−n·log2(n),t]\n\u001a\nΓ(s)\nα2 ≤2c′\nϵ′α2\n· n\n\u001b \u0015\n≥1 −2 · c · n−(8+ℓ).\nProof. The proof is almost identical to that of Lemma 5.6 in [LS21]. In fact the modifications\nimmediately follow from those made to Lemma 5.5 (here: Lemma C.2).\n31\n\nWe are now in a position to finish the proof of Theorem C.1, a part that essentially mimics\nSection 5.2, with just some few slightly adapted coefficients. We do of course not wish to copy a\npage or two from another paper, and therefore just briefly highlight the modifications.\nThe application of Lemma 5.6 in [LS21] (here Lemma C.3) gives a lower bound of 1 −2cn−(8+ℓ)\ninstead of 1 −2cn−8 for the probability. Further down, the application of the union bound gives\nPr\n\n\n\\\nu∈[t−n log2 n,t+n log5 n]\n{Γ(u)\n1\n≤cn9+ℓ}\n\n≥1 −2 log5 n\nn7+ℓ.\nThen, the application of Theorem A.4 in [LS21] gives\nPr\nh\nX(u)\nr\n≥X(r)\nr\n+ cn\n2\ni\n≤exp\n\u0012\n−\nc2n2/4\n2 · (2n log5 n) · (2n1/3)\n\u0013\n+ 2 log5 n\nn7+ℓ\n≤3 log5 n\nn7+ℓ.\nFurther down we find that if a red phase starts at time r, then with probability 1 −3 log5 n\nn7+ℓ, Γ(u)\n2\nwill\nalways be ≤3cn\n2 . Taking union bounds as in Section 5.2 in [LS21] yields\nPr\n\n\n[\nr∈[s,t+n log5 n]\n[\nu∈[r,t+n log5 n]\n{X(u)\nr\n> 3cn\n2 }\n\n≤3 · log5 n\nn7+ℓ· (4n2 log10 n) ≤1\n2n−4+ℓ.\nConcluding as in [LS21], with probability 1 −1\n2n−4+ℓ−2cn−8+ℓ≥1 −n−4+ℓ, it holds that Γ(r)\n2\n≤\n3cn\n2\nfor all time-steps r which are within a red phase in [s, t + n log5 n] ⊆[t, t + n log5 n]. Since\nΓ(r)\n2\n≤T ≤cn holds (deterministically) by definition for all time-steps r within a green phase, the\ntheorem follows.\nThis concludes the proof sketch of the Lemma.\nD\nOmitted Material from Section 3.\nWe first present the proof of Lemma 3.2. We restate it for convenience:\nLemma 3.2. Consider an h-valid configuration x and let x′ denote the (random) configuration\nafter one step of our process with the insertion probability β(t) ≤ˆβ < 1.\n(a) If level ℓ∈{ 1, 2, . . . , ℓ∗} of configuration x is h-critical, then\nPr\nh\nm(h)\nℓ(x′) < m(h)\nℓ(x)\ni .\nPr\nh\nm(h)\nℓ(x′) > m(h)\nℓ(x)\ni\n≥2.\n(4)\n(b) If level ℓ= ℓ∗+ 1 of configuration x is h-critical and if n is large enough, then\nPr\nh\nm(h)\nℓ(x′) < m(h)\nℓ(x)\ni .\nPr\nh\nm(h)\nℓ(x′) > m(h)\nℓ(x)\ni\n≥√n.\n(5)\nProof. For the sake of readability, we abuse notation in this proof by omitting the base height h\nwhen talking about the number of balls that have at least some level ℓ(i.e., we write mℓ(•) instead\nof m(h)\nℓ(•)). Similarly, we refer to the notations h-valid and h-critical simply as valid/critical within\nthe proof.\n32\n\n(a) Consider a critical level ℓ∈{ 1, 2, . . . , ℓ∗} of configuration x. Since x is valid, level ℓ−1 and\nℓ+ 1 are valid. Using Lemma 3.1 and Observation 3.1 we calculate\nPr [ mℓ(x′) < mℓ(x) ]\nPr [ mℓ(x′) > mℓ(x) ] ≥1 −β(t)\nβ(t)\n·\nmℓ(x) −mℓ+1(x)\n\u0000mℓ−1(x) −mℓ(x)\n\u00012 · n\n≥1 −β\nβ\n·\nαℓ/2 −αℓ+1\n(αℓ−1 −αℓ/2)2 · n ≥1 −β\nβ\n· αℓ/4\nα2\nℓ−1\n· n ≥2.\nThe first inequality follows from Lemma 3.1. The second inequality uses the bound on β(t),\nthat level ℓis critical, and that levels ℓ−1 and ℓ+1 are valid. The third and final inequalities\nfollow from Observation 3.1.\n(b) For level ℓ∗+ 1 we similarly calculate\nPr [ mℓ∗+1(x′) < mℓ∗+1(x) ]\nPr [ mℓ∗+1(x′) > mℓ∗+1(x) ] ≥1 −β(t)\nβ(t)\n·\nmℓ∗+1(x) −mℓ∗+1+1(x)\n\u0000mℓ∗+1−1(x) −mℓ∗+1(x)\n\u00012 · n\n≥1 −β\nβ\n· 1\nα2\nℓ∗\n· n = 1 −β\nβ\n·\nn\n144 log2(n) = Ω\n\u0012\nn\nlog2(n)\n\u0013\n≥√n.\nCompared to the proof of the previous statement, we bound the number of balls at level at\nleast ℓ∗+1 in the numerator by 1 (level ℓ∗+1 being critical means it is non-empty, so there is\nat least one bin with a ball on level ℓ∗+ 1). The rest of the calculation applies the definition\nof αℓ∗(see Equation (2)) and simplifies.\nWe next present the proof of Lemma 3.4. We restate it for convenience:\nLemma 3.4. Let h(t) := ⌈mmax(t)/n⌉+ γ denote the base height at time t. Consider an initial\nconfiguration x(0) = x that is h(0)-safe. Fix the time T = n4 and assume n to be large enough. If\nβ(t) ≤ˆβ < 1 for all t ∈{ 0, 1, . . . , T −1 }, then\nPr [ ∀t ∈{ 0, 1, . . . , T −1 } : x(t) is h(t)-valid ] ≥1 −n−1.\n(6)\nProof. Define the stopping time τ := min({ T } ∪{ t ∈N | x(t) is not h(t)-valid }). With this, we\nclearly have Pr [ ∀t ∈{ 0, 1, . . . , T −1 } : x(t) is h(t)-valid ] = Pr [ τ ≥T ], so it is sufficient to prove\nthat Pr [ τ ≥T ] ≥1 −n−1. In the following, we say level ℓis valid at time t (critical/safe) if level\nℓof configuration x(t) is h(t)-valid (-critical/-safe). We use the shorthand ˜mℓ(t) := m(h(t))\nℓ\n(x(t)).\nSince the base height h(t) is monotonously non-decreasing in t, we have ˜mℓ(t+1) = mh(t+1)\nℓ\n(x(t+\n1)) ≤mh(t)\nℓ\n(x(t + 1)) for any t ∈N0. Thus, if x(t) is h(t)-valid and level ℓis critical at time t,\nLemma 3.2 implies\nPr [ ˜mℓ(t + 1) < ˜mℓ(t) ]\nPr [ ˜mℓ(t + 1) > ˜mℓ(t) ] ≥\nPr\nh\nm(h(t))\nℓ\n(x(t + 1)) < m(h(t))\nℓ\n(x(t))\ni\nPr\nh\nm(h(t))\nℓ\n(x(t + 1)) > m(h(t))\nℓ\n(x(t))\ni\n≥\n(\n2\nif ℓ∈{ 0, 1, . . . , ℓ∗} and\n√n\nif ℓ= ℓ∗+ 1.\n(31)\nIf τ < T, there must be some level ℓthat is not valid at time τ and a minimal time t′ < τ such\nthat level ℓis critical at any integral time t ∈[t′, τ). In particular, using the minimality of t′, we\nhave for all integral t ∈[t′, τ), ˜mℓ(t′) = αℓ/2 ≤˜mℓ(t) < ˜mℓ(τ) = αℓ. That is, between time step t′\n33\n\nand τ, the number of balls at level at least ℓstarted at αℓ/2 and increased to αℓwithout decreasing\nbelow αℓ/2. If this happens, we say level ℓcrosses [αℓ/2, αℓ) during [t′, τ), and denote this event\nas Cℓ,[t′,τ). We will show that for any level ℓand any time t′ event Cℓ,[t′,τ) is unlikely to happen.\nUsing a union bound, this yields that, w.h.p., τ ≥T, as desired. To bound Pr\n\u0002\nCℓ,[t′,τ)\n\u0003\n, we define\na random walk W (ℓ,t′) that has an even larger probability to cross [αℓ/2, αℓ) during [t′, τ) and show\nthat this larger probability is still small enough.\nTo this end, fix a level ℓ∈{ 0, 1, . . . , ℓ∗+ 1 } and integral time t′ ∈[0, τ] ⊆[0, T]. Let p(ℓ,t) :=\nPr [ ˜mℓ(t + 1) = ˜mℓ(t) | x(t) ] denote the probability that the number of balls at level at least ℓ\ndoes not change from time t to time t+1 (note that this probability is a random variable depending\non the random configuration x(t)). We define the random walk W (ℓ,t′) = (W (ℓ,t′)\nt\n)t∈[t′,T) over the\ndiscrete time interval [t′, T). The random walk’s start position is W (ℓ,t′)\nt′\n:= ˜mℓ(t′) = αℓ/2 and its\nmovement from time t ∈[t′, T) to time t + 1 is defined via\nW (ℓ,t′)\nt+1\n=\n\n\n\n\n\nW (ℓ,t′)\nt\nwith probability p(ℓ,t),\nW (ℓ,t′)\nt\n−1\nwith probability\n\u00001 −p(ℓ,t)\u0001\n· 2/3, and\nW (ℓ,t′)\nt\n+ 1\nwith probability\n\u00001 −p(ℓ,t)\u0001\n· 1/3.\n(32)\nIn other words, the random walk W (ℓ,t′) starts at time t′ at position ˜mℓ(t′) = αℓ/2, maintains\nits position whenever ˜mℓ(•) does not change, and the probability that it moves left is twice the\nprobability that it moves right. For a time t ≥t′, we say W (ℓ,t′) crosses [αℓ/2, αℓ) during [t′, t) if\nW (ℓ,t′) reaches αℓbefore decreasing below αℓ/2, and denote this event as CW\nℓ,[t′,t).\nGiven the current configuration x(t) = x at time t ≥t′ and depending on how ˜mℓ(•) changes\nfrom time t to time t + 1, we can easily couple the random walk W (ℓ,t′) with our process as follows:\n• if ˜mℓ(•) does not change, W (ℓ,t′) remains stationary;\n• if ˜mℓ(•) increases, W (ℓ,t′) moves right; and\n• if ˜mℓ(•) decreases, W (ℓ,t′) moves left with prob. min { 1, 2\n3/Pr [ ˜mℓ(t + 1) < ˜mℓ(t) | x(t) = x ] }\nand right otherwise.\nBy this coupling, the random walk is never “left of” ˜mℓ(•) (i.e., for all t ≥t′ we have W (ℓ,t′)\nt\n≥˜mℓ(t)).\nWe now consider the crossing probabilities for the different levels ℓ∈{ 0, 1, . . . , ℓ∗+ 1 } for some\nintegral time t′ ∈[0, τ) at which level 0 just became critical. For level ℓ= 0, Theorem 2.2 can be\nused to see that, w.h.p, this level will not become invalid at any time during the discrete interval\n[0, τ). For any other level ℓ, we use the fact that the ˜mℓ-value is upper-bounded by the position\nof the random walk W (ℓ,t′) during [t′, τ) (by the coupling above) together with the fact that the\nleft-biased random walk W (ℓ,t′) is very unlikely to reach the right border before becoming again safe\n(by Lemma 3.3).\nCase 1: For level ℓ= 0 we do not need to considered the random walk.\nInstead, note that\nTheorem 2.2 allows us to choose a constant γ = γ(α0) such that the number of balls above\nheight ⌈m(t)/n⌉+ γ is, with (arbitrarily) high probability, at most α0 = 1−β\n128β · n. Thus, by\nchoosing the probability high enough and by using a union bound over the T = poly(n) many\ntime steps, we can ensure that, with probability at least 1 −n−6, level 0 does not become\ninvalid at any time ∈{ 1, 2, . . . , T }. In particular, this yields\nPr\n\u0002\nC0,[t′,τ)\n\u0003\n≤Pr\n\u0002\nC0,[t′,T)\n\u0003\n≤n−6.\n(33)\n34\n\nCase 2: For any level ℓ∈{ 1, 2, . . . , ℓ∗} and large enough n we get\nPr\n\u0002\nCℓ,[t′,τ)\n\u0003\n≤Pr\nh\nCW\nℓ,[t′,τ)\ni\n≤Pr\nh\nCW\nℓ,[t′,T)\ni\nLem. 3.3\n≤\n21 −1\n21+αℓ/2 −1 ≤\n2\n21+αℓ/2 =\n1\n2αℓ/2 ≤\n1\n2αℓ∗/2 = n−6.\n(34)\nCase 3: For level ℓ= ℓ∗+ 1 we get\nPr\n\u0002\nCℓ∗,[t′,τ)\n\u0003\n≤Pr\nh\nCW\nℓ∗,[t′,τ)\ni\n≤Pr\nh\nCW\nℓ∗,[t′,T)\ni\nLem. 3.3\n≤\n√n1 −1\n√n1+αℓ∗+1/2 −1\n≤\n√n\n√n1+αℓ∗+1/2 = n−6.\n(35)\nBy this case distinction, we know that the probability that any single level that just became\ncritical at time t′ ∈[0, τ) has probability at most n−6 to become critical within [0, τ). Taking a\nunion bound over the ℓ∗+ 2 = log log(n) + Θ(1) many levels and T = n4 many time steps finishes\nthe proof of Lemma 3.4.\nE\nLower Discrepancy Bound when Deletions are Dominant.\nTheorem E.1 gives a lower bound that shows that our result from Theorem 2.1 is tight in the sense\nthat there are insertion probability sequences that result in a logarithmic discrepancy. Our lower\nbound statement assumes that the potential Φα (see Section 2) is linear in n, which holds with\nhigh probability (see Lemma C.1). Similar to the results of [PTW15] the discrepancy is heavily\ndependent on the one-choice operations, ball deletions in our case. A long sequence of ball deletions\ncreates deep holes in the bins and the frequency of ball insertions must be high enough to re-balance\nthe load.\nTheorem E.1. Fix a step t and assume Φα(t) = O(n) for some constant c. Let T = t + n ln(n)\n2+4ϵ\nand assume that for every τ with t ≤τ ≤T we have β(τ) = 1/2 −ϵ with ϵ < 1/2 being an\narbitrary positive constant. Then the expected number of bins at the end of step T with load at least\n⌊m(T)/n⌋+ ln(n)/2 is Ω(√n).\nWe note that the above result also holds (with basically the same proof) for varying insertion\nprobabilities as long as fewer than (T −t)·(1/2−ϵ) balls are inserted during the time interval [t, T].\nThe proof idea for Theorem E.1 is as follows: We first show that, as long as in step t we have\nΦ(t) ≤c·n for some constant c (which holds with high probability), there exists a constant fraction\nof bins with a load which is at least m(t)/n (Lemma E.1). If now during the next O(n log n) steps\nmuch more balls are deleted than inserted, the average load will go down and one of the bins with\nload at least m(t)/n will receive no ball deletions at all. This bin will have a discrepancy of O(log n).\nThis holds even under the assumption that the bin does not receive any additional ball.\nLemma E.1. Assume Φ ≤c · n for some constant c and let r = r(n) denote the fraction of bins\nwith a load of at least ⌊m(t)/n⌋. Then r = Θ(1), i.e., a constant fraction of bins has load at least\n⌊m(t)\nn ⌋.\nProof. Let B1 denote the set of bins with a load of at least ⌊m(t)/n⌋and B2 the set of bins with\nfewer than ⌊m(t)/n⌋balls. Since r is defined as the fraction of bins in B1, we have |B1| = r · n and\n35\n\n|B2| = (1 −r) · n. This implies that there are at least (1 −r) · n holes below the level ⌊m(t)\nn ⌋and,\nin turn, at least (1 −r) · n balls above the average load of m(t)/n.\nLet m(t)/n+ℓbe defined as the average load of the bins in B1. Note that ℓ≥0 because m(t)/n\nis the average load and the bins in B1 have a higher load than the ones in B2. From the definition,\nit follows that\nℓ≥n · (1 −r)\nr · n\n= 1 −r\nr\n.\nSince the potential function Φ is minimal when all bins in B1 have load ℓ, we obtain\nr·n\nX\ni=1\neα· 1−r\nr\n≤Φ ≤c · n\nwhich implies r · eα· 1−r\nr\n≤c. If we view r = r(n) as function in n and assume r(n) = o(1) then in\nr(n) · eα\n\u0010\n1\nr(n)−1\n\u0011\n= r(n) · f(n)\nwith\nf(n) = eα\n\u0010\n1\nr(n)−1\n\u0011\nthe exponent is a function growing as 1/r(n) = ω(1). This implies that regardless of the size of α\n(as long as it is positive) the f(n) grows exponentially quicker than the r(n) = o(1) decays, and so\nthe whole r(n) · f(n) is in ω(1) and in particular not upper-bounded by the constant c.\nSince r(n) ≤1 ∈O(1), we conclude that r(n) = Θ(1).\nWith this, we can prove Theorem E.1:\nProof of Theorem E.1. First of all, a simple application of Chernoff bounds proves that, w.h.p.,\n⌊m(t)/n⌋≥⌊m(T)/n⌋+ ln(n)/2. Fix a bin i and define ℓ= T −t = n ln(n)/(2 + 4ϵ). Then for\nlarge n, the probability that during all steps τ with t ≤τ ≤T no ball is deleted from bin i can be\nbounded by (using the inequality 1 −x ≥e−2x for x ∈[0, 0.795])\n\u0012\n1 −1\nn\n\u0013(1−(1/2−ϵ))ℓ\n=\n\u0012\n1 −1\nn\n\u0013(1/2+ϵ)ℓ\n≥exp\n\u0012\n−(1 + 2ϵ) ℓ\nn\n\u0013\n= exp (−ln(n)/2) = n−1/2.\n(36)\nFrom Lemma E.1 we get that, at time t, a constant fraction of bins have load at least ⌊m(t)/n⌋.\nAnd by the above calculation, the expected number of those bins from which no ball is deleted\nis Ω(√n).\nThus, at time T, in expectation there are Ω(√n) bins that have a load of at least\n⌊m(t)/n⌋≥⌊m(T)/n⌋+ ln(n)/2.\n36\n"},"cleaned":{"kind":"string","value":"Balls and Bins and the Infinite Process with Random Deletions Petra Berenbrink∗ Tom Friedetzky† Peter Kling‡ Lars Nagel§ Abstract We consider an infinite balls-into-bins process with deletions where in each discrete step t a coin is tossed as to whether, with probability β(t) ∈(0, 1), a new ball is allocated using the Greedy[2] strategy (which places the ball in the lower loaded of two bins sampled uniformly at random) or, with remaining probability 1 -β(t), a ball is deleted from a non-empty bin chosen uniformly at random. Let n be the number of bins and m(t) the total load at time t. We are interested in bounding the discrepancy xmax(t) -m(t)/n (current maximum load relative to current average) and the overload xmax(t) -mmax(t)/n (current maximum load relative to highest average observed so far). We prove that at an arbitrarily chosen time t the total number of balls above the average is O(n) and that the discrepancy is O(log(n)). For the discrepancy, we provide a matching lower bound. Furthermore we prove that at an arbitrarily chosen time t the overload is log log(n) + O(1). For \"good\" insertion probability sequences (in which the average load of time intervals with polynomial length increases in expectation) we show that even the discrepancy is bounded by log log(n) + O(1). One of our main analytical tools is a layered induction, as per [ABKU99]. Since our model allows for rather more general scenarios than what was previously considered, the formal analysis requires some extra ingredients as well, in particular a detailed potential analysis. Furthermore, we simplify the setup by applying probabilistic couplings to obtain certain \"recovery\" properties, which eliminate much of the need for intricate and careful conditioning elsewhere in the analysis. 1 Introduction. Balls-into-bins processes have been formalized in the mathematical community since at least the early 18th century [dM18]. Their objective is, e.g., to allocate at random a set of balls into a number of bins such that the maximum number of balls per bin is minimized. These processes have been very well studied, with many results being largely folklore. However, even seemingly minor modifications of the protocol or the model tend to result in vastly different and often far more complex analyses as well as, occasionally, very surprising results. Apart from the undoubted usefulness as probabilistic paradigms, balls-into-bins processes have many applications, often under the bonnet as primitive black-box operations. Among others, this includes areas like (the obvious) load balancing and allocation of jobs to servers [CL97], routing in networks [CMMadH+98], hashing [CRSW13], dictionary data structures [DW05], design of file systems [LCB13], peer-to-peer networks [SMK+01], and many more. The most simple balls-into-bins process, which allocates m balls independently and uniformly at random to n bins, is very well understood. For example, if m = n then the maximum number of ∗ ( ). †Durham University, U.K. ( ). ‡Darmstadt ( ). §Loughborough University, U.K. ( ). 1 16 Oct 2025 balls (load) in any bin is, w.h.p.1, log(n)/ log log(n) · (1 + o(1)) [RS98]. In [ABKU99] the authors introduced the Greedy[d] process. Here, balls are allocated sequentially; each ball chooses d ∈N bins uniformly at random and is allocated to a least loaded of them. Greedy[1] is identical to the simple process described above. But if d ≥2, the maximum load is, w.h.p., reduced to log log(n)/ log(d) + O(1) [ABKU99] for m = n. This exponential decrease in the maximum load is often referred to as the power of two choices. Greedy[2] was later analyzed for m ≥n (the heavily loaded case), which led to the surprising result that, in contrast to Greedy[1], the difference between the maximum and average load is independent of m [BCSV06] (namely, w.h.p. it is log log(n)/ log(d) + O(1)). Balls-into-Bins with Deletions. We study Greedy[2] in the heavily loaded case where balls are not only inserted but also deleted. Specifically, we ask whether allocation via Greedy[2] can overcome the negative effect of random deletions (effectively \"holes\" allocated according to Greedy[1], which without insertions would cause a rift of depth p m n log n for large m). One can differentiate balls-into-bins with deletions depending on whether, each time a ball is inserted, the ball gets d fresh choices (insertion/deletion model) or whether the ball might be a previously deleted ball that is reinserted with its original d choices (reinsertion/deletion model). Recently, the authors of [BK22] showed that, somewhat surprisingly, even in the insertion/deletion model, an oblivious adversary2 can make sure that Greedy[2] does not overcome the negative effect of deletions: already for n = 4 bins, it can cause a maximum load of m/4+Ω(√m) with probability Ω(1). Intuitively, the lower bound uses the fact that if there is ever a bin i that contains far fewer (say k) balls than other bins, Greedy[2] is biased toward i for the next k throws. After filling the system to m balls with at least k insertions, the adversary exploits the insertion bias to perform k deletions that are now biased towards bin i. Adding replacement balls for the just deleted balls, one can show that enough of these land on bins ̸= i to cause the desired overload. Results in a Nutshell. We consider the insertion/deletion model for n ∈N bins with random but dynamic deletions, starting from an initially empty system. Each time step t, with probability β(t) a new ball with fresh 2 choices is inserted by Greedy[2]. Otherwise, with probability 1-β(t), a ball is deleted from a bin chosen uniformly at random among all non-empty bins. (All our results also hold for the protocol that does not delete from a random bin, but a ball chosen uniformly at random among all balls, see Appendix A.) Our main result can be seen as complementing this unfortunate lower bound of [BK22] for Greedy[2] with a positive result: In a model with random deletions (but still dynamic, that is, resulting in a fluctuating average load), Greedy[2]'s allocation overcomes the negative deletion effect as long as insertions dominate deletions ever so slightly in the recent past of a given time step t. This highlights a fundamental difference between our model and the one of [BK22]. Our result suggests that in many practical applications, where deletions (if at all) occur non-adversarially, Greedy[2] easily overcomes random Greedy[1]-type deletions. This holds true even if the system utilization (ratio between insertions and deletions) fluctuates significantly. See Section 1.1 for a full and formal description of our results. 1We say an event E happens with high probability (w.h.p.) if Pr [ E ] ≥1 -n-Ω(1). 2The adversary knows only the d choices and insertion time of each ball. Each time step it can either (a) insert a new ball with d fresh random bin choices or (b) delete the r-th most recently inserted ball (r chosen freely). In particular, it is oblivious to the exact load situation, since it cannot see to which of the d choices the allocation strategy (like Greedy[d]) assigned a given ball. 2 1.1 Our Contribution. The system state at any time can be modeled by a load vector x = (x1, . . . , xn) ∈Nn 0, where xi is the load of the i-th fullest bin. We use m(x) := ∥x∥1 to denote the total load of x and define, for clarity, xmax := x1 and xmin := xn as the maximum and minimum load of x, respectively. We say x is perfectly balanced if xmin, xmax ∈{ ⌊m(x)/n⌋, ⌈m(x)/n⌉}. The discrepancy disc(x) := xmax-m(x)/n of x is the difference between the current maximum and average load. In some cases, we also use the stronger absolute discrepancy adisc(x) := max { xmax -m(x)/n, m(x)/n -xmin } of x (the largest difference of any load to the current average load). With this, our process can be described as a random sequence x(t) t∈N0 of load vectors with total load m(t) := m x(t) , discrepancy disc(t) := disc x(t) , and absolute discrepancy adisc(t) := adisc x(t) at time t. We use mmax(t) := max { m(0), . . . , m(t) } to denote the maximum total load up to time t. The overload ovld(t) := xmax(t) -mmax(t)/n at time t is the difference between the current maximum and the highest average load up to time t. The height of a ball is its position in the bin modelled as a stack of balls counting from bottom to top. We assume a dynamic insertion probability β(t) that may change from step to step to any value in [0, 1]. In other words, insertions and deletions are governed by an infinite insertion probability sequence β(t) t∈N giving the (independent) probabilities of an insertion/deletion in each time step. We say value x ∈(0, 1) is bounded away from 0 (from 1) if x ≥0 + Ω(1) (x ≤1 -Ω(1)). While balls-into-bins processes typically aim to bound the discrepancy, this cannot work in general when deletions are involved. For example, the insertion probability sequence could contain a stretch that deletes m(t)/2 balls after some time t. Translating a folklore example (also mentioned in [BK22]) to our setting, deletions can be regarded as random \"holes\" allocated via Greedy[1]. This would cause a discrepancy of order ̃Θ( p m(t)/n), a bound trivially achieved by allocations via Greedy[1], even holds if the system starts out perfectly balanced. Thus, in this paper we aim to bound the overload. We characterize which (possibly local) properties an insertion probability sequence requires for Greedy[2] to achieve a small discrepancy. Also, in an infinite process it is quite impossible to always have a small discrepancy (or even overload). Thus, our main results (given below) all hold with high probability at an arbitrary (also super-exponentially large) point t in time: (a) Assume all insertion probabilities up to time t are bounded away from 0. Then the discrepancy at time t is O(log(n)) (Theorem 2.1) and the number of balls above average at time t is O(n) (Theorem 2.2). The bound on the discrepancy is tight if the insertion probabilities are bounded away from 1/2 from above in the previous O(n log(n)) time steps (see Appendix E). (b) Assume we are in a well (e.g., perfectly) balanced situation at time t. Then we can maintain an overload of log log(n) + O(1) for poly(n) steps as long as the insertion probabilities during that time are bounded away from 0 and 1. If, additionally, the insertion probabilities ensure an expected load increase for polynomially long subintervals (see definition of c-good time intervals in Section 3), we can maintain a discrepancy (instead of only overload) of log log(n) + O(1) (see Theorem 3.1 and Lemma 3.4 in Section 3 for the full, formal statements). (c) Our final result shows that we achieve an overload of log log(n) + O(1) at an arbitrary time t ∈N0 as long as the insertion probability sequence till that time is bounded away from 0 and 1 (Theorem 4.1). Similar to the previous result, if the insertion probabilities ensure an expected (possibly quite small) load increase for long subintervals during the previous n4 time steps, we even get a discrepancy of log log(n) + O(1) at time t. To the best of our knowledge, these are the first results for the discrete, heavily-loaded insertion/deletion setting that provide characterizations for when Greedy[2] achieves strong guarantees 3 on the discrepancy instead of only on the overload. Main Technical Contributions and Proof Idea. To prove that the discrepancy is O(log(n)) (see (a) above) we use a potential function argument similar to [PTW15, TW14]. To reduce the discrepancy to double-logarithmic (see (c) above) we use the excellent recovery properties of Greedy[2]: In [Czu00], Czumaj studied the recovery of Greedy[2] for a process that alternates between ball insertions and deletions, maintaining the total load invariant. He showed that the process is able to recover, i.e., reach a steady state distribution, in polynomial time. However, this does not reveal any properties (e.g., the discrepancy) of the steady state. In contrast to [Czu00] we cannot have a unique steady state as the total load varies over time. However, we can show via a path coupling argument that when we start two copies of our process in any two configurations of logarithmic discrepancy with the same number of balls - which we have at any time via (a) - it causes both configurations to converge to each other quickly (in o(n4) steps). This can be interpreted as a \"typical\" state to which all configurations with the same number of balls converge. Again, as in [Czu00] this does not yet reveal any further information about such a typical state. This is where (b) comes into play, which uses a layered induction to show that, starting from a certain class of configurations (with discrepancy at most log log(n)+c), the process remains in such a well-balanced configuration for a long time, Ω(n4) steps. That is, we maintain well-balancedness for longer than it takes to reach the typical state, implying that such typical states lie in this class of well-balanced configurations. Note that this deviates significantly from previous usages of layered induction in this area: For example, in [TW14] the authors used layered induction to reduce the discrepancy from O(log n) to O(log log n). Our layered induction is \"merely\" used to establish that a typical state (reached quickly due to the good recovery) has a low discrepancy. Apart from being flexible enough to deal with a rather general deletion model, our approach also allows for stronger probability guarantees (of order 1 -1/poly(n) instead of 1 -1/polylog(n) in [TW14]). In our layered induction we have to cope with two new problems in comparison with previous work: The total load m(t) changes over time, and we may not assume it to be linear in n. The classical layered induction proof of [ABKU99] heavily relies on the base case, which by means of a simple pigeon-hole argument states that there are at most n/c bins with c balls. This holds because in their setting the total load cannot exceed n. In our case, however, the total load is unbounded. We deal with this first problem by performing our layered induction only on the highest log log(n)+O(1) many load levels. This leads us to the second challenge, since the changing total load m(t) also implies that the (absolute) height of the considered levels changes over time. To deal with this, we carefully formulate our layered induction relative to the maximum average and use a flexible random walk argument to prove strong bounds on the number of balls above a certain level over the maximum average. This yields a bound on the overload measure, from which we derive the discrepancy bound - under the assumption that there was no large, recent drop of the total load. Note that in the case where such a large drop occurred, our lower bound mentioned in (a) shows that we cannot hope for a good discrepancy. To the best of our knowledge, this is the first layered induction for the case m ≫n as well as for a process with a changing load. 1.2 Related Work. There is a vast amount of literature on balls-into-bins games in a multitude of settings. For an overview see [Wie17] and for results with one random choice per ball see [BCN+15, BFK+18, LS23]. After a very brief overview of results with multiple choices we focus on processes with deletions. 4 Multiple Choice. The Greedy[d] process was introduced and analyzed in [ABKU99], although a similar result in a different setting had been shown in [KLMadH92]. In [BCSV06] the authors generalize the results to the heavily loaded case m ≫n. [TW14] presents a simpler analysis for the heavily loaded case, albeit showing a slightly worse upper bound on the maximum load. The author of [V ̈03] considers the always-go-left process Left[d] where bins are divided into d clusters of n/d bins each. Every ball randomly chooses one bin per clusters and is allocated to a least loaded among the chosen bins; ties are broken in favor of the leftmost cluster. This results in a maximum load of log log(n)/(d·φd) with φd ≤2 (φd ∈(1, 2) is the generalized golden ratio), w.h.p., a surprising improvement over [ABKU99]. For more recent results see [LSS22, LSS23, LSS24]. The potential function analysis for Part (a) is inspired by methods introduced in [PTW15] where the authors consider a very interesting variant of Greedy[d], which they call (1 + β)-process. Here each ball is allocated using Greedy[1] with probability 1 -β, and using Greedy[2] with the remaining probability β. The authors showed that the difference between maximum and average load is Θ(log(n)/β) for each β n. The infinite insertion/deletion model with adversarial deletions for the moderately loaded case is considered by [V ̈03] for the Left[d] process where the maximum system load is always bounded by h · n. He proves that the maximum (bin) load is ln ln(n)/(d · ln(φd)) + O(h) w.h.p. (again, φd ∈(1, 2) is the generalized golden ratio). Note that for h = Ω(log(n)), this is not better than the bound achieved by the trivial Greedy[1] allocation. The authors of [BCF+23] consider the moderately loaded case but for the reinsertion/deletion model. They introduce the Iceberg[d] process, combining Greedy[1] and Left[d]. Slightly simplified, the Iceberg[d] process allocates a ball via Greedy[1] if the target bin contains at most h + ̃O( √ h) balls that were allocated by Greedy[1]. Otherwise, either Left[d] is used (if there are currently fewer than n/d balls allocated by Left[d]) or the ball is allocated to bin 1. This process achieves a maximum load of h + log log(n)/(d log φd)) + ̃O( √ h). Note that this process must know h and keep track of the number of balls placed with specific strategies, making it difficult to use in distributed settings. In contrast, Greedy[d] places balls solely based on the selected bins' loads. The focus of [Czu00] is the analysis of the recovery time (the time it takes a process to move from an arbitrary to a \"typical\" state) in the heavily loaded case (m arbitrary). The author considers a setting starting with m arbitrarily placed balls followed by alternating deletions (either a random ball or from a random bin) and insertions. The recovery time under Greedy[d] allocation is proven to be polynomial in m and n. While parts of our analysis (Section 4) are inspired by this result, note that in our case the system load m fluctuates over time and that we do not only show fast recovery but actually characterize what the \"typical\" state looks like. As mentioned at the beginning of the introduction, [BK22] give lower bounds for Greedy[2] in the heavily loaded, adversarial insertion/deletion model. They also provide a more general lower 5 bound for any oblivious allocation strategy (like Greedy[2] or Left[2]) in the reinsertion/deletion model (with the system load capped at m): an oblivious adversary can force a maximum load of m/4 + mΩ(1) for n = 4 bins within the first poly(m) steps. On the positive side, they introduce the allocation strategy ModulatedGreedy for the insertion/deletion model, which achieves an overload of O(log(m)). To overcome their lower bound for Greedy[2], ModulatedGreedy decides probabilistically between two bin candidates, based on their current load. The algorithm must know global guarantees on the load situation (in particular, the maximum number of balls that will ever be present and the current maximum and minimum load). Queuing Systems. A prominent example of a queuing system is the supermarket model considered in, e.g., [VDK96, BCFV00, Mit01, LN05, LM05]. It can be seen as a variant of balls-into-bins process in continuous time. Here, balls typically arrive in a Poisson stream of rate λn and are allocated according to Greedy[d]. Bins process (delete) balls in first-in first-out order, with a processing time that is exponentially distributed with mean 1. These systems and their results can often be readily mapped to the balls-into-bins settings yielding results similar to those discussed above. However, the focus in queuing systems lies typically on properties like the balls' service times (the average time a ball spends in the system), how long it takes to reach an equilibrium state, or how the system behaves in the equilibrium. Note that the (expected) unit processing time together with λ 0. It states that the absolute discrepancy at any time t is, w.h.p., at most O(log n) higher than it was at the start. In particular, if the process starts with an absolute discrepancy of O(log n), it will almost certainly be O(log n) at time t. Theorem 2.1. Consider an insertion probability sequence β(t) t∈N bounded away from 0 by a constant β > 0. If α ≤β/16, then for any t ∈N0 we have E[Γα(x(t))] = O(n) · eα·adisc(0). Moreover, for any a ≥0 we have Pr adisc(t) -adisc(0) ≥a+2 α · ln n ≤n-a. In particular, if we start in a perfectly balanced situation with discrepancy 0 (e.g., empty system), the expected potential at an arbitrary time t is only linear. Equipped with the above, we can prove a strong upper bound on the number of balls that lie above the average at a given time t: Theorem 2.2. Fix an arbitrary step t. Let β ∈(0, 1] be a constant. Assume β(τ) ≥β for all 1 ≤τ ≤t and α ≤β/16. For any fixed non-negative land any fixed μ ∈(0, 1), there is a γ = γ(μ) so that, with probability at least 1 -n-(3+l), the number of balls above height ⌈m(t) n ⌉+ γ is at most μn. The proof is based on Lemma C.1 in Appendix C, an adaptation of a result in [LS22, LS21], which gives us a high-probability guarantee for the potential Γα(x(t)) being linear. In fact, the purpose of the adaptation of [LS22, LS21]'s results is two-fold: (i) to incorporate our deletions and their respective probability bounds, and (ii) to improve the high-probability guarantee from the original 1 -1/n3 (which was sufficient for the purposes of [LS22, LS21]). Note that the probability stated in Theorem 2.2 is inherited from Lemma C.1; apart from that, the theorem's actual proof is an entirely deterministic counting argument. The proof details may be found in Appendix C. 3 Maintaining a Well-balanced State. Before we state this section's main result (Theorem 3.1), we introduce the notion of c-good time intervals I for a given insertion probability sequence. Intuitively, such an interval guarantees us that the total system load increases in expectation in any sufficiently long subinterval of I. Definition 3.1 (c-good interval). Given a sequence β(t) (t ∈N) of insertion probabilities, we call a discrete time interval I c-good if for each subinterval (t1, t2] ⊆I with t2 -t1 ≥c · n we have Pt2 t=t1+1 β(t) /(t2 -t1) ≥1 2(1 + ε) for a constant ε > 0. Theorem 3.1 states that, w.h.p., our process, when started in a perfectly balanced configuration, remains well-balanced for at least n4 steps.3 Here, well-balanced (this is formalized later in Definition 3.2 and Lemma 3.4) is measured relative to the maximum average observed so far for general insertion probability sequences and relative to the current average if the considered time window is O(1)-good. Theorem 3.1. Consider an insertion probability sequence β(t) t∈N bounded away from 0 and 1. Let x(0) be a perfectly balanced configuration. Then, w.h.p., xmax(t) ≤mmax(t)/n+log log n+O(1) for any t ≤n4. Moreover, if there is a constant c ∈N such that the time interval [0, n4] is c-good, then, w.h.p., xmax(t) ≤m(t)/n + log log n + O(1) for any t ≤n4. 3In fact, our proof shows that starting from any well-balanced configuration, the situation remains well-balanced for polynomial many steps. 7 To prove Theorem 3.1, we conduct a layered induction. A major difficulty stems from the fact that we have to deal with a permanently changing average load. For this an important ingredient is Theorem 2.2, which gives us a high probability bound on the number of balls above the current average. This enables us to derive a base case for our layered induction, independent of the changing average load. The layered induction works roughly as follows: For each level l∈{ 0, 1, . . . , log log(n) + Θ(1) } we consider the number of balls at height greater than average +l. For each level we define a critical threshold αlthat decreases doubly exponentially with l. Moreover, each level has a safe threshold αl/2. We show that, as long as all levels remain below their critical threshold, any level that is above its safe threshold has a strong drift back towards the safe threshold. A standard result on random walks then shows that it is unlikely that during the poly(n) many considered time steps, any of the log log(n) + Θ(1) many levels (that all start below their respective safe threshold) crosses the critical interval [αl/2, αl). Formalizing Well-balanced Configurations. In the following, let ˆβ ∈(0, 1) be a constant upper bound on the insertion probability of our process, such that β(t) ≤ˆβ for all considered time steps t. For a height h ∈N0 and a configuration x define mh(x) as the number of balls that have height at least h in configuration x. For our process starting in an initial configuration x(0), we use the shorthand mh(t) := mh(x(t)). In our analysis, it will be helpful to measure the number of balls at or above some level l, which, in turn, is above a given base height h (which might vary over time). To this end, we introduce the notation m(h) l(x) := mh+l(x) and m(h) l(t) := mh+l(t). We consider l∗+ 2 = log log(n) + Θ(1) many levels l∈{ 0, 1, . . . , l∗+ 1 } with their critical thresholds αldefined as follows: αl:= 1-ˆβ 128ˆβ · n if l= 0, 32ˆβ 1-ˆβ · α2 l-1/n if αl-1 > r 3(1-ˆβ) 2ˆβ · n log(n), 12 log(n) if l= l∗:= min l αl-1 ≤ r 3(1-ˆβ) 2ˆβ · n log(n) , and 24 if l= l∗+ 1. (2) Note that the recursive definition ensures l∗= log log(n) + Θ(1). Moreover, it implies the following relations between the critical thresholds αland αl-1: Observation 3.1. For any l∈{ 1, 2, . . . , l∗+ 1 } and large enough n we have 8ˆβ 1 -ˆβ · α2 l-1 n ≤αl≤1 4 · αl-1. (3) Using the critical threshold αl, we define the valid interval of level l∈{ 0, 1, . . . , l∗+ 1 } as Vl:= [0, αl). Similarly, we define the level's safe interval as Sl:= [0, αl/2) and its critical interval as Cl:= [αl/2, αl). Definition 3.2. For a base height h ∈N0 and level l∈{ 0, 1, . . . , l∗+ 1 } we say level lof configuration x is h-valid if m(h) l(x) ∈Vl, h-safe if m(h) l(x) ∈Sl, and h-critical if m(h) l(x) ∈Cl. Configuration x is h-valid/ -safe/ -critical if all levels 0 to l∗+ 1 are h-valid/-safe/-critical. In proofs, we sometimes omit the base height if it is clear from the context. 8 Analysis. We start with a simple lemma that provides the probabilities for the increase and decrease of the number of balls at or above a given height h. Lemma 3.1. For any configuration x and any height h ∈N we have (a) Pr [ mh(t + 1) > mh(t) | x(t) = x ] = β(t) · mh-1(t) -mh(t) 2/n2 and (b) Pr [ mh(t + 1) m(h) l(x) i ≥2. (4) (b) If level l= l∗+ 1 of configuration x is h-critical and if n is large enough, then Pr h m(h) l(x′) m(h) l(x) i ≥√n. (5) The proof of Lemma 3.2 may be found in Appendix D. Next we formulate an auxiliary lemma about the probability that a random walk on integers reaches position b > 0 before position -a 0. We will apply this lemma to the values m(h) l(•) (basically corresponding to the random walks) to show that these values are unlikely to cross the critical interval Cl(on which, by Lemma 3.2, m(h) l(•) is biased by a factor of at least 2 towards the safe left side). Lemma 3.3. Consider a (not necessarily i.i.d.) random walk (St)t∈N0 on Z with initial position S0 = 0 and position St = Pt i=1 Xi after t steps with step width Xi ∈{ -1, 0, +1 } at time i. Let F = (Ft)t∈N0 denote the random walk's natural filtration. Assume there is r ∈R>0 \\ { 1 } such that for all t ∈N, Pr [ Xt = -1 | Ft-1 ] Pr [ Xt = +1 | Ft-1 ] = r. For x ∈Z define the stopping time τx := inf { t ∈N0 | St = x } indicating when the random walk reaches x for the first time. Then, for any a, b > 0, Pr [ τ+b 0. Consider the random variable Y = Pt2 t=t1+1 Yt where Yt = 1 with probability β(t) and Yt = 0 otherwise. Note that Y counts the number of balls added in the time interval (t1, t2]. By c-goodness we have E[Y ] ≥(1 + ε) · t2-t1 2 . A simple application of Chernoff yields Pr Y 0. (8) Thus, we can take a union bound over the poly(n) many subintervals (t1, t2] ⊆[0, n4] of length at least c·n to get that, w.h.p., the average load at the end of the subinterval is at least the average load at its beginning. All other subintervals have length 0 in Definition 3.1 is an artifact of our analysis technique and can be reduced to o(1) or even eliminated. The same holds for our assumption that the insertion probability must be bounded away from 1 (note that for β = 1 the result follows from [BCSV06]). We note that the first of the two results in the theorem also holds when first m ≥n balls are inserted and the process then alternates between insertions and deletions. Analysis Overview. The proof idea of Theorem 4.1 is as follows: For any time t ∈N0, Theorem 2.1 implies (w.h.p.) a logarithmic absolute discrepancy at time t′ := t -n4. We then use a path coupling argument similar to [Czu00] to show that our process causes any load situation with logarithmic absolute discrepancy to recover quickly to a \"typical\" state. More exactly, any two (different) such load situations at time t′ can be coupled such that (w.h.p.) they become identical at time t. At this point however, we don't know how such a \"typical\" looks like (and whether it has small maximal load). This is where Theorem 3.1 comes into play. It tells us that if we start in a perfectly balanced load situation at time t′, our process maintains (w.h.p.) a double-logarithmic maximal load for at least n4 steps. A simple union bound over these two high probability bounds then implies that the \"typical\" load situation at time t must also have a double-logarithmic maximal load. Analysis. We now formalize the above idea. To this end, we first introduce a measure for the similarity of two given load situations. Definition 4.1. Consider two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 = ∥y∥1. The transformation distance between x and y is ∆(x, y) := ∥x -y∥1/2. Note that the transformation distance obeys the identity ∆(x, y) = Pn i=1 max { 0, xi -yi } and corresponds to the minimal number of ball movements to transform x into y (and vice versa). Moreover, any load vector x can be transformed into the perfectly balanced load vector by moving the at most n · disc(x) ≤n · adisc(x) balls above average. As an immediate consequence, we get the following observation. Observation 4.1. For two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 = ∥y∥1, we have ∆(x, y) ≤2n · max { adisc(x), adisc(y) }. The following lemma leverages a result from [Czu00] to show that one step of our process tends to reduce transformation distance between two given load vectors. In [Czu00], the author considers basically the same setting as ours, but with guaranteed alternating insertions/deletions. Lemma 4.1 basically shows that their proof transfers directly to our more general setting with insertion probability sequences. 11 Lemma 4.1. Consider two load vectors x, y over n ∈N bins with identical total loads ∥x∥1 = ∥y∥1. Assume ∆(x, y) = 1 and let x′ and y′ denote the load vectors after applying one step of our process with insertion probability β ∈[0, 1] to x and y, respectively. There is a coupling of x′ and y′ such that (a) E[∆(x′, y′)] ≤∆(x, y) = 1 and (b) Pr [ ∆(x′, y′) ̸= ∆(x, y) ] ≥(1 -β)/n. Proof. For the random decision whether to insert or delete a ball in x and y, we use the identity coupling that either inserts a ball in both processes (with probability β(t)) or deletes a ball in both processes (with probability 1-β(t)). It remains to couple the random bin choices in a deletion step and the two random bin choices of Greedy[2] in an insertion step. Assume the considered step is a deletion step. Then we use the first coupling described in [Czu00, Section 6] (who consider alternating insertions/deletions instead of our more flexible insertion probability sequences). In this case, [Czu00, Claims 6.1 and 6.2] together yield E[∆(x′, y′) | Deletion Step] ≤∆(x, y) = 1 and Pr [ ∆(x′, y′) ̸= ∆(x, y)|Deletion Step ] ≥1/n. Note that this already yields the lemma's second statement, since a deletion step occurs with probability 1 -β. Similarly, for an insertion step, we use the second coupling described in [Czu00, Section 6], where the exact same insertion rule is used (again, in a setting with alternating insertions/deletions). In this case, the proof of [Czu00, Lemma 5.1] implies E[∆(x′, y′) | Insertion Step] ≤∆(x, y) = 1. Together, we get the desired result. The next lemma characterizes the expected and high probability crossing time of a simple, αlazy reflecting random walk on { 0, 1, . . . , D } (i.e., a random walk that has the same probability to move left/right and that stays put with probability α or if it tries to move out of bounds). We use the lemma for our coupling result in Lemma 4.3. Lemma 4.2. Let D ∈N0 and α ∈[0, 1). Consider a simple, α-lazy reflecting random walk (Wt)t∈N0 on { 0, 1, . . . , D } started in W0 = D. Let T := min { t ∈N | Wt = 0 }. Then E[T] = D·(D+1)/(1α). Moreover, for any n ∈N and a > 0 we have Pr [ T ≥a · log(n) · 2E[T] ] ≤1/n-a. Proof. Let Ei denote the expected time for the random walk to reach 0 if started at position i ∈{ 0, 1, . . . , D }. Clearly, E0 = 0 and ED = E[T]. Note that we have the recurrence relation Ei = 1+Ei ·α+(Ei-1 +Ei+1)·(1-α)/2 for i ∈{ 1, 2, . . . , D -1 } and ED = 1+ED-1 ·(1-α)/2+ ED ·(1+α)/2. From this one can first deduce Ei = 2(D-i+1)/(1-α)+Ei-1 for i ∈{ 1, 2, . . . , D } and, then, E[T] = ED = D · (D + 1)/(1 -α). For the high probability bound, we observe that Markov's inequality implies Pr [ T ≥2E[T] ] ≤ 1/2. Thus, after at most a · log(n) repetitions of such phases of length 2E[T], the random walk has reached 0 with probability at least (1/2)a·log(n) = n-a. The following lemma shows that it is possible to couple two instances of our process with a logarithmic absolute discrepancy; they will be in the same distribution after polynomial time. Lemma 4.3. Consider two instances of our random processes x(t) t∈N0 and y(t) t∈N0 with ∥x(0)∥1 = ∥y(0)∥1 and adisc(x(0)), adisc(y(0)) ≤3 α · log n. Assume both processes use the same insertion probability sequence β(t) t∈N0 bounded away from 0 and 1. Then there is a coupling of x(t) t∈N0 and y(t) t∈N0 for whose coupling time τ := min t ∈N0 ∆ x(t), y(t) = 0 we have, w.h.p., τ = O n3 · (log n)3 . 12 Proof. Consider any time t ∈N0 and define ∆(t) := ∆ x(t), y(t) . There is a sequence x(t) = x(0)(t), x(1)(t), . . . , x(∆(t))(t) = y(t) of load vectors such that for all indices i ∈{ 1, 2, . . . , ∆(t) } we have ∆(i)(t) := ∆ x(i-1)(t), x(i)(t) = 1. By Lemma 4.1(a), there is a coupling of each pair x(i-1)(t + 1), x(i)(t + 1) such that E ∆(i)(t + 1) ≤1. This implies a coupling of x(t + 1) and y(t + 1) with E ∆(t + 1) ≤E ∆(t) X i=1 ∆(i)(t + 1) = ∆(t) X i=1 E h ∆(i)(t + 1) i ≤∆(t). (9) Similarly, Lemma 4.1(b) implies Pr [ ∆(t + 1) ̸= ∆(t) | ∆(t) > 0 ] ≥1 -β(t) n . (10) Consider the random process ∆(t) t∈N0. Let D := n · 10 α · log n = O(n · log n) and define the stopping time T1 := min { t ∈N0 | ∆(t) > D } . (11) Let β := sup { β(t) | t ∈N0 } T2, the majorization implies ∆(T2) ≤W(T2) = 0 and, thus, τ ≤T2. By applying Theorem 2.1 with a = 5 to the first n4 time steps and using a union bound, w.h.p. both x(t) t∈N0 and y(t) t∈N0 maintain an absolute discrepancy of at most adisc(0) + 7 α · log n = 10 α ·log n during the first n4 steps. By Observation 4.1, this implies a transformation distance of at most n· 10 α ·log n = D and, thus, T1 > n4. By Lemma 4.2, w.h.p., T2 = O n3 ·(log n)3 . Thus, by a union bound we have, w.h.p., T1 ≥T2 and T2 = O n3 · (log n)3 . As argued above, together these imply τ ≤T2 = O n3 · (log n)3 . With the coupling at hand we are ready to prove our main result. Proof of Theorem 4.1. Note that for t ≤n4, the statement follows immediately from Theorem 3.1. For larger t, Theorem 2.1 gives us, w.h.p., xmax(t -n4) -xmin(t -n4) ≤2c · log n. By this and by shifting each time step forward by n4 steps, it is sufficient to prove the theorem's statement for t = n4 starting from an initial load vector x(0) with xmax(0) -xmin(0) ≤2c · log n. Define an initial load vector y(0) with total load ∥y(0)∥1 = m(0) that is perfectly balanced. By Lemma 4.3 there is a coupling of x(t) t∈N0 and y(t) t∈N0 such that, w.h.p., x(t) = y(t) for all t = ̃Ω(n3) (so, in particular, for t = n4). Note that Theorem 3.1 applies to y(t) t∈N0. This gives us the corresponding high probability guarantees from Theorem 3.1 for y(n4) which, by leveraging the coupling, hold w.h.p. for x(n4), finishing the proof. Acknowledgments. Petra Berenbrink's research was funded by the DFG project Distributed and Collaborative Systems of Agents (project number 411362735) as part of the Research Unit (Forschungsgruppe) Algorithms, Dynamics, and Information Flow in Networks (FOR 2975). 13 References [ABKU99] Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. 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In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 979-990. Springer, 2014. Full version available on arXiv. [V ̈03] Berthold Vöcking. How asymmetry helps load balancing. Journal of the ACM, 50(4):568-589, 2003. [VDK96] Nikita Vvedenskaya, Roland Dobrushin, and Fridrikh. Karpelevich. Queueing system with selection of the shortest of two queues: An asymptotic approach. Problemy Peredachi Informatsii, 32(1):20-34, 1996. [Wie17] Udi Wieder. Hashing, load balancing and multiple choice. Found. Trends Theor. Comput. Sci., 12(3-4):275-379, 2017. 16 Appendix. A Deletion of a Random Ball. Consider an insertion probability sequence β(t) t∈N. Let X = (x(t))t∈N0 be the process where balls are deleted from randomly chosen non-empty bins during a deletion step. Remember that configuration x(t) is the load vector after step t with xi(t) denoting the i-th fullest bin. Similarly, let Y = (y(t))t∈N0 be the process in which every ball is deleted with the same probability during a deletion step. Our goal is to show that process Y is not worse (w.r.t. discrepancy/overload) than process X. We formalize this using the standard notion of majorization: For a configuration x let Sk(x) = Pk i=1 xi be the total load of the k fullest bins. We say a configuration x majorizes configuration y if Sk(y) ≤Sk(x) for all k ∈{ 1, 2, . . . , n }. Note that this immediately implies that y has a smaller maximal load than x. Thus, we can formalize the desired result via the following theorem: Theorem A.1. Starting from an empty system, there exists a coupling between processes X and Y such that x(t) majorizes y(t) for all t ∈N0. Coupling. Let us define a suitable coupling between processes X and Y . To this end, we use the identity coupling for the decision whether a ball is allocated (with probability β(t)) or deleted (with probability 1 -β(t)). As a result, the total load in both processes remains the same at any time; we denote it by m(t). It remains to couple the random choices for allocation and for deletion steps from configuration x(t) to configuration x(t + 1): Allocation Step: Here we use the identity coupling. That is, both processes use the same two random bin choices for Greedy[2]. Deletion Step: Here we number the m(t) balls in both configurations x(t) and y(t) from left to right (and arbitrarily within each bin). We then draw a uniformly random number z(t) from [0, 1) and use it to determine the ball deletions as follows: (a) Y deletes the ball i ∈{ 1, . . . , m(t) } if and only if i -1 m(t) ≤z(t) Sr(x(t)) would imply that ball j lies in a bin > r. The inequality i ≤j is a simple consequence of our coupling, since the probabilities for ball i being deleted in Y are constant (w.r.t. i) while the probabilities for ball j being deleted in X are monotonously non-decreasing (w.r.t. j). With this, we are ready to prove the main result of this section. Proof of Theorem A.1. We use the coupling defined above and prove the statement via induction over t, with the trivial base case Sk(x(0)) = 0 = Sk(y(0)) for all k. For the inductive step we show that the coupling maintains majorization when going from x(t) to x(t + 1) during time step t + 1. If a ball is inserted during time step t + 1, this follows immediately from [ABKU99, Lemma 3.4]. Thus, in the rest of this proof we consider a deletion during time step t + 1. Moreover, as in Observation A.1, we use (a) r and s to denote the bins from which X and Y delete a ball in step t, respectively; and (b) j and i to denote the balls deleted by X and Y , respectively. A plateau of a configuration x is a maximal set of consecutive bins having the same load. Note that when going from x(t) to x(t+1) by deleting a ball from bin r, we have to re-sort the bins after the deletion. This will swap bin r with the rightmost bin r′ of the plateau in which r lies. In other words, we can think of X as deleting a ball from some bin r′ ≥r instead of bin r. Formally we get x(t + 1) = x(t) -er′, where ei denotes the i-th unit vector in Rn. Similarly, Y deletes a ball from the rightmost bin s′ ≥s in the plateau containing bin s, yielding y(t + 1) = y(t) -es′ So assume x(t) majorizes y(t). That is, Sk(y(t)) ≤Sk(x(t)) for all k ∈{ 1, 2, . . . , n }. We show by case distinction that Sk(y(t + 1)) ≤Sk(x(t + 1)) for all k ∈{ 1, 2, . . . , n }: Case 1: k Sk(x(t + 1)) and Sk-1(y(t + 1)) ≤Sk-1(x(t + 1)).4 This implies yk(t + 1) > xk(t + 1). Now note that k lies in the plateau of y(t) between s and s′, such that for all q ∈{ k + 1, k + 2, s′ -1 } we have yq(t + 1) = yk(t + 1) > xk(t + 1) ≥xq(t + 1) (20) (the last inequality using that x(t + 1) is sorted non-increasingly) and, similarly, ys′(t + 1) = yk(t + 1) -1 ≥xk(t + 1) ≥xq(t + 1). (21) This implies Ss′(y(t)) = Ss′(y(t + 1)) + 1 > Ss′(x(t + 1)) + 1 = Ss′(x(t)), contradicting the induction hypothesis. Together, this shows that x(t + 1) majorizes y(t + 1). B Omitted Material from Proof of Theorem 2.1. In this section we prove Theorem 2.1, which we restate for convenience: Theorem 2.1. Consider an insertion probability sequence β(t) t∈N bounded away from 0 by a constant β > 0. If α ≤β/16, then for any t ∈N0 we have E[Γα(x(t))] = O(n) · eα·adisc(0). Moreover, for any a ≥0 we have Pr adisc(t) -adisc(0) ≥a+2 α · ln n ≤n-a. We use the potential functions from [PTW15] and show that, at an arbitrary point of time, the expected potential is linear in n. Since the authors of [PTW15] considered a process without ball deletions, we extend the potential function analysis to capture ball deletions. Recall that the process works as follows: In every time step a ball is inserted with probability β(t) using two choices. Otherwise, with probability 1 -β(t), a random non-empty bin is picked and a ball from that bin is deleted. The three potential functions defined in [PTW15] are: Φα(x(t)) = n X i=1 eα·(xi(t)-m(t)/n); Ψα(x(t)) = n X i=1 e-α·(xi(t)-m(t)/n); Γα(x(t)) = Φα(x(t)) + Ψα(x(t)) where x(t) = (x1(t), ..., xn(t)) is the normalized load vector. For convenience, we use yi(t) := xi(t) -m(t)/n for the difference between the load in the ith bin and the average. The remainder of the section is dedicated to prove Lemma B.8 which directly implies Theorem 2.1. For this proof, we require insertion and deletion probabilities. Given that a ball is deleted in round t, we denote by qi(t) the probability that it is removed from the ith bin. And given that a ball is inserted, we denote by pi(t) the probability that it is inserted into the ith bin. If no bin is empty, then qi(t) = 1/n because the protocol deletes a ball from a non-empty bin chosen uniformly at random. If there are s empty bins, then the empty bins, which come last in the normalized array, have probability qi(t) = 0, and the remaining bins have probability qi(t) = 1/(n -s). pi(t), on the other hand, is independent of t. pi(t) = pi = i n 2 - i-1 n 2 because for the ith bin to be chosen in the normalized array, the ith bin must be among the choices next to a bin with the same or a smaller index. (W.l.o.g. we assume that if bins of equal load are chosen, the bin with the larger index receives the ball.) From this we can derive the following simple observation: 4For k = s this holds by the previous case. 19 Observation B.1. P i≤n/4 pi ≤1/4 -ε and P i≥3n/4 pi ≥1/4 + ε for any ε ≤3 16. Proof. The probability for any of the first n/4 bins receiving the new ball is X i≤n/4 pi = X i≤n/4 i n 2 - i -1 n 2 = n/4 n 2 = 1 16. The probability for any of the last n/4 bins receiving the new ball is X i>3n/4 pi = X i>3n/4 i n 2 - i -1 n 2 = n n 2 - 3n/4 n 2 = 1 - 3 4 2 = 7 16 Let ˆy(t) = (ˆy1(t), . . . , ˆyn(t)) denote the load vector above average before normalization. By combining the probabilities β, pi and qi, we can express ˆyi(t + 1), i.e. the ith bin's load above average before normalization at t + 1, as a function of its previous load yi(t): ˆyi(t + 1) = yi + 1 -1/n, with probability β(t) · pi (ball is allocated to ith bin), yi -1/n, with probability β(t) · (1 -pi) (ball is allocated to other bin), yi -1 + 1/n, with probability (1 -β(t)) · qi (ball is deleted from ith bin), yi + 1/n, with probability (1 -β(t)) · (1 -qi) (ball is deleted from other bin) Let Φi α(x(t)) = eα·yi(t) and Ψi α(x(t)) = e-α·yi(t) be the local potentials in bin i. And let ∆ˆΦi α := Φi α(ˆx(t+1))-Φi α(x(t)) and ∆ˆΨi α := Ψi α(ˆx(t+1))-Ψi α(x(t)) be the local potential changes in the ith bin (before normalization). Then: ∆ˆΦi α = eα(yi+1-1/n) -eαyi = eαyi · (eα(1-1/n) -1), with probability β(t) · pi, eα(yi-1/n) -eαyi = eαyi · (e-α/n -1), with probability β(t) · (1 -pi), eα(yi-1+1/n) -eαyi = eαyi · (eα(-1+1/n) -1), with probability (1 -β(t)) · qi, eα(yi+1/n) -eαyi = eαyi · (eα/n -1), with probability (1 -β(t)) · (1 -qi) ∆ˆΨi α = e-α(yi+1-1/n) -eαyi = e-αyi · (e-α(1-1/n) -1), with probability β(t) · pi, e-α(yi-1/n) -eαyi = e-αyi · (eα/n -1), with probability β(t) · (1 -pi), e-α(yi-1+1/n) -eαyi = e-αyi · (e-α(-1+1/n) -1), with probability (1 -β(t)) · qi, e-α(yi+1/n) -eαyi = e-αyi · (e-α/n -1), with probability (1 -β(t)) · (1 -qi) We use these local changes to prove different bounds on the overall potential changes ∆Φα = Φα(x(t + 1)) -Φα(x(t)) (in Lemma B.1, Lemma B.2, Corollary B.1 and Lemma B.5) and ∆Ψα = Ψα(x(t + 1)) -Ψα(x(t)) (in Lemma B.3, Lemma B.4, Corollary B.2 and Lemma B.6). Lemma B.1 (similar to a part of Lemma A.5 in [TW14]). If, for a constant β, β(t) ≥β for all t and if α ≤ε·β 3 ≤1 16, then E[∆Φα | x(t)] ≤ n X i=1 eαyi β(t) pi(eα -1) -α n + (1 -β(t)) 1 n(e-α -1 + α) + 2α2 n2 . 20 Proof. We compute the expected potential difference: E[∆Φα | x(t)] = n X i=1 β(t)pi · (eα(1-1/n) -1) · eαyi + β(t)(1 -pi) · (eα(-1/n) -1) · eαyi +(1 -β(t))qi · (eα(-1+1/n) -1) · eαyi + (1 -β(t))(1 -qi) · (eα(1/n) -1) · eαyi = n X i=1 β(t)eαyi pi · e-α/n · (eα -1) + e-α/n -1 +(1 -β(t))eαyi qieα/n(e-α -1) + eα/n -1 ≤ n X i=1 β(t)eαyi pi(eα -1) + e-α/n -1 + (1 -β(t))eαyi qi(e-α -1) + eα/n -1 = n X i=1 eαyi β(t)pi(eα -1) + (1 -β(t))qi(e-α -1) + eα/n -1 + β(t)(e-α/n -eα/n) ≤ n X i=1 eαyi β(t)pi(eα -1) + (1 -β(t))qi(e-α -1) + α n + 2α2 n2 -2β(t)α n ≤ n X i=1 eαyi β(t)pi(eα -1) + (1 -β(t)) 1 n(e-α -1) + α n + 2α2 n2 -2β(t)α n (22) = n X i=1 eαyi β(t) pi(eα -1) -α n + (1 -β(t)) 1 n(e-α -1 + α) + 2α2 n2 Inequality (22) holds because, if no bin is empty, all qi = 1/n, and if bins are empty, then the qi are decreasing while the term (1 -β(t)) · (e-α -1) · eαyi is increasing as e-α -1 0 and E[∆Φα | x(t)] ≥ -α·β·ε 4·n · Φα. Then either Φα n/3 eαyi β(t) 3 2n(eα -1) -α/n + (1 -β(t)) 1 n(e-α -1 + α) + 2α2 n2 ≤ Φα,≤n/3 n β(t) ((1 -4ε)(eα -1) -α) + (1 -β(t))(e-α -1 + α) + 2α2 n +Φα,>n/3 n β(t) 3 2(eα -1) -α + (1 -β(t))(e-α -1 + α) + 2α2 n ≤ -3 · β(t) · α · ε n · Φα + α n · Φα,>n/3 ≤ -3 · β · α · ε n · Φα + α n · Φα,>n/3 Recall that we assume E[∆Φα | x(t)] ≥-α·β·ε 4·n · Φα. -α · β · ε 4 · n · Φα ≤ -3 · α · β · ε n · Φα + α n · Φα,>n/3 ⇒ 2 · β · ε · Φα ≤ Φα,>n/3 If Φα 0 implies Ψα ≥n 4 · e4·α·B/n, we get: n · ε 16 · e4·α·B/n ≤ε 4 · Ψα ≤Φα ≤ Φα,>n/3 · 1 2 · β · ε ≤ n 3 · β · ε · e3·α·B/n =⇒ eα·B/n ≤ 6 β · ε2 Finally, it follows that: Γα = Φα + Ψα ≤5 ε · Φα ≤5 ε · n 3 · β · ε · e3·α·B/n ≤5 ε · n 3 · β · ε · 6 β · ε2 3 = 360 · n ε8 · β4 Lemma B.6 (similar to Lemma A.8 in [TW14]). Suppose β(t) ≥β, yn/4 1+ε n for i > 2n/3 and e-α -1 2n/3 e-αyi β(t) 1 + ε n (e-α -1) + α n + (1 -β(t)) 1 n(eα -1 -α) + 2α2 n2 ≤ Ψα,≤2n 3 · α n + Ψα,> 2n 3 n β(t)ε(e-α -1) + α2 + α3 2 ≤ Ψα,≤2n 3 · α n + Ψα,> 2n 3 n -αβ(t)ε + α2 ≤ Ψα,≤2n 3 · α n + Ψα,> 2n 3 · -αβ(t)ε 2n ≤ Ψα,≤2n 3 · α n + Ψα,> 2n 3 · -αβε 2n Using E[∆Ψα | x(t)] ≥-α·β·ε 4·n · Ψα, we get: -α · β · ε 4 · n · Ψα ≤ Ψα,≤2n 3 · α n -Ψα · αβε 2n Ψα ≤ 4 β · ε · Ψα,≤2n 3 If Ψα 0. If E[∆Φα | x(t)] ≤-ε·β·α 4·n · Φα, apply Lemma B.4: E[∆Γα | x(t)] = E[∆Φα | x(t)] + E[∆Ψα | x(t)] ≤ -ε · β · α 4 · n · Φα + α -Ψα · α · β · ε n + α ≤ -α · β · ε 4 · n · Γα + α If this is not the case, apply Lemma B.5 which distinguishes two cases: If Φα ⌈m(t) n ⌉ πb(t) = eα + k X j=2 eα·j -eα·(j-1) = eα + (e2α -eα) + (e3α -e2α) + · · · + (e(k-1)α -e(k-2)α) + (ekα -e(k-1)α). 29 This is a telescoping sum evaluating to Πi(t) = ekα. This however is precisely this bin's potential contribution in the potential function Φ, so that we can conclude that Π(t) = X ball b πb(t) = X bin i X ball b in bin i πb(t) = X bin i Πi(t) = X bin i eαX+ i (t) = Φ(t). Consider a height ⌈m(t) n ⌉+ k for some k ≥2; the precise value will be fixed below. Assume that there is a set B of at least μn balls at height ⌈m(t) n ⌉+ k or higher (if no such set exists the proof is finished); the precise distribution of the balls in bins is irrelevant. Each of those at least μn many balls contributes a potential of at least eαk -eα(k-1) = eαk · 1 -1 eα ≥ε · eαk for some constant ε = ε(α) > 0, for a total potential contribution of those balls of at least ΠB(t) ≥ε · μn · eαk. Now ΠB(t) > c · n whenever k > 1 α ln c εμ -1, so letting d = 1 α ln c εμ proves the theorem (recall that the probability is inherited from Lemma C.1 - the actual argument hee in the proof building on top of it is entirely deterministic). Lemma C.1. Assume the premise of Theorem 2.2 holds. Then Γα(x(t)) = O(n) with probability at least 1 -n-(3+l) for any fixed non-negative l, where the constant factor hidden in the big-Oh depends on l. Proof. The proof is a minor adaptation of Theorem 5.3 on page 18 in Los and Sauerwald's [LS21], which is a more detailed version in arXiv of their [LS22]. All references in Appendix C to their lemmas etc. are with respect to the numbering in their arXiv version [LS21]. As mention before, the purpose of the adaptation of [LS21]'s results is two-fold: (i) to incorporate our deletions and their respective probability bounds, and (ii) to improve the high-probability guarantee from the original 1 -1/n3 (which was sufficient for the purposes of [LS22, LS21]) to 1 -1/n3+lfor any arbitrary but fixed l> 0. To show this result we adapt Theorem 5.3 in [LS21] as follows. The lin the exponent of the statement comes from Theorem 2.2. Theorem C.1 (cf. Theorem 5.3 in [LS21]). Consider any probability vector p that is (i) nondecreasing in i, i.e., pi ≤pi+1 and (ii) for constant ε, P i≥3n/4 pi ≥1/4+ε and P i≤n/4 pi ≥1/4-ε. Then, for any t ≥0 and α2 ≤β·ε 110, c = cε,α2 = 2 · 40 · 1283 · ε-7 · 4 · α-1 2 , Pr h ∩s∈[t,t+n log5(n)]{Γ(s) α2 ≤2cn} i ≥1 -n-(3+l). Proof. The proof is almost identical to that of Theorem 5.3 in [LS21]. The chain of modifications starts with Lemma 5.5 in [LS21] where the restriction Γ(t) 1 ≤cn9 occurs. An application of Markov's inequality turns the n9 into a probability 1/n8, which subsequent lemmas etc. further decay to 1/n3. The main idea to make the result work for us is to modify the restriction on Γ(t) 1 to Γ(t) 1 ≤cn9+l, which accordingly will turn into a probability 1/n8+l, only to subsequently be eroded to 1/n3+l. Basically we have the exact same results with the exact same proofs as in in [LS21] but with some explicit constant parameters replaced with their respective original values plus l. We will need to 30 make some equally straightforward modification of [LS21]'s results to make them compatible with our potential function analysis. We will use two instances of the potential function Γα which only differ in α. The first one, Γα1, uses α1 ≤β·ε 3 , the second one, Γα2, uses α2 ≤ α1 4·9.1. (Explanation: α1 and α2 must fulfil the requirements of the Potential Function Analysis in the previous section and of Lemma C.2.) From Lemma B.8 we can derive, for n and c large enough, i.e. c ≥4160002·n α·β3·ε9 , that Pr Γα1(x(t)) ≥c · n9+l 0, ε′ α2 > 0 defined as above, Pr ∪s∈[t-n·log2(n),t] Γ(s) α2 ≤2c′ ε′α2 · n ≥1 -2 · c · n-(8+l). Proof. The proof is almost identical to that of Lemma 5.6 in [LS21]. In fact the modifications immediately follow from those made to Lemma 5.5 (here: Lemma C.2). 31 We are now in a position to finish the proof of Theorem C.1, a part that essentially mimics Section 5.2, with just some few slightly adapted coefficients. We do of course not wish to copy a page or two from another paper, and therefore just briefly highlight the modifications. The application of Lemma 5.6 in [LS21] (here Lemma C.3) gives a lower bound of 1 -2cn-(8+l) instead of 1 -2cn-8 for the probability. Further down, the application of the union bound gives Pr \\ u∈[t-n log2 n,t+n log5 n] {Γ(u) 1 ≤cn9+l} ≥1 -2 log5 n n7+l. Then, the application of Theorem A.4 in [LS21] gives Pr h X(u) r ≥X(r) r + cn 2 i ≤exp - c2n2/4 2 · (2n log5 n) · (2n1/3) + 2 log5 n n7+l ≤3 log5 n n7+l. Further down we find that if a red phase starts at time r, then with probability 1 -3 log5 n n7+l, Γ(u) 2 will always be ≤3cn 2 . Taking union bounds as in Section 5.2 in [LS21] yields Pr [ r∈[s,t+n log5 n] [ u∈[r,t+n log5 n] {X(u) r > 3cn 2 } ≤3 · log5 n n7+l· (4n2 log10 n) ≤1 2n-4+l. Concluding as in [LS21], with probability 1 -1 2n-4+l-2cn-8+l≥1 -n-4+l, it holds that Γ(r) 2 ≤ 3cn 2 for all time-steps r which are within a red phase in [s, t + n log5 n] ⊆[t, t + n log5 n]. Since Γ(r) 2 ≤T ≤cn holds (deterministically) by definition for all time-steps r within a green phase, the theorem follows. This concludes the proof sketch of the Lemma. D Omitted Material from Section 3. We first present the proof of Lemma 3.2. We restate it for convenience: Lemma 3.2. Consider an h-valid configuration x and let x′ denote the (random) configuration after one step of our process with the insertion probability β(t) ≤ˆβ m(h) l(x) i ≥2. (4) (b) If level l= l∗+ 1 of configuration x is h-critical and if n is large enough, then Pr h m(h) l(x′) m(h) l(x) i ≥√n. (5) Proof. For the sake of readability, we abuse notation in this proof by omitting the base height h when talking about the number of balls that have at least some level l(i.e., we write ml(•) instead of m(h) l(•)). Similarly, we refer to the notations h-valid and h-critical simply as valid/critical within the proof. 32 (a) Consider a critical level l∈{ 1, 2, . . . , l∗} of configuration x. Since x is valid, level l-1 and l+ 1 are valid. Using Lemma 3.1 and Observation 3.1 we calculate Pr [ ml(x′) ml(x) ] ≥1 -β(t) β(t) · ml(x) -ml+1(x) ml-1(x) -ml(x) 2 · n ≥1 -β β · αl/2 -αl+1 (αl-1 -αl/2)2 · n ≥1 -β β · αl/4 α2 l-1 · n ≥2. The first inequality follows from Lemma 3.1. The second inequality uses the bound on β(t), that level lis critical, and that levels l-1 and l+1 are valid. The third and final inequalities follow from Observation 3.1. (b) For level l∗+ 1 we similarly calculate Pr [ ml∗+1(x′) ml∗+1(x) ] ≥1 -β(t) β(t) · ml∗+1(x) -ml∗+1+1(x) ml∗+1-1(x) -ml∗+1(x) 2 · n ≥1 -β β · 1 α2 l∗ · n = 1 -β β · n 144 log2(n) = Ω n log2(n) ≥√n. Compared to the proof of the previous statement, we bound the number of balls at level at least l∗+1 in the numerator by 1 (level l∗+1 being critical means it is non-empty, so there is at least one bin with a ball on level l∗+ 1). The rest of the calculation applies the definition of αl∗(see Equation (2)) and simplifies. We next present the proof of Lemma 3.4. We restate it for convenience: Lemma 3.4. Let h(t) := ⌈mmax(t)/n⌉+ γ denote the base height at time t. Consider an initial configuration x(0) = x that is h(0)-safe. Fix the time T = n4 and assume n to be large enough. If β(t) ≤ˆβ ̃ml(t) ] ≥ Pr h m(h(t)) l (x(t + 1)) m(h(t)) l (x(t)) i ≥ ( 2 if l∈{ 0, 1, . . . , l∗} and √n if l= l∗+ 1. (31) If τ < T, there must be some level lthat is not valid at time τ and a minimal time t′ < τ such that level lis critical at any integral time t ∈[t′, τ). In particular, using the minimality of t′, we have for all integral t ∈[t′, τ), ̃ml(t′) = αl/2 ≤ ̃ml(t) < ̃ml(τ) = αl. That is, between time step t′ 33 and τ, the number of balls at level at least lstarted at αl/2 and increased to αlwithout decreasing below αl/2. If this happens, we say level lcrosses [αl/2, αl) during [t′, τ), and denote this event as Cl,[t′,τ). We will show that for any level land any time t′ event Cl,[t′,τ) is unlikely to happen. Using a union bound, this yields that, w.h.p., τ ≥T, as desired. To bound Pr Cl,[t′,τ) , we define a random walk W (l,t′) that has an even larger probability to cross [αl/2, αl) during [t′, τ) and show that this larger probability is still small enough. To this end, fix a level l∈{ 0, 1, . . . , l∗+ 1 } and integral time t′ ∈[0, τ] ⊆[0, T]. Let p(l,t) := Pr [ ̃ml(t + 1) = ̃ml(t) | x(t) ] denote the probability that the number of balls at level at least l does not change from time t to time t+1 (note that this probability is a random variable depending on the random configuration x(t)). We define the random walk W (l,t′) = (W (l,t′) t )t∈[t′,T) over the discrete time interval [t′, T). The random walk's start position is W (l,t′) t′ := ̃ml(t′) = αl/2 and its movement from time t ∈[t′, T) to time t + 1 is defined via W (l,t′) t+1 = W (l,t′) t with probability p(l,t), W (l,t′) t -1 with probability 1 -p(l,t) · 2/3, and W (l,t′) t + 1 with probability 1 -p(l,t) · 1/3. (32) In other words, the random walk W (l,t′) starts at time t′ at position ̃ml(t′) = αl/2, maintains its position whenever ̃ml(•) does not change, and the probability that it moves left is twice the probability that it moves right. For a time t ≥t′, we say W (l,t′) crosses [αl/2, αl) during [t′, t) if W (l,t′) reaches αlbefore decreasing below αl/2, and denote this event as CW l,[t′,t). Given the current configuration x(t) = x at time t ≥t′ and depending on how ̃ml(•) changes from time t to time t + 1, we can easily couple the random walk W (l,t′) with our process as follows: • if ̃ml(•) does not change, W (l,t′) remains stationary; • if ̃ml(•) increases, W (l,t′) moves right; and • if ̃ml(•) decreases, W (l,t′) moves left with prob. min { 1, 2 3/Pr [ ̃ml(t + 1) < ̃ml(t) | x(t) = x ] } and right otherwise. By this coupling, the random walk is never \"left of\" ̃ml(•) (i.e., for all t ≥t′ we have W (l,t′) t ≥ ̃ml(t)). We now consider the crossing probabilities for the different levels l∈{ 0, 1, . . . , l∗+ 1 } for some integral time t′ ∈[0, τ) at which level 0 just became critical. For level l= 0, Theorem 2.2 can be used to see that, w.h.p, this level will not become invalid at any time during the discrete interval [0, τ). For any other level l, we use the fact that the ̃ml-value is upper-bounded by the position of the random walk W (l,t′) during [t′, τ) (by the coupling above) together with the fact that the left-biased random walk W (l,t′) is very unlikely to reach the right border before becoming again safe (by Lemma 3.3). Case 1: For level l= 0 we do not need to considered the random walk. Instead, note that Theorem 2.2 allows us to choose a constant γ = γ(α0) such that the number of balls above height ⌈m(t)/n⌉+ γ is, with (arbitrarily) high probability, at most α0 = 1-β 128β · n. Thus, by choosing the probability high enough and by using a union bound over the T = poly(n) many time steps, we can ensure that, with probability at least 1 -n-6, level 0 does not become invalid at any time ∈{ 1, 2, . . . , T }. In particular, this yields Pr C0,[t′,τ) ≤Pr C0,[t′,T) ≤n-6. (33) 34 Case 2: For any level l∈{ 1, 2, . . . , l∗} and large enough n we get Pr Cl,[t′,τ) ≤Pr h CW l,[t′,τ) i ≤Pr h CW l,[t′,T) i Lem. 3.3 ≤ 21 -1 21+αl/2 -1 ≤ 2 21+αl/2 = 1 2αl/2 ≤ 1 2αl∗/2 = n-6. (34) Case 3: For level l= l∗+ 1 we get Pr Cl∗,[t′,τ) ≤Pr h CW l∗,[t′,τ) i ≤Pr h CW l∗,[t′,T) i Lem. 3.3 ≤ √n1 -1 √n1+αl∗+1/2 -1 ≤ √n √n1+αl∗+1/2 = n-6. (35) By this case distinction, we know that the probability that any single level that just became critical at time t′ ∈[0, τ) has probability at most n-6 to become critical within [0, τ). Taking a union bound over the l∗+ 2 = log log(n) + Θ(1) many levels and T = n4 many time steps finishes the proof of Lemma 3.4. E Lower Discrepancy Bound when Deletions are Dominant. Theorem E.1 gives a lower bound that shows that our result from Theorem 2.1 is tight in the sense that there are insertion probability sequences that result in a logarithmic discrepancy. Our lower bound statement assumes that the potential Φα (see Section 2) is linear in n, which holds with high probability (see Lemma C.1). Similar to the results of [PTW15] the discrepancy is heavily dependent on the one-choice operations, ball deletions in our case. A long sequence of ball deletions creates deep holes in the bins and the frequency of ball insertions must be high enough to re-balance the load. Theorem E.1. Fix a step t and assume Φα(t) = O(n) for some constant c. Let T = t + n ln(n) 2+4ε and assume that for every τ with t ≤τ ≤T we have β(τ) = 1/2 -ε with ε < 1/2 being an arbitrary positive constant. Then the expected number of bins at the end of step T with load at least ⌊m(T)/n⌋+ ln(n)/2 is Ω(√n). We note that the above result also holds (with basically the same proof) for varying insertion probabilities as long as fewer than (T -t)·(1/2-ε) balls are inserted during the time interval [t, T]. The proof idea for Theorem E.1 is as follows: We first show that, as long as in step t we have Φ(t) ≤c·n for some constant c (which holds with high probability), there exists a constant fraction of bins with a load which is at least m(t)/n (Lemma E.1). If now during the next O(n log n) steps much more balls are deleted than inserted, the average load will go down and one of the bins with load at least m(t)/n will receive no ball deletions at all. This bin will have a discrepancy of O(log n). This holds even under the assumption that the bin does not receive any additional ball. Lemma E.1. Assume Φ ≤c · n for some constant c and let r = r(n) denote the fraction of bins with a load of at least ⌊m(t)/n⌋. Then r = Θ(1), i.e., a constant fraction of bins has load at least ⌊m(t) n ⌋. Proof. Let B1 denote the set of bins with a load of at least ⌊m(t)/n⌋and B2 the set of bins with fewer than ⌊m(t)/n⌋balls. Since r is defined as the fraction of bins in B1, we have |B1| = r · n and 35 |B2| = (1 -r) · n. This implies that there are at least (1 -r) · n holes below the level ⌊m(t) n ⌋and, in turn, at least (1 -r) · n balls above the average load of m(t)/n. Let m(t)/n+lbe defined as the average load of the bins in B1. Note that l≥0 because m(t)/n is the average load and the bins in B1 have a higher load than the ones in B2. From the definition, it follows that l≥n · (1 -r) r · n = 1 -r r . Since the potential function Φ is minimal when all bins in B1 have load l, we obtain r·n X i=1 eα· 1-r r ≤Φ ≤c · n which implies r · eα· 1-r r ≤c. If we view r = r(n) as function in n and assume r(n) = o(1) then in r(n) · eα 1 r(n)-1 = r(n) · f(n) with f(n) = eα 1 r(n)-1 the exponent is a function growing as 1/r(n) = ω(1). This implies that regardless of the size of α (as long as it is positive) the f(n) grows exponentially quicker than the r(n) = o(1) decays, and so the whole r(n) · f(n) is in ω(1) and in particular not upper-bounded by the constant c. Since r(n) ≤1 ∈O(1), we conclude that r(n) = Θ(1). With this, we can prove Theorem E.1: Proof of Theorem E.1. First of all, a simple application of Chernoff bounds proves that, w.h.p., ⌊m(t)/n⌋≥⌊m(T)/n⌋+ ln(n)/2. Fix a bin i and define l= T -t = n ln(n)/(2 + 4ε). Then for large n, the probability that during all steps τ with t ≤τ ≤T no ball is deleted from bin i can be bounded by (using the inequality 1 -x ≥e-2x for x ∈[0, 0.795]) 1 -1 n (1-(1/2-ε))l = 1 -1 n (1/2+ε)l ≥exp -(1 + 2ε) l n = exp (-ln(n)/2) = n-1/2. (36) From Lemma E.1 we get that, at time t, a constant fraction of bins have load at least ⌊m(t)/n⌋. And by the above calculation, the expected number of those bins from which no ball is deleted is Ω(√n). Thus, at time T, in expectation there are Ω(√n) bins that have a load of at least ⌊m(t)/n⌋≥⌊m(T)/n⌋+ ln(n)/2. 36"}}},{"rowIdx":188,"cells":{"id":{"kind":"string","value":"2510.14793"},"raw":{"kind":"string","value":"Reflections of quantum educators on strategies to diversify the second quantum \nrevolution \nApekshya Ghimire* and Chandralekha Singh \nDepartment of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA, 15260 \n \n*apg61@pitt.edu \n \nIntroduction \nThe second quantum revolution focuses on quantum information science and technology \n(QIST). It promises to bring about unprecedented improvements in computing, \ncommunication and sensing due to our ability to exquisitely control and manipulate \nquantum systems [1, 2]. This fast-growing field presents equally unprecedented \nopportunities and challenges for preparing students to be future leaders in this area [3, \n4]. \n \nOne major challenge pertains to how to diversify the second quantum revolution. The first \nquantum revolution, which began nearly a century ago and laid the foundation for the \ntechnological breakthroughs of the mid-20th century, reflects the broader lack of diversity \nthat characterized the scientific landscape of that era. Since the second quantum \nrevolution is in its infancy, it is critical to contemplate these issues to ensure that people \nfrom historically marginalized groups in physics and related fields can contribute equitably \n[5, 6]. \n \nThis paper focuses on reflections and suggestions of five college quantum educators from \nfour different institutions (two from the same institution) regarding what can be done to \ndiversify the second quantum revolution. They are leading QIST researchers, and very \npassionate about improving quantum education. The educators were also asked about \ntheir thoughts on whether the interdisciplinary nature of the field, in which nobody can \nclaim to be an expert in all aspects of QIST, may make it easier to create a better culture \nfrom the beginning, supportive of equitable participation of diverse groups unlike physics. \nThis is because disciplines such as physics have an ingrained inequitable culture based \non brilliance attribution [7] that is a major impediment to diversity, equity and inclusion. \nEducators were interviewed on Zoom using a semi-structured think-aloud protocol about \nvarious issues related to QIST education including those pertaining to how to diversify the \nsecond quantum revolution discussed here. \n \nBelow, we discuss some excerpts that exemplify interviewed quantum educators’ \nreflections and suggestions in their own words. They are insightful and can help other \neducators adapt and implement strategies to diversify QIST. We mention below that one \nof the educators was a woman because her gender identity may be useful to understand \nher point of view. \n \nQuantum educators in their own words \nSome educators emphasized the importance of teaching quantum concepts early, e.g., \nin high school and early in college to students from diverse backgrounds. This exposure \nis important because targeting diverse high schools has the potential to introduce exciting \n\nquantum concepts and careers to students from diverse backgrounds considering \ndiversity is severely lacking among college STEM majors. Early introduction is also \nvaluable because learning quantum concepts early may make the subject matter less \nmysterious. Students may also better learn to discern the contexts in which classical and \nquantum concepts are applicable. This is similar to the facility children have in discerning \nwhich language to use in which contexts and switching between them as needed \ndepending upon contexts, when they simultaneously learn two different languages early. \nEmphasis on potential QIST applications focusing on solving societal problems that \nresonate with students and discussing career pathways with students can be useful. \nFor example, one male educator emphasized the importance of teaching quantum and \nrelevant mathematical concepts early saying, “If I had the magic wand that could reform, \nyou know, high school curriculum, I would probably change a few [parts] of the \nmathematics teaching. Like, I think, data science and statistics at this point should have \nat least as much space as calculus does. Well, once you understand statistics…then \nquantum is a small step from there and [can even be taught with] linear algebra. I would \nteach…coding, programming, statistics... And once you've got that, you can teach any \nquantum you want…even in high school…My course, that I am teaching second year \n[college students] here, with minor modifications, I could teach in high school.” \nHe also emphasized the importance of helping students from diverse backgrounds \ndevelop confidence along with competence from early on while learning QIST concepts \nsaying, “in every field, no matter how developed or under-developed [like QIST] it is, you \nwill find people who have a lot of confidence, you know, becoming the most visible \nones…It's something that students should learn a little in high school…”. \n \nAnother male educator who is passionate about early teaching as well as early research \nexperiences related to QIST for first- and second-year college students in science and \nengineering reflected on the importance of supporting all students in foundational courses \nespecially those not particularly interested in physics saying, “let's say that we [with \nphysics background] are teaching a college course on foundations of quantum \ncomputing…we found some advisors who said to students in electrical engineering or \ncomputer science that this might be a good thing if they were thinking about…going into \nthe workforce in this area. Now when they landed in those courses…they're like, oh, this \nsounds too much like physics to me. So what can you do?...We want all these students if \nthey're going to take an intro course…to succeed…there should be something for \neverybody...They shouldn't be punished for not being a physicist”. He acknowledged that \nthere are many ways of structuring interdisciplinary QIST courses for early learners, but \nfelt educators need to emphasize applications in quantum computing, communication and \nsensing and how they could impact society through examples that resonate with students \nfrom diverse backgrounds as well as discuss career pathways. He also noted that while \nthese early courses should mainly focus on learners with diverse background \npreparations and interests, students can be provided enrichment options through \nprojects/special modules as part of the course that dive more deeply into the sub-field of \nQIST, e.g., engineering or computer science, they may be more interested in. This \n\neducator also noted leading a first experiences in quantum research program for first-\nyear college students currently that he hopes will play a role in diversifying QIST. \n \nHe also stated that diversifying issues are not only in QIST but in many STEM disciplines \nincluding physics stating, “simply acknowledging this…and trying to make sure that \npeople have support that they need throughout their…education to be able to…walk into \na room where people don't look like them and succeed” is important. He did not think that \nQIST being a new field makes it easier to diversify than the sub-fields such as physics, “I \ndo think that there's in some sense…we could just like start over afresh, because this \n[QIST] is a new field, and we don't carry over the baggage…I'm not sure that it's \nsucceeding so far”. He felt that the current diversity in QIST is dismal saying, “I don't see \nthings as automatically getting fixed just because we rebranded. And now it's quantum \n2.0. I think that it's…very important to push back against these…And I think that the earlier \nthe positive experiences people have in doing that [learning quantum concepts] \nsuccessfully, the better the outcome could be, that you have to…realize that, you know, \nimposter syndrome is like, almost everybody feels that…has that kind of feeling at some \nstage or another and that you…can embrace it and say, well, I'm new to this field, but I \nstill belong here…[do] some type of self-affirmation”. \n \nHe also reflected on the highly interdisciplinary nature of QIST playing a role in sense of \nbelonging and emphasized the role of educators and researchers in contemplating ways \nto integrate students in QIST, who are trained differently. He felt that since the preparation \nof students is siloed in physics, engineering, computer science etc., some students may \nfeel uncomfortable being part of this new interdisciplinary field of QIST saying, “what's \ninteresting about the quantum is…there are dimensions to [it] where people feel like they \ndon't belong…they have to do with their professional upbringing, right? So, somebody \ncan feel that they don't belong…because they weren't trained in a certain way and I think \nthat it's important to address all of these issues, right? Whatever it is that is inhibiting \npeople from learning and doing [QIST]”. \n \nA female educator reflected on these issues saying, “I think, in general…physics has been \nvery sort of depressingly bad…I'd say that quantum does not have good representation, \nyou know it's not so inclusive…”. \n \nShe also felt that unless the culture of each subfield, e.g., physics, that makes up QIST \nchanges, she disagreed with “this idea that there are these different fields with somewhat \ndifferent cultures and maybe that could provide an avenue to improving [QIST]” \nemphasizing that a major barrier to diversity is, “this thing of like, if you get a guy, and he \nknows, you know, half of something, [he’ll say] oh, yeah, I know that. And you get a \nwoman, and she knows more, [she’ll say] oh, I don't know…everybody's faking it. But \nsomehow people's internalization of what they know relative to everyone else is very \nculturally dependent”. She felt that unless we can do something to eradicate these types \nof dichotomies that are systemic in the culture, e.g., overly confident men making \nstatements bragging about more than what they know and women undermining \nthemselves, the new field of QIST will continue to be intimidating for many women and \n\nother underrepresented people similar to physics and they would decide not to be part of \na field with such an inequitable culture. \n \nShe gave several concrete examples of men being overconfident and acting as if they \nwere pinnacles of brilliance based upon her own interactions with them in physics. She \nexemplified specific cases in which they knew significantly less than her but pretended to \nbe on top of everything and took the opportunity to promote themselves. She reiterated \nthat she is worried that this kind of inequitable culture that pervades physics will seep into \nthis new field of QIST since the same people who have dominated the space in physics \nwill continue to do so in QIST, “I'm saying just like having that confidence of, you know, I \nbelong here...[I disagree that] one of the great things about a field that's as broad as this \none [QIST] is that nobody knows everything. But you don't feel that way [if you are an \nunderrepresented person such as a woman], you feel like everybody knows it but you \n[because of the way many men portray themselves]”. She felt that the only way to diversify \nQIST is to change the systemic cultural issues in physics and other related disciplines \nthat make up QIST in which brilliance is over-emphasized and many individuals from the \ndominant group are pretending and portraying to know more than they do, thereby \nintimidating others and even driving them out of the discipline. \n \nAnother male educator strongly believed that early outreach to students from diverse \nbackgrounds in various settings can play a critical role in diversifying QIST saying, “we've \ngot a massive issue with diversity in physics and…in electrical engineering and other \nengineering disciplines…in computer science…and I think we, as physicists, have to be \nvery active in going out to the general public”. \n \nHe also emphasized the importance of increasing the visibility and impact of \nunderrepresented people in QIST saying, “And I think that it's important to emphasize and \nunderline the contributions of those role models and to show to people that this is a field \nfor everyone”. He thought it was important that “when we organize events, we represent \nthe diversity of people who are involved…there are people from a lot of minority ethnic \ngroups involved in this area. There are a lot of women involved in this area really doing \ninternationally leading research”. To diversify QIST, he stressed the need to change the \nculture of physics and related disciplines comprising QIST that currently favors the \ndominant group, “we need to keep looking at our own research teams, our ways of \ninteracting with people. We need to look at our recruiting practices. We need to look at \nthe ways in which we organize meetings, and we need to be constantly questioning \nwhether we have tried to understand all of this best practice from what's been done in this \nfield around the world, making sure that we implement it in our own teams and make sure \nthat we essentially spread the message and take leadership roles in saying that diversity, \nequity, and inclusion, these things are extremely important to us, and extremely important \nto the health of the field, going forth”. \n \nHe added, “we lose out still a lot. A lot of very talented physicists choose to go into other \nareas…they have bad experiences with a workplace culture...I also think that when you \nget a diverse range of people, people with different ethnic backgrounds, people of \ndifferent nationalities, people from different gender backgrounds, people from…all \n\ndifferent ways you can be, I find that teams tend to work much better. They tend to be \nmore creative because people coming in with different ideas, different ways [of] thinking \nlead to different types of creativity, but also the culture of teams change…most inefficient \ncommittees I've ever been on, we're essentially all made up of white men and the most \nproductive committees I was on, we're typically always characterized by having a very \ndiverse range of people. And I think for the health of our field, it's important that we take \nleadership roles and that we do everything we can in this area”. \n \nAnother male educator, who acknowledged that the physics culture is a detriment to \ndiversifying, was concerned that this burgeoning QIST field can morph into a non-diverse \nfield like physics saying, “physics, of course, has just long problems with \nunderrepresented minorities, which because quantum technology is physics led, is a \nproblem that [can] propagate into that”. However, he expressed optimism that “attitude in \nphysics might not hopefully propagate to wider quantum technology, and so might be a \nway to kind of get…minorities in…so there's that sense of optimism there that with the \ninvention of a new field [QIST], you can invent a new culture hopefully, free from some of \nthe mistakes of physics”. He also felt that whatever strategies are currently being used \nto increase diversity, equity and inclusion in physics should be employed in QIST since \nthey may be useful for QIST as well. \n \nDiscussion and Summary \nThe interdisciplinary field of QIST is fast-growing with major implications for the future \nworkforce. This paper focuses on reflections and suggestions from five college quantum \neducators, who are passionate about teaching quantum concepts, regarding how to \ndiversify the second quantum revolution. Their suggestions can be invaluable for other \neducators interested in similar pursuits and can be summarized as follows: \n• Focus on changing the culture of disciplines such as physics since the \ninequitable culture in these sub-disciplines comprising QIST can impact \noutcomes in QIST, which is interdisciplinary. \n• Acknowledge the need for diversity, equity, and inclusion in QIST and support \nstudents from diverse backgrounds as appropriate. \n• Conduct quantum outreach in both formal (K-12) and informal settings to get \nstudents from diverse backgrounds aware of and interested in QIST concepts \nand career opportunities. \n• Teach quantum concepts early, e.g., in high school and early in college to \nstudents from diverse backgrounds. \no More focus in high school on statistics, data science, and programming \nmay be helpful for introducing quantum early. \no Focus on developing student confidence along with competence. \no Focus on not teaching early QIST courses as physics courses \no Tailor courses to benefit diverse backgrounds and interests emphasizing \napplications in quantum computing, communication and sensing that \nhave the potential to solve major societal problems as well as career \npathways. \n• Give students early experience in QIST research. \n\n• Increase the number of role models in QIST. \n• Increase the visibility and impact of existing underrepresented QIST \nresearchers, e.g., by including them as invited speakers and on committees \nthat make major decisions. \n• Try similar approaches to those that have proved useful for diversifying physics. \n \nBased upon these suggestions, the involvement of early learners, e.g., in high school, \ncollege and via outreach is important for ensuring that students from diverse backgrounds \ncan learn about exciting quantum concepts and careers particularly because college \nSTEM majors lack diversity. There is a need to ensure that students develop an \nappropriate level of confidence and competence commensurate with these courses. Early \nexposure at the appropriate level can make quantum concepts less spooky and ensure \nthat students recognize the differences between situations in which classical and \nquantum frameworks are useful to understand physical phenomena. It would help to \nemphasize QIST applications and how they can help solve major societal problems such \nas making nitrogen fixation more energy efficient for agricultural innovation and drug \ndiscovery. In addition, discussion of career pathways and the relevance of QIST to \nnational security can be valuable. Giving students opportunities for early experiences in \nQIST research as appropriate, commensurate with their prior preparation, can also get \nstudents excited about pursuing QIST related careers. \n \nEducators have begun to develop educational tools for early quantum learners. For \nexample, tools have been developed to visualize QIST concepts [8-10], explore physical \nrealizations of qubits [11], as well as approaches that allow for kinesthetic [12] and game-\nbased learning [13-15] to understand entanglement, decoherence, Bell’s inequality and \nquantum cryptography. Educators can engage students by using other interactive \napproaches that use “compare and contrast” activities involving quantum and classical \nconcepts [16, 17], diagrammatic tools [18], evidence-based teaching-learning sequences \n[19] and optics experiments [20, 21] for early learners to get students excited about QIST \nconcepts [22]. \n \nSome organizations such as National Q-12 partnership [23] and Quantum for All [24, 25] \nare providing a community to K-12 educators interested in incorporating quantum \nconcepts early. For example, National Q-12 partnership [23] is promoting and supporting \nquantum education by connecting those with interest in K-12 quantum education with \neach other as well as hosting a framework for K-12 quantum education that can be used \nto develop QIST curricula. They also host a variety of QuanTime activities, which are \nready-to-use activities, that K-12 educators can use in their classes without having prior \nexperience with teaching QIST concepts. For example, in 2024 year, there were 27 \nactivities available to the K-12 teachers [23] and teachers were encouraged to devote at \nleast one class period around the World Quantum Day on April 14 on one of the activities \nthey selected. Quantum for All [24, 25] has been a leader in providing an inclusive \ncommunity for K-12 educators interested in incorporating quantum education in their \nclasses in a conceptual manner. They provide intensive professional development \nactivities to K-12 educators to ensure that they feel like they are part of a community with \nsimilar goals and feel confident infusing quantum concepts into their classes [24, 25]. \n\nOther organizations such as Womanium [26] are for all levels (high school through \ngraduate school and postdoc). Womanium is founded and run by women, who are QISE \nresearchers and educators, and it is working on bringing in students of all genders, \nnationalities, and levels to the QIST field[26]. Qubitbyqubit [27] and IBM [28] focus on K-\n16 QIST education and conduct workshops and courses suited for various levels. \n \nWhile the efforts of these organizations and others to support QIST education is important \nparticularly due to the increased career opportunities in both quantum-proficient and \nquantum-aware/adjacent jobs[3], further efforts to diversify QIST is critical for harnessing \nthe talents of demographic groups that have historically been marginalized in the first \nquantum revolution. Increasing the visibility and impact of existing underrepresented \nQIST researchers as well as increasing the number of role models would help with \ndiversification. Changing the culture of disciplines such as physics would help since the \ninequitable culture in these sub-disciplines can impact the culture of QIST and the extent \nto which it ends up being diverse, equitable and inclusive as the field matures. \n \nFinally, the interdisciplinary nature of QIST, in which nobody can claim to be an expert, \ncan be an opportunity for diversifying this new field whose culture can be shaped and \nmade more equitable than disciplines such as physics, electrical engineering, or computer \nscience that already have a set culture, which is an impediment to diversity, equity and \ninclusion. However, the interdisciplinary nature of QIST can also present challenges since \nstudent preparation is often siloed into different disciplines, and people with a particular \ntype of training may not feel comfortable being part of an interdisciplinary community in \nwhich it is difficult to communicate with people with different types of training. Without \nadequate support, these issues, e.g., related to sense of belonging [29] in such a \ncommunity consisting of those with training in physics, engineering, computer science, \nchemistry, etc., may be particularly challenging for people from historically marginalized \ngroups. Coming up with a shared QIST language that is not heavy on quantum physics \njargon (unimportant for many stakeholders with non-physics training) can be useful. \nUltimately, forging an empathetic and supportive community culture in which those with \ndifferent types of training feel comfortable and safe sharing their ideas can go a long way \nto diversify QIST. \n \nAcknowledgment \nWe thank the NSF for award PHY-2309260. \nReferences \n[1] Raymer M G and Monroe C 2019 The US national quantum initiative. Quantum Science and Technology 4 (2) \n020504. \n[2] https://qt.eu/. \n[3] Asfaw A, et al. 2022 Building a quantum engineering undergraduate program. IEEE Transactions on Education \n65 (2) 220-242.10.1109/TE.2022.3144943. \n[4] Fox M F J, et al. 2020 Preparing for the quantum revolution: What is the role of higher education? Physical Review \nPhysics Education Research 16 (2) 020131. \n[5] Maries A and Singh C 2024 Towards meaningful diversity, equity and inclusion in physics learning environments. \nNature Physics 20 (3) 367-375.10.1038/s41567-024-02391-6. \n\n[6] Meyer J C, et al. 2024 Disparities in access to U.S. quantum information education. Physical Review Physics \nEducation Research 20 (1) 010131.10.1103/PhysRevPhysEducRes.20.010131. \n[7] Leslie S-J, et al. 2015 Expectations of brilliance underlie gender distributions across academic disciplines. Science \n347 (6219) 262-265.10.1126/science.1261375. \n[8] Dür W and Heusler S 2014 Visualization of the invisible: the qubit as key to quantum physics. The Physics Teacher \n52 (8) 489-492.10.1119/1.4897588. \n[9] McKagan S B, et al. 2008 Developing and researching PhET simulations for teaching quantum mechanics. Am. J. \nPhys. 76 (4) 406-417. \n[10] Kohnle A, et al. 2014 A new introductory quantum mechanics curriculum. European Journal of Physics 35 (1) \n015001.10.1088/0143-0807/35/1/015001. \n[11] Dür W and Heusler S 2016 The Qubit as Key to Quantum Physics Part II: Physical Realizations and Applications. \nThe Physics Teacher 54 (3) 156-159.10.1119/1.4942137. \n[12] Hahn K and Gire E 2022 Waving arms around to teach quantum mechanics. Am. J. Phys. 90 778-\n786.10.1119/5.0073946. \n[13] López-Incera A and Dür W 2019 Entangle me! A game to demonstrate the principles of quantum mechanics. Am. \nJ. Phys. 87 (2) 95-101.10.1119/1.5086275. \n[14] López-Incera A, et al. 2020 Encrypt me! A game-based approach to Bell inequalities and quantum cryptography. \nEuropean Journal of Physics 41 (6) 065702.10.1088/1361-6404/ab9a67. \n[15] Marckwordt J, et al. 2021 Entanglement ball: using dodgeball to introduce quantum entanglement. The Physics \nTeacher 59 613-616.10.1119/5.0019871. \n[16] Oss S and Rosi T 2015 A bit of quantum mechanics. The Physics Teacher 53 (4) 230-233.10.1119/1.4914565. \n[17] Singh C, et al. 2022 Preparing precollege students for the second quantum revolution with core concepts in \nquantum information science. The Physics Teacher 60 (8) 639-641.10.1119/5.0027661. \n[18] Rudolph T (2017). Q is for Quantum, Terence Rudolph.https://www.amazon.com/Q-Quantum-Terry-\nRudolph/dp/0999063502. \n[19] Hu P, et al. 2024 Investigating and improving student understanding of the basics of quantum computing. Physical \nReview Physics Education Research 20 (2) 020108.10.1103/PhysRevPhysEducRes.20.020108. \n[20] Walsh J A, et al. 2022 Piloting a full-year, optics-based high school course on quantum computing. Physics \nEducation 57 (2) 025010.10.1088/1361-6552/ac3dc2. \n[21] Bitzenbauer P and Meyn J-P 2020 A new teaching concept on quantum physics in secondary schools. Physics \nEducation 55 (5) 055031.10.1088/1361-6552/aba208. \n[22] Darienzo M, et al. 2024 Student attitudes toward quantum information science and technology in a high school \noutreach program. Phys. Rev. PER 20 020126. https://doi.org/10.1103/PhysRevPhysEducRes.20.020126. \n[23] https://q12education.org/quantime. \n[24] https://quantumforall.org/. \n[25] Matsler K, et al. 2024 Applying Classroom Practices Learned from Virtual Professional Development During a \nPandemic. The Physics Teacher 62 41-46.10.1119/5.0107084. \n[26] https://www.womanium.org/. \n[27] https://www.qubitbyqubit.org/. \n[28] Singh C, et al. 2021 Preparing students to be leaders of the quantum information revolution. Physics Today \nhttps://doi.org/10.1063/PT.6.5.20210927a. \n[29] Binning K R, et al. 2020 Changing social contexts to foster equity in college science courses: An ecological-\nbelonging intervention. Psychological Science 31 (9) 1059-1070.10.1177/0956797620929984. \n \n \n \n \n"},"cleaned":{"kind":"string","value":"Reflections of quantum educators on strategies to diversify the second quantum revolution Apekshya Ghimire* and Chandralekha Singh 3941 O'Hara St, Pittsburgh, PA, 15260 * Introduction The second quantum revolution focuses on quantum information science and technology (QIST). It promises to bring about unprecedented improvements in computing, communication and sensing due to our ability to exquisitely control and manipulate quantum systems [1, 2]. This fast-growing field presents equally unprecedented opportunities and challenges for preparing students to be future leaders in this area [3, 4]. One major challenge pertains to how to diversify the second quantum revolution. The first quantum revolution, which began nearly a century ago and laid the foundation for the technological breakthroughs of the mid-20th century, reflects the broader lack of diversity that characterized the scientific landscape of that era. Since the second quantum revolution is in its infancy, it is critical to contemplate these issues to ensure that people from historically marginalized groups in physics and related fields can contribute equitably [5, 6]. This paper focuses on reflections and suggestions of five college quantum educators from four different institutions (two from the same institution) regarding what can be done to diversify the second quantum revolution. They are leading QIST researchers, and very passionate about improving quantum education. The educators were also asked about their thoughts on whether the interdisciplinary nature of the field, in which nobody can claim to be an expert in all aspects of QIST, may make it easier to create a better culture from the beginning, supportive of equitable participation of diverse groups unlike physics. This is because disciplines such as physics have an ingrained inequitable culture based on brilliance attribution [7] that is a major impediment to diversity, equity and inclusion. Educators were interviewed on Zoom using a semi-structured think-aloud protocol about various issues related to QIST education including those pertaining to how to diversify the second quantum revolution discussed here. Below, we discuss some excerpts that exemplify interviewed quantum educators' reflections and suggestions in their own words. They are insightful and can help other educators adapt and implement strategies to diversify QIST. We mention below that one of the educators was a woman because her gender identity may be useful to understand her point of view. Quantum educators in their own words Some educators emphasized the importance of teaching quantum concepts early, e.g., in high school and early in college to students from diverse backgrounds. This exposure is important because targeting diverse high schools has the potential to introduce exciting quantum concepts and careers to students from diverse backgrounds considering diversity is severely lacking among college STEM majors. Early introduction is also valuable because learning quantum concepts early may make the subject matter less mysterious. Students may also better learn to discern the contexts in which classical and quantum concepts are applicable. This is similar to the facility children have in discerning which language to use in which contexts and switching between them as needed depending upon contexts, when they simultaneously learn two different languages early. Emphasis on potential QIST applications focusing on solving societal problems that resonate with students and discussing career pathways with students can be useful. For example, one male educator emphasized the importance of teaching quantum and relevant mathematical concepts early saying, \"If I had the magic wand that could reform, you know, high school curriculum, I would probably change a few [parts] of the mathematics teaching. Like, I think, data science and statistics at this point should have at least as much space as calculus does. Well, once you understand statistics...then quantum is a small step from there and [can even be taught with] linear algebra. I would teach...coding, programming, statistics... And once you've got that, you can teach any quantum you want...even in high school...My course, that I am teaching second year [college students] here, with minor modifications, I could teach in high school.\" He also emphasized the importance of helping students from diverse backgrounds develop confidence along with competence from early on while learning QIST concepts saying, \"in every field, no matter how developed or under-developed [like QIST] it is, you will find people who have a lot of confidence, you know, becoming the most visible ones...It's something that students should learn a little in high school...\". Another male educator who is passionate about early teaching as well as early research experiences related to QIST for first- and second-year college students in science and engineering reflected on the importance of supporting all students in foundational courses especially those not particularly interested in physics saying, \"let's say that we [with physics background] are teaching a college course on foundations of quantum computing...we found some advisors who said to students in electrical engineering or computer science that this might be a good thing if they were thinking about...going into the workforce in this area. Now when they landed in those courses...they're like, oh, this sounds too much like physics to me. So what can you do?...We want all these students if they're going to take an intro course...to succeed...there should be something for everybody...They shouldn't be punished for not being a physicist\". He acknowledged that there are many ways of structuring interdisciplinary QIST courses for early learners, but felt educators need to emphasize applications in quantum computing, communication and sensing and how they could impact society through examples that resonate with students from diverse backgrounds as well as discuss career pathways. He also noted that while these early courses should mainly focus on learners with diverse background preparations and interests, students can be provided enrichment options through projects/special modules as part of the course that dive more deeply into the sub-field of QIST, e.g., engineering or computer science, they may be more interested in. This educator also noted leading a first experiences in quantum research program for firstyear college students currently that he hopes will play a role in diversifying QIST. He also stated that diversifying issues are not only in QIST but in many STEM disciplines including physics stating, \"simply acknowledging this...and trying to make sure that people have support that they need throughout their...education to be able to...walk into a room where people don't look like them and succeed\" is important. He did not think that QIST being a new field makes it easier to diversify than the sub-fields such as physics, \"I do think that there's in some sense...we could just like start over afresh, because this [QIST] is a new field, and we don't carry over the baggage...I'm not sure that it's succeeding so far\". He felt that the current diversity in QIST is dismal saying, \"I don't see things as automatically getting fixed just because we rebranded. And now it's quantum 2.0. I think that it's...very important to push back against these...And I think that the earlier the positive experiences people have in doing that [learning quantum concepts] successfully, the better the outcome could be, that you have to...realize that, you know, imposter syndrome is like, almost everybody feels that...has that kind of feeling at some stage or another and that you...can embrace it and say, well, I'm new to this field, but I still belong here...[do] some type of self-affirmation\". He also reflected on the highly interdisciplinary nature of QIST playing a role in sense of belonging and emphasized the role of educators and researchers in contemplating ways to integrate students in QIST, who are trained differently. He felt that since the preparation of students is siloed in physics, engineering, computer science etc., some students may feel uncomfortable being part of this new interdisciplinary field of QIST saying, \"what's interesting about the quantum is...there are dimensions to [it] where people feel like they don't belong...they have to do with their professional upbringing, right? So, somebody can feel that they don't belong...because they weren't trained in a certain way and I think that it's important to address all of these issues, right? Whatever it is that is inhibiting people from learning and doing [QIST]\". A female educator reflected on these issues saying, \"I think, in general...physics has been very sort of depressingly bad...I'd say that quantum does not have good representation, you know it's not so inclusive...\". She also felt that unless the culture of each subfield, e.g., physics, that makes up QIST changes, she disagreed with \"this idea that there are these different fields with somewhat different cultures and maybe that could provide an avenue to improving [QIST]\" emphasizing that a major barrier to diversity is, \"this thing of like, if you get a guy, and he knows, you know, half of something, [he'll say] oh, yeah, I know that. And you get a woman, and she knows more, [she'll say] oh, I don't know...everybody's faking it. But somehow people's internalization of what they know relative to everyone else is very culturally dependent\". She felt that unless we can do something to eradicate these types of dichotomies that are systemic in the culture, e.g., overly confident men making statements bragging about more than what they know and women undermining themselves, the new field of QIST will continue to be intimidating for many women and other underrepresented people similar to physics and they would decide not to be part of a field with such an inequitable culture. She gave several concrete examples of men being overconfident and acting as if they were pinnacles of brilliance based upon her own interactions with them in physics. She exemplified specific cases in which they knew significantly less than her but pretended to be on top of everything and took the opportunity to promote themselves. She reiterated that she is worried that this kind of inequitable culture that pervades physics will seep into this new field of QIST since the same people who have dominated the space in physics will continue to do so in QIST, \"I'm saying just like having that confidence of, you know, I belong here...[I disagree that] one of the great things about a field that's as broad as this one [QIST] is that nobody knows everything. But you don't feel that way [if you are an underrepresented person such as a woman], you feel like everybody knows it but you [because of the way many men portray themselves]\". She felt that the only way to diversify QIST is to change the systemic cultural issues in physics and other related disciplines that make up QIST in which brilliance is over-emphasized and many individuals from the dominant group are pretending and portraying to know more than they do, thereby intimidating others and even driving them out of the discipline. Another male educator strongly believed that early outreach to students from diverse backgrounds in various settings can play a critical role in diversifying QIST saying, \"we've got a massive issue with diversity in physics and...in electrical engineering and other engineering disciplines...in computer science...and I think we, as physicists, have to be very active in going out to the general public\". He also emphasized the importance of increasing the visibility and impact of underrepresented people in QIST saying, \"And I think that it's important to emphasize and underline the contributions of those role models and to show to people that this is a field for everyone\". He thought it was important that \"when we organize events, we represent the diversity of people who are involved...there are people from a lot of minority ethnic groups involved in this area. There are a lot of women involved in this area really doing internationally leading research\". To diversify QIST, he stressed the need to change the culture of physics and related disciplines comprising QIST that currently favors the dominant group, \"we need to keep looking at our own research teams, our ways of interacting with people. We need to look at our recruiting practices. We need to look at the ways in which we organize meetings, and we need to be constantly questioning whether we have tried to understand all of this best practice from what's been done in this field around the world, making sure that we implement it in our own teams and make sure that we essentially spread the message and take leadership roles in saying that diversity, equity, and inclusion, these things are extremely important to us, and extremely important to the health of the field, going forth\". He added, \"we lose out still a lot. A lot of very talented physicists choose to go into other areas...they have bad experiences with a workplace culture...I also think that when you get a diverse range of people, people with different ethnic backgrounds, people of different nationalities, people from different gender backgrounds, people from...all different ways you can be, I find that teams tend to work much better. They tend to be more creative because people coming in with different ideas, different ways [of] thinking lead to different types of creativity, but also the culture of teams change...most inefficient committees I've ever been on, we're essentially all made up of white men and the most productive committees I was on, we're typically always characterized by having a very diverse range of people. And I think for the health of our field, it's important that we take leadership roles and that we do everything we can in this area\". Another male educator, who acknowledged that the physics culture is a detriment to diversifying, was concerned that this burgeoning QIST field can morph into a non-diverse field like physics saying, \"physics, of course, has just long problems with underrepresented minorities, which because quantum technology is physics led, is a problem that [can] propagate into that\". However, he expressed optimism that \"attitude in physics might not hopefully propagate to wider quantum technology, and so might be a way to kind of get...minorities in...so there's that sense of optimism there that with the invention of a new field [QIST], you can invent a new culture hopefully, free from some of the mistakes of physics\". He also felt that whatever strategies are currently being used to increase diversity, equity and inclusion in physics should be employed in QIST since they may be useful for QIST as well. Discussion and Summary The interdisciplinary field of QIST is fast-growing with major implications for the future workforce. This paper focuses on reflections and suggestions from five college quantum educators, who are passionate about teaching quantum concepts, regarding how to diversify the second quantum revolution. Their suggestions can be invaluable for other educators interested in similar pursuits and can be summarized as follows: • Focus on changing the culture of disciplines such as physics since the inequitable culture in these sub-disciplines comprising QIST can impact outcomes in QIST, which is interdisciplinary. • Acknowledge the need for diversity, equity, and inclusion in QIST and support students from diverse backgrounds as appropriate. • Conduct quantum outreach in both formal (K-12) and informal settings to get students from diverse backgrounds aware of and interested in QIST concepts and career opportunities. • Teach quantum concepts early, e.g., in high school and early in college to students from diverse backgrounds. o More focus in high school on statistics, data science, and programming may be helpful for introducing quantum early. o Focus on developing student confidence along with competence. o Focus on not teaching early QIST courses as physics courses o Tailor courses to benefit diverse backgrounds and interests emphasizing applications in quantum computing, communication and sensing that have the potential to solve major societal problems as well as career pathways. • Give students early experience in QIST research. • Increase the number of role models in QIST. • Increase the visibility and impact of existing underrepresented QIST researchers, e.g., by including them as invited speakers and on committees that make major decisions. • Try similar approaches to those that have proved useful for diversifying physics. Based upon these suggestions, the involvement of early learners, e.g., in high school, college and via outreach is important for ensuring that students from diverse backgrounds can learn about exciting quantum concepts and careers particularly because college STEM majors lack diversity. There is a need to ensure that students develop an appropriate level of confidence and competence commensurate with these courses. Early exposure at the appropriate level can make quantum concepts less spooky and ensure that students recognize the differences between situations in which classical and quantum frameworks are useful to understand physical phenomena. It would help to emphasize QIST applications and how they can help solve major societal problems such as making nitrogen fixation more energy efficient for agricultural innovation and drug discovery. In addition, discussion of career pathways and the relevance of QIST to national security can be valuable. Giving students opportunities for early experiences in QIST research as appropriate, commensurate with their prior preparation, can also get students excited about pursuing QIST related careers. Educators have begun to develop educational tools for early quantum learners. For example, tools have been developed to visualize QIST concepts [8-10], explore physical realizations of qubits [11], as well as approaches that allow for kinesthetic [12] and gamebased learning [13-15] to understand entanglement, decoherence, Bell's inequality and quantum cryptography. Educators can engage students by using other interactive approaches that use \"compare and contrast\" activities involving quantum and classical concepts [16, 17], diagrammatic tools [18], evidence-based teaching-learning sequences [19] and optics experiments [20, 21] for early learners to get students excited about QIST concepts [22]. Some organizations such as National Q-12 partnership [23] and Quantum for All [24, 25] are providing a community to K-12 educators interested in incorporating quantum concepts early. For example, National Q-12 partnership [23] is promoting and supporting quantum education by connecting those with interest in K-12 quantum education with each other as well as hosting a framework for K-12 quantum education that can be used to develop QIST curricula. They also host a variety of QuanTime activities, which are ready-to-use activities, that K-12 educators can use in their classes without having prior experience with teaching QIST concepts. For example, in 2024 year, there were 27 activities available to the K-12 teachers [23] and teachers were encouraged to devote at least one class period around the World Quantum Day on April 14 on one of the activities they selected. Quantum for All [24, 25] has been a leader in providing an inclusive community for K-12 educators interested in incorporating quantum education in their classes in a conceptual manner. They provide intensive professional development activities to K-12 educators to ensure that they feel like they are part of a community with similar goals and feel confident infusing quantum concepts into their classes [24, 25]. Other organizations such as Womanium [26] are for all levels (high school through graduate school and postdoc). Womanium is founded and run by women, who are QISE researchers and educators, and it is working on bringing in students of all genders, nationalities, and levels to the QIST field[26]. Qubitbyqubit [27] and IBM [28] focus on K16 QIST education and conduct workshops and courses suited for various levels. While the efforts of these organizations and others to support QIST education is important particularly due to the increased career opportunities in both quantum-proficient and quantum-aware/adjacent jobs[3], further efforts to diversify QIST is critical for harnessing the talents of demographic groups that have historically been marginalized in the first quantum revolution. Increasing the visibility and impact of existing underrepresented QIST researchers as well as increasing the number of role models would help with diversification. Changing the culture of disciplines such as physics would help since the inequitable culture in these sub-disciplines can impact the culture of QIST and the extent to which it ends up being diverse, equitable and inclusive as the field matures. Finally, the interdisciplinary nature of QIST, in which nobody can claim to be an expert, can be an opportunity for diversifying this new field whose culture can be shaped and made more equitable than disciplines such as physics, electrical engineering, or computer science that already have a set culture, which is an impediment to diversity, equity and inclusion. However, the interdisciplinary nature of QIST can also present challenges since student preparation is often siloed into different disciplines, and people with a particular type of training may not feel comfortable being part of an interdisciplinary community in which it is difficult to communicate with people with different types of training. Without adequate support, these issues, e.g., related to sense of belonging [29] in such a community consisting of those with training in physics, engineering, computer science, chemistry, etc., may be particularly challenging for people from historically marginalized groups. Coming up with a shared QIST language that is not heavy on quantum physics jargon (unimportant for many stakeholders with non-physics training) can be useful. Ultimately, forging an empathetic and supportive community culture in which those with different types of training feel comfortable and safe sharing their ideas can go a long way to diversify QIST. Acknowledgment We thank the NSF for award PHY-2309260. References [1] Raymer M G and Monroe C 2019 The US national quantum initiative. Quantum Science and Technology 4 (2) 020504. [2] https://qt.eu/. [3] Asfaw A, et al. 2022 Building a quantum engineering undergraduate program. IEEE Transactions on Education 65 (2) 220-242.10.1109/TE.2022.3144943. [4] Fox M F J, et al. 2020 Preparing for the quantum revolution: What is the role of higher education? Physical Review Physics Education Research 16 (2) 020131. [5] Maries A and Singh C 2024 Towards meaningful diversity, equity and inclusion in physics learning environments. Nature Physics 20 (3) 367-375.10.1038/s41567-024-02391-6. [6] Meyer J C, et al. 2024 Disparities in access to U.S. quantum information education. Physical Review Physics Education Research 20 (1) 010131.10.1103/PhysRevPhysEducRes.20.010131. [7] Leslie S-J, et al. 2015 Expectations of brilliance underlie gender distributions across academic disciplines. Science 347 (6219) 262-265.10.1126/science.1261375. [8] Dür W and Heusler S 2014 Visualization of the invisible: the qubit as key to quantum physics. The Physics Teacher 52 (8) 489-492.10.1119/1.4897588. [9] McKagan S B, et al. 2008 Developing and researching PhET simulations for teaching quantum mechanics. Am. J. Phys. 76 (4) 406-417. [10] Kohnle A, et al. 2014 A new introductory quantum mechanics curriculum. European Journal of Physics 35 (1) 015001.10.1088/0143-0807/35/1/015001. [11] Dür W and Heusler S 2016 The Qubit as Key to Quantum Physics Part II: Physical Realizations and Applications. The Physics Teacher 54 (3) 156-159.10.1119/1.4942137. [12] Hahn K and Gire E 2022 Waving arms around to teach quantum mechanics. Am. J. Phys. 90 778786.10.1119/5.0073946. [13] López-Incera A and Dür W 2019 Entangle me! A game to demonstrate the principles of quantum mechanics. Am. J. Phys. 87 (2) 95-101.10.1119/1.5086275. [14] López-Incera A, et al. 2020 Encrypt me! A game-based approach to Bell inequalities and quantum cryptography. European Journal of Physics 41 (6) 065702.10.1088/1361-6404/ab9a67. [15] Marckwordt J, et al. 2021 Entanglement ball: using dodgeball to introduce quantum entanglement. The Physics Teacher 59 613-616.10.1119/5.0019871. [16] Oss S and Rosi T 2015 A bit of quantum mechanics. The Physics Teacher 53 (4) 230-233.10.1119/1.4914565. [17] Singh C, et al. 2022 Preparing precollege students for the second quantum revolution with core concepts in quantum information science. The Physics Teacher 60 (8) 639-641.10.1119/5.0027661. [18] Rudolph T (2017). Q is for Quantum, Terence Rudolph.https://www.amazon.com/Q-Quantum-TerryRudolph/dp/0999063502. [19] Hu P, et al. 2024 Investigating and improving student understanding of the basics of quantum computing. Physical Review Physics Education Research 20 (2) 020108.10.1103/PhysRevPhysEducRes.20.020108. [20] Walsh J A, et al. 2022 Piloting a full-year, optics-based high school course on quantum computing. Physics Education 57 (2) 025010.10.1088/1361-6552/ac3dc2. [21] Bitzenbauer P and Meyn J-P 2020 A new teaching concept on quantum physics in secondary schools. Physics Education 55 (5) 055031.10.1088/1361-6552/aba208. [22] Darienzo M, et al. 2024 Student attitudes toward quantum information science and technology in a high school outreach program. Phys. Rev. PER 20 020126. https://doi.org/10.1103/PhysRevPhysEducRes.20.020126. [23] https://q12education.org/quantime. [24] https://quantumforall.org/. [25] Matsler K, et al. 2024 Applying Classroom Practices Learned from Virtual Professional Development During a Pandemic. The Physics Teacher 62 41-46.10.1119/5.0107084. [26] https://www.womanium.org/. [27] https://www.qubitbyqubit.org/. [28] Singh C, et al. 2021 Preparing students to be leaders of the quantum information revolution. Physics Today https://doi.org/10.1063/PT.6.5.20210927a. [29] Binning K R, et al. 2020 Changing social contexts to foster equity in college science courses: An ecologicalbelonging intervention. Psychological Science 31 (9) 1059-1070.10.1177/0956797620929984."}}},{"rowIdx":189,"cells":{"id":{"kind":"string","value":"2510.14797"},"raw":{"kind":"string","value":"Grain volume distribution alters the critical phenomena in complex granular systems\nTeng Man,1, ∗Yimin Lu,2 Zhongrong Wang,3 Herbert Huppert,4 Alessio Zaccone,5 and Honglei Sun1, †\n1College of Civil Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China\n2Department of Civil, Environmental, and Construction Engineering,\nTexas Tech University, Lubbock, Texas 79409, United States\n3School of Engineering, Royal Melbourne Institute of Technology, Victoria, 3001, Australia\n4Institute of Theoretical Geophysics, King’s College, University of Cambridge,\nKing’s Parade, Cambridge CB2 1ST, United Kingdom\n5Department of Physics “A. Pontremoli,” University of Milan, via Celoria 16, 20133 Milan, Italy\n(Dated: October 17, 2025)\nThe grain size distribution (GSD) plays an important role in the mechanical properties of amor-\nphous disordered systems and complex granular materials. Varying GSD causes segregation issues\nand alters critical behaviors. This work used the discrete element method (DEM) to investigate the\nrheological and critical behaviors of sheared granular flows with various GSDs. The results show\nthat, while a unified rheological relation can be obtained, a characteristic length scale, which is as-\nsociated with the contact probability and can be obtained from any GSD, is embedded within such\na polydisperse disordered system. We further acquire a correlation function between critical solid\nfractions and dimensionless grain volume distributions. This work elucidates the effect of particle\nvolumes on the rheology and micromechanics of dry granular systems and provides further insights\nin better incorporating the influence of other particle properties into a unified framework, which is\nhelpful and critical for the corresponding engineering and geophysical problems.\nGranular materials and disordered systems are com-\nmonly encountered in natural systems and industrial pro-\ncesses, such as landslides, pyroclastic flows, fresh con-\ncrete, and pharmaceutical and chemical particulates [1–\n4]. In recent decades, extensive investigations have been\nconducted to describe the rheology[5, 6], compaction\n[7, 8], and jamming transition [9, 10] of granular ma-\nterials. The proposal of the inertial number Ic and the\nviscous number Iv suggests that the dynamics of granular\nflows is governed by the combination of three time scales:\nti = dp/\np\nσn/ρp, tv = ηf/σn, and tM = 1/˙γ, where ti,\ntv, and tM are inertial, viscous, and macroscopic time\nscales, dp and ρp are the particle diameter and density,\nσn is the pressure, and ηf denotes the fluid viscosity if\nthe system is submerged or fully saturated.\nHowever, the current rheological framework cannot\nfully capture the influence of either particle shapes or size\ndistributions.\nThe grain size distribution (GSD) plays\nan important role in the rheology of granular systems,\nbut previous studies often provide seemingly contradic-\ntory opinions, where some studies suggested that the ex-\nistence of small particles initiates a lubrication effect that\nhelps the system increase its mobility [11], while others\nbelieved that a wider GSD induces a denser packing frac-\ntion that increases the shear strength of granular systems\nand makes the system easier to transition to a jammed\nstate [12].\nHill and Yohannes [13] studied the rheology of bidis-\npersed granular flows using the discrete element model\n(DEM) [13, 14] and suggested that the pressure of a\ngranular system is determined not only by its size dis-\n∗manteng0520@zjut.edu.cn\n† sunhonglei@zju.edu.cn\ntribution, but also by the effective free volume per par-\nticle, which was adopted from the jammed packing of\nmonosized particles.\nThey stated that the dimension-\nless pressure, (1/Ic)2, scales as a function of the coor-\ndination number, Zc.\nPolan´ıa et al. [15] studied the\nrheological behavior of polydispersed granular systems,\nwhich concluded that such systems still behave similarly\nto monodispersed systems if the right length scale is cho-\nsen, but the solid fraction can be quite different due to\npolydispersity. Recently, Ding et al. [11] also studied the\nbehavior of granular systems with fractal distributions,\nand found that the shear strength of confined flow de-\ncreases at large fractal dimensions, which can be further\nlinked to help decipher the mechanisms within the highly\nlocalized basal zones and the high mobility of general ge-\nological avalanches.\nIn this Letter, we focus on disordered physical sys-\ntems with various continuous GSDs, where sheared gran-\nular materials with three different cumulative distribu-\ntion functions (CDF) are examined using DEM. We show\nthat, unlike the length scale proposed in Polan´ıa et al.\n[15], a simply defined “average” particle size, which has\na clear physical root in the contact probability, already\nunifies the rheological behavior of dry granular systems\nwith a wide range of different particle size distributions.\nFurther analyses are devoted to better understand the\ncritical phenomena when a granular system with differ-\nent GSDs and frictional properties is approaching the\n“jamming” transition.\nMethodology and simulation setup In this work, the\nclassical DEM is used to simulate sheared granular sys-\ntems, where the particles are spherical and follow a lin-\near spring-dashpot contact model. The normal contact\nforce F n\nij is calculated as a Hookean contact with energy\ndamping. The magnitude of the tangential contact force\narXiv:2510.14797v1 [cond-mat.soft] 16 Oct 2025\n\n2\nFIG. 1. Rheological behaviors: (a-c) The µeff ∼Ica and ϕs ∼Ica relationships plotted for systems with size distribution\nD1, D2, and D3, where Ica is the conventional inertial number that uses system averaged particle diameter as the characteristic\nlength. The inset of (b) shows the relationship between two length scales, dac and daw, where dac is the average particle\ndiameter calculated based on contact number of each particle and daw is the average particle diameter calculated based on\nvolume of each particle. (e-f) The µeff ∼Icw and ϕs ∼Icw relationships plotted for systems with size distribution D1, D2, and\nD3, where Icw is the new dimensionless number calculated with daw. Inset of (d) presents a simulation snapshop, and the inset\nof (f) is the relationship between ϕs/ϕc and the proposed new inertial number Icw (see Equation 1.\n|ktδt\nij| cannot exceed its friction limit, µp|F n\nij|, where µp\nis the particle frictional coefficient and for most cases we\nset µp = 0.3. The system has no rolling resistance. The\ngranular system is located between two sawteeth plates\nand is sheared with constant pressure and shear rate. For\ndifferent simulations, we vary the pressure σ from 50 to\n1000 Pa and the shear rate ˙γ from 0.01 to 1.0 s−1, and\nmeasure the mean value of the key variables when the\nsheared system is already in steady state, shown in Ap-\npendix A, where its solid fraction ϕs, pressure σ, shear\nstress τ, and velocity profile become stable. We can ob-\ntain the effective frictional coefficient µeff = τ/σ, the\neffective viscosity ηeff = τ/˙γ, and the average inertial\nnumber Ic from each simulation for further investigation.\nThe plot of a shear cell is shown in the inset of FIG.\n1(d). The particle size distributions used in this work,\ni.e., Fuller distribution and two power-law distributions,\nare elaborated in detail in Appendix B.\nRheological behaviors According to the classical µ(I)\nrheology of dry granular systems, both µeff = τ/σ and ϕs\nshould scale with the inertial number Ica = ˙γdp/\np\nσ/ρp,\nwhere dp is the particle diameter and also a characteristic\nlength scale. Ge et al. [6] proposed that the length scale\nplays an important role in determining the rheological\nbehavior of granular systems, and we further argue that\nthis length scale should be adjusted to reflect changes\nin particle size distributions. FIGs. 1(a-b) demonstrate\nthe µeff ∼Ica relationship for systems with GSDs of D1,\nD2, and D3, which are clarified in Appendix B, whereas\nFIG. 1(c) shows the ϕs ∼Ica relationship, where Ica rep-\nresents the conventional inertial number that uses the\naverage particle diameter of the system as the character-\nistic length.\nFIGs. 1(a-c) shows that, even though the µeff ∼Ica\nand ϕs ∼Ica relationships collapse for systems with any\nspecific GSD, changing the particle gradation leads to\ndeviations in both µeff and ϕs.\nFIGs.\n1(a) and (b)\nshow that systems with different particle gradations re-\nsult in different transitional Ica, which denotes a transi-\ntion from quasi-static to inertial flow. The change in this\ntransitional Ica implies that the characteristic length of\nthe system may also change due to changes of GSDs.\nHill and Yohannes [13] reported that the average particle\nsize should be calculated based on each particle volume.\nThus, we define the characteristic length of the system\nas daw = [PNp\ni=1(dpiVpi)]/(PNp\nj=1 Vpj) and particles with\ndifferent volumes are assumed to have different weight\nin calculating this “average” particle size. Thus, a new\ninertial number can be defined as\nIcw = ˙γdaw(ρp/σ)1/2 .\n(1)\nAfter adopting the new inertial number, as shown in\nFIGs. 1(d,e), the relationships between µeff and Icw col-\nlapse onto a single master curve, indicating that the char-\nacteristic length, daw, reflects some physical and topolog-\nical nature of the sheared granular systems, and choos-\ning an appropriate length scale helps unify the frictional\nrheology of dry granular systems with any size distri-\nbution. However, we admit that Icw fails to unify the\nϕs ∼Icw relationship, although FIG. 1(f) looks much\nbetter than FIG. 1(c). For systems with different GSDs,\ntheir transitional Icw seems to be unified, and the only\n\n3\nFIG. 2. Relationship between the effective viscosity of gran-\nular systems, ηeff = τ/ ˙γ, and the solid fraction, ϕs. In the\ninset of this figure, we change the horizontal and vertical axes\ninto |ϕs −ϕc| and ηeff\n−1/1.3, respectively.\nThe dashed line\nrepresents a linear relationship.\ndifference lies in the ϕs−axis, i.e., moving the ϕs ∼Icw\nrelationships upward or downward can also unify this re-\nlation. For a series of simulations with the same GSD,\nthe relationship between ϕs and Icw can be written as\nϕs = ϕc/(1 + αϕIcw), where αϕ ≈0.32.\nThus, if we\nchange the vertical coordinate into ϕs/ϕc, where ϕc is\nthe critical solid fraction for each GSD, we can obtain\na master curve of ϕs/ϕc = 1/(1 + αϕIcw), shown as the\nblue dashed curve in the inset of FIG. 1(f).\nCritical length scale and critical solid fraction Our\nchoices of characteristic length and critical solid frac-\ntion have deep physical and geometrical reasons, even\nthough they seem to be deliberately fitted. On one hand,\ndaw can be related to the coordination number of each\npolydispersed granular system. The characteristic length\nshould be linked to particle volumes because they are\ndirectly related to the number of contacts. Intuitively,\na particle with larger size has a greater chance to con-\ntact with more particles, indicating that such a particle\nplays a more important role in bearing strong force net-\nworks and determining the macroscopic behavior of the\nwhole system. To validate this, we calculated another\nlength scale, dac, based on contact numbers of each par-\nticle, where dac = [P\ni(dpiNci)]/(P\nj Ncj) and Nci is the\ncontact number of particle i, and plotted its relation-\nship with daw in the inset of FIG. 1(b). This shows that\ndaw and dac are linearly related, which verifies that daw,\nwhich can be easily obtained from GSD, has a deep root\nin contact statistics.\nϕc varies with the GSD and frictional properties of\nparticles. It is clearly neither the jamming solid fraction,\nϕJ [10], nor the random close packing fraction, ϕRCP [16].\nHowever, Zaccone [16] and his follow-up research on the\nrandom close packing of the polydisperse granular system\n[17] encourage us to argue that ϕc is a function of ϕRCP\nand µp. Since we kept µp constant, ϕc is predominantly\ncontrolled by ϕRCP , which should be related to the GSD\nof a granular system. Thus, we could obtain a functional\nrelationship between ϕc and the GSD.\nWe note that ϕc for each GSD should not be ob-\ntained from fitting of the ϕs ∼Icw relationship, since\nthe functional form of the ϕs ∼Icw relationship is\nbased on phenomena, not physics. Fortunately, Zaccone\n[16] derived the relationship between effective viscosity\nηeff = τ/˙γ and ϕs from first principles, which yields\nηeff ∼|ϕc −ϕs|−1.3. Thus, if we plot ηeff −1/1.3 against\n|ϕc −ϕs|, we should be able to obtain a linear relation-\nship. In FIG. 2, we plot ηeff against ϕs for all simulations\nand find that ηeff always diverges at relatively high solid\nfractions (which are different for systems with different\nGSDs). The solid fraction, where a granular system is\napproaching infinite viscosity, indicates a critical solid\nfraction, ϕc. For each GSD, we can obtain a specific ϕc,\nand ηeff −1/1.3 and |ϕc −ϕs| are linearly related, as shown\nin the inset of FIG. 2.\nOur above analyses indicate that the length scale daw,\nwhich implies contact statistics, helps to unify the µeff ∼\nIcw relationship but leaves a problem of ϕc for us to solve.\nThis implies that, for systems with different GSDs but ex-\nactly the same daw, their stress-strain rate relationships\nbehave similarly, but their solid fractions differ from each\nother. Since stresses in granular materials are strongly\nrelated to force networks (especially the strong force net-\nwork), granular systems with different GSDs may have\ndifferent amounts of rattler particles, which do not alter\nthe strong force network, but occupy different amounts\nof volumes, resulting in different solid fractions.\nWe define particles with zero contact or only one con-\ntact as rattlers, while particles with more than one con-\ntact as non-rattlers. While calculating the contact num-\nber of each particle, Nc, and the coordination number,\nZc ≡⟨Nc⟩, of each simulation, we can also obtain the\nvolumetric percentage of rattlers, Vp01/Vp, and the coor-\ndination number of non-rattlers, Zc≥2. FIG. 3(a) plots\nthe histograms for a set of simulations with GSD of D3\nand η3 = 1.0, where the legend provides the information\nof Icw. Increasing Icw leads to a decrease in the num-\nber of particles with large contact numbers and increases\nthe chances of finding particles with only one contact,\nindicating that increasing the inertial number results in\nmore binary collisions. With such a wide range of parti-\ncle sizes, a sheared granular system naturally has many\n“small” particles, which do not play a role in the force\nnetwork.\nFIG. 3(b) indicates that, for some cases, Zc\nis even below 1.0, but for each set of simulations with\nthe same GSD, the data collapse onto a Zc ∼ϕs master\ncurve. However, it is difficult to figure out how changing\nthe GSD alters Zc.\nNeglecting particles with only zero and one contact\nnumber and changing the vertical axis into the average\ncoordination number of particles with more than one con-\ntact, Zc≥2, in FIG. 3(c), the relationship Zc≥2 ∼ϕs\nshows power-law features with systematic changes intro-\nduced by changing the GSD. The inset of FIG. 3(c) shows\nthat dividing ϕs by ϕc helps to get a unified Zc≥2 ∼ϕs/ϕc\nrelationship, which is similar to the ηeff ∼ϕs/ϕc relation-\nship. We argue that it is mostly particles with Nc ≤2\nthat determine the effective viscosity of the system. All\nof these imply that changing the GSD leads to a dif-\n\n4\nFIG. 3. (a) Histogram of the number of contact while changing Icw. This figure only plots data from the system of D3 and ηs =\n1.0. (b) The Zc ∼ϕs relationship for systems with different GSDs, and Zc is the average coordination number. (c) The volume\npercentage of particles with only 0 or 1 contact is plotted against Icw. (d) Relationship between the dimensionless granular\ntemperature, Tg/( ˙γdaw)2, and ϕs (inset: Tg/( ˙γdaw)2 ∼|ϕc −ϕs|). (e) The relationship between ϕc and the dimensionless index\nof the grain volume distribution, IGV D.\nferent percentage of particles that participate in bearing\nforces in granular systems. However, FIG. 3(d) contra-\ndicts our expectation and shows that changing the GSD\ndoes not change the volume percentage of particles much,\nas Nc ≤1.\nWe further plot the relationship between the normal-\nized granular temperature Tg/(˙γdaw)2 and ϕs in FIG.\n3(e), where Tg ≡⟨(⃗u −⟨⃗u⟩)2⟩and ⃗u is the particle ve-\nlocity.\nAgain, the critical solid fraction ϕc, which is\nheavily influenced by the GSD, plays a key role in the\nscaling of Tg/(˙γdaw)2 ∼ϕs, and Tg/(˙γdaw)2 scales with\n∼|ϕc −ϕs|−0.6.\nWe argue that ϕc, as a critical solid\nfraction, marks the transition from a system with finite\nviscosity to that with infinite viscosity. Physically, it has\nsimilar characteristics as random close packing fractions\nϕRCP , which are affected by both µp and GSD.\nZaccone [17] and Anzivino et al. [18] suggested that\nϕRCP can be expressed as a function of the standard de-\nviation of GSD, while Desmond and Weeks [12] argued\nthat they could draw a unified relation between ϕRCP\nand a combination of skewness and polydispersity of GSD\nwith the data obtained from simulations. However, we\nfind that both methods do not work for our system. We\nadopted the idea of Desmond and Weeks [12] and calcu-\nlated both the skewness Sv and the polydispersity δv of\nthe normalized grain volume distribution (GVD), instead\nof the GSD, so that\n˜V i\np =\n\u0002\nV i\np −min(Vp)\n\u0003\n/ [max(Vp) −min(Vp)] , (2a)\nSv = ⟨∆˜V 3\np ⟩/\n\u0010\n⟨∆˜V 2\np ⟩\n\u00113/2\n, δv =\nq\n⟨∆˜V 2\np ⟩/⟨˜Vp⟩, (2b)\nwhere ˜Vp is the normalized particle volume, ∆˜Vp = ˜Vp −\n⟨˜Vp⟩, ⟨˜V n\np ⟩=\nR ˜V n\np · P( ˜Vp)d ˜Vp, and ⟨∆˜V (n)\np\n⟩=\nR\n∆˜V n\np ·\nP( ˜Vp)d ˜Vp. To obtain a universal relationship, we varied\nboth the size ranges ([1, 10] cm, [2, 10] cm, and [3.16, 10]\ncm) and the particle frictional coefficient (µp =0.0001,\n0.1, 0.3 and 0.9). With a combined dimensionless index\nof the GVD, IGV D = δv +Svδ2\nv, we obtained an empirical\nrelationship between ϕc and IGV D so that\nϕc =1 −(1 −ϕc0)/ [1 + β ln (1 + IGV D/Ig0)] ,\n(3)\nwhere β ≈0.087 and Ig0 = 4.5 are fitting constants, ϕc0 is\nthe base ϕc and varies with µp, and the functional form\nof Eq. 3 is inspired by the time evolution of the solid\nfraction of granular system under tapping compactions\nin Knight et al. [19]. FIG. 3(f) suggested that we only\nneed to adjust ϕc0 to get a reasonable ϕc ∼IGV D re-\nlationship for systems with different µp, and Eq. 3, al-\nthough being empirical, introduces a logarithmic depen-\ndence that ensures the eventual “flattening” of ϕc with\nincreasing IGV D. Ultimately, ϕc should flatten out to a\n“maximum” packing fraction that is lower than or equal\nto 1 when IGV D →+∞, as also suggested in previous\npublications [12, 18, 20].\nIn Appendix C, we plot the\nϕRCP ∼IGV D relationship for data extracted from pre-\nvious publications to show that Eq.\n3 also works for\npredicting the random close packing fractions.\nThis Letter focuses on the rheology of granular sys-\ntems with different grain size distributions and identifies\na characteristic length scale that can be associated with\nthe contact probability. We find that varying the GSD\nleads to the change in criticality behavior, i.e. the critical\n\n5\nFIG. 4. (a) The time evolution of the shear stress, τ, pres-\nsure, σn, and the solid fraction, ϕs. (b) the profile of particle\nvelocities in the x−direction, Vpx. The inset of (b) shows the\nconfiguration of a simple shear simulation of the granular sys-\ntem.\nsolid fraction ϕc that marks the transition from a finite\nηeff to an infinite one. The analyses of Zc and Tg verify\nthe importance of determining ϕc based on the GVD, and\nwe have introduced a dimensionless index, IGV D, to suc-\ncessfully acquire a unified solution for granular systems\nwith different particle gradations, which has the poten-\ntial to be adopted to other athermal disordered systems\nwith various polydispersities.\nACKNOWLEDGMENTS\nAcknowledgments- We wish to acknowledge the finan-\ncial support of Grant No.\n12202367 from NSFC and\nthe start-up grant from Zhejiang University of Tech-\nnology.\nT.M. thanks Prof.\nK.M. Hill, Prof.\nJ.-L.\nLe, Dr.\nZ. Ge, and Prof.\nS.A. Galindo-Torres for\nhelpful discussions. The simulations in this work were\nconducted on an open-access multiphysics simulation li-\nbrary: MECHSYS, which can be acquired from Github\n(https://github.com/Axtal/mechsys.git)\nAppendix A: Simulations and stress calculations\nThe discrete element method (DEM) solves the mo-\nmentum balance equation for both translational and ro-\ntational motions. The governing equations of DEM can\nbe written in the form of the Newton’s second law, where\nthe calculation of contact forces is of vital importance. In\neach simulation, the governing equations are integrated\nusing a Verlet method [21], which is commonly used in\nmolecular dynamics. Particles are spherical and follow a\nlinear spring-dashpot contact model, where the normal\ncontact force F n\nij is calculated as a Hookean contact with\nenergy damping, so that F n\nij = −knδn\nij −cn ˙δn\nij, where\nkn is the contact stiffness, cn is the damping coefficient,\nand δn\nij is the contact overlap. F n\nij has a viscous term for\nenergy dissipation, where cn is calculated based on the\ncoefficient of restitution, en, so that\ncn = 2ξn\np\nmeffkn , ξn = −ln en/\np\nπ2 + (ln en)2 , (A1)\nwhere meff = (1/mi + 1/mj)−1 is the reduced mass and\nwe set en = 0.5. The magnitude of the tangential con-\ntact force |ktδt\nij| cannot exceed its frictional limit, µp|F n\nij|,\nwhere µp is the friction coefficient of particles.\nThe\ngranular system is located within and sheared by two\nsawteeth plates at constant pressure and shear rate, as\nshown in the inset of FIG. 4(b). At each timestep, the\nsheared granular system reaches a steady state when ϕs,\nτ, and σn become stable. Meanwhile, instead of measur-\ning stresses from the upper and lower plates, we calcu-\nlated the average stress tensor for each output timestep.\nThe average stress tensor σij can be obtained by\nσij = (1/V )\nNct\nX\nc\nf i\nclj\nc ,\n(A2)\nwhere V is the volume of the granular system, f i\nc is the\nith component of a contact force vector (a summation\nof normal and tangential contact forces of the cth con-\ntact), and lj\nc is the jth component of a contact branch\nvector that links centroids of two contacting particles.\nFIG. 4(a) presents typical time-lapse evolutions of ϕs, τ,\nand σn, showing a clear steady state for determining the\nrepresentative average values. Since we utilized sawteeth\nplates as upper and lower boundaries, the velocity pro-\nfile across height is always linear [shown in FIG. 4(b)],\nindicating a constant shear rate.\nAppendix B: Particle size distributions\nIn this letter, we use two different types of particle size\ndistributions, i.e., Fuller distribution (D1) and power-law\ndistributions (D2 and D3). The cumulative distribution\nfunction (CDF) of D1 can be written as\nC1(dp) = [(dp −dmin)/(dmax −dmin)]η1 .\n(B1)\nThen, a random particle diameter that follows this CDF\ncan be obtained by imposing a uniformly distributed ran-\ndom number within [0, 1], Rand(0, 1), on the left-hand\nside of Eq. B1 so that\ndp = dmin + [Rand(0, 1)]1/η1 · (dmax −dmin).\n(B2)\n\n6\nFIG. 5. (a-d) show both the PDF and CDF of different GSDs for particle size ranging from 1 to 10 cm, while (c) and (f)\nshow the PDF and CDF for systems with dp ∈[3.16, 10] cm. In Figs. (g,h), we plot the rheological properties of systems with\ndp ∈[3.16, 10] cm.\nFIG. 6.\nData extracted from published papers, where the\nparticle size in Farr and Groot [22] follows a lognormal dis-\ntribution and that in Anzivino et al. [18] follows a Gamma\ndistribution. The published data are fitted with Eq. 3.\nConsidering the possible slopes of the PDF of power-law\ndistributions, we take two different functional forms:\nC2(dp) = (dp/dmax)η2 ,\n(B3a)\ndp = dmax · [Rand(0, 1)]1/η2 ,\n(B3b)\nC3(dp) = 1 −(dp/dmin)−η3 ,\n(B3c)\ndp = dmin · [1 −Rand(0, 1)]−1/η3 .\n(B3d)\nwhere we impose the dp ∈[dmin, dmax] condition every\ntime we generate a random particle diameter. Figures\n5(a-f) show both the PDF and CDF of systems with\ndifferent GSDs, and FIGs.\n5(g, h) show the rheologi-\ncal properties of granular systems with a different size\nrange, which indicate that changing the particle size\nrange would not change the rheological behavior, espe-\ncially when we consider Icw and eliminate the effect of\nϕc.\nAppendix C: ϕRCP extracted from previous\npublications\nWe mentioned that previous research has focused on\npredicting the random close packing solid fraction, ϕRCP ,\nwhich is closely related to ϕc in our present work. How-\never, we make clear that ϕRCP is different from ϕc. The\nstudies of ϕRCP are mostly focused on frictionless sys-\ntems.\nFurthermore, DEM simulations, where particles\nare in loading conditions, allow for the existence of over-\nlaps among contacting particles, which will result in rel-\natively larger solid fractions. To validate the ϕc ∼IGV D\nrelationship, we turn to published works to also link\nϕRCP to IGV D. In FIG. 6, we plot the data extracted\nfrom both Farr and Groot [22] with lognormal size dis-\ntribution and Anzivino et al. [18] with Γ distribution.\nWe converted their particle size distributions into parti-\ncle volume distributions and obtained the corresponding\nstatistical parameters to calculate IGV D. FIG. 6 shows\nthat our empirical relationship of Eq. 3 also works for\npredicting ϕRCP for frictionless spheres with a single fit-\nting parameter of ϕc0 ≈0.645.\n[1] T. Man, Mathematical modeling of pavement gyratory\ncompaction: A perspective on granular-fluid assemblies,\nMathematics 11, 2096 (2023).\n\n7\n[2] T. Man, H. E. Huppert, and S. A. Galindo-Torres, Run-\nout scaling of granular column collapses on inclined\nplanes, J. Fluid Mech. 1002, A50 (2025).\n[3] Y. Lu, W. Jin, J. Klinger, T. L. Westover, and S. Dai,\nFlow characterization of compressible biomass particles\nusing multiscale experiments and a hypoplastic model,\nPow. Technol. 383, 396 (2021).\n[4] Y. Lu, W. 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Phys. 131, 244104\n(2009).\n"},"cleaned":{"kind":"string","value":"Grain volume distribution alters the critical phenomena in complex granular systems Teng Man,1, ∗Yimin Lu,2 Zhongrong Wang,3 Herbert Huppert,4 Alessio Zaccone,5 and Honglei Sun1, † 1 310023, China 2 79409, United States 3 3001, Australia 4 's College, 's Parade, Cambridge CB2 1ST, United Kingdom 5 \"A. Pontremoli,\" 16, 20133 Milan, Italy (Dated: October 17, 2025) The grain size distribution (GSD) plays an important role in the mechanical properties of amorphous disordered systems and complex granular materials. Varying GSD causes segregation issues and alters critical behaviors. This work used the discrete element method (DEM) to investigate the rheological and critical behaviors of sheared granular flows with various GSDs. The results show that, while a unified rheological relation can be obtained, a characteristic length scale, which is associated with the contact probability and can be obtained from any GSD, is embedded within such a polydisperse disordered system. We further acquire a correlation function between critical solid fractions and dimensionless grain volume distributions. This work elucidates the effect of particle volumes on the rheology and micromechanics of dry granular systems and provides further insights in better incorporating the influence of other particle properties into a unified framework, which is helpful and critical for the corresponding engineering and geophysical problems. Granular materials and disordered systems are commonly encountered in natural systems and industrial processes, such as landslides, pyroclastic flows, fresh concrete, and pharmaceutical and chemical particulates [14]. In recent decades, extensive investigations have been conducted to describe the rheology[5, 6], compaction [7, 8], and jamming transition [9, 10] of granular materials. The proposal of the inertial number Ic and the viscous number Iv suggests that the dynamics of granular flows is governed by the combination of three time scales: ti = dp/ p σn/ρp, tv = ηf/σn, and tM = 1/ ̇γ, where ti, tv, and tM are inertial, viscous, and macroscopic time scales, dp and ρp are the particle diameter and density, σn is the pressure, and ηf denotes the fluid viscosity if the system is submerged or fully saturated. However, the current rheological framework cannot fully capture the influence of either particle shapes or size distributions. The grain size distribution (GSD) plays an important role in the rheology of granular systems, but previous studies often provide seemingly contradictory opinions, where some studies suggested that the existence of small particles initiates a lubrication effect that helps the system increase its mobility [11], while others believed that a wider GSD induces a denser packing fraction that increases the shear strength of granular systems and makes the system easier to transition to a jammed state [12]. Hill and Yohannes [13] studied the rheology of bidispersed granular flows using the discrete element model (DEM) [13, 14] and suggested that the pressure of a granular system is determined not only by its size dis- ∗ † tribution, but also by the effective free volume per particle, which was adopted from the jammed packing of monosized particles. They stated that the dimensionless pressure, (1/Ic)2, scales as a function of the coordination number, Zc. Polan ́ıa et al. [15] studied the rheological behavior of polydispersed granular systems, which concluded that such systems still behave similarly to monodispersed systems if the right length scale is chosen, but the solid fraction can be quite different due to polydispersity. Recently, Ding et al. [11] also studied the behavior of granular systems with fractal distributions, and found that the shear strength of confined flow decreases at large fractal dimensions, which can be further linked to help decipher the mechanisms within the highly localized basal zones and the high mobility of general geological avalanches. In this Letter, we focus on disordered physical systems with various continuous GSDs, where sheared granular materials with three different cumulative distribution functions (CDF) are examined using DEM. We show that, unlike the length scale proposed in Polan ́ıa et al. [15], a simply defined \"average\" particle size, which has a clear physical root in the contact probability, already unifies the rheological behavior of dry granular systems with a wide range of different particle size distributions. Further analyses are devoted to better understand the critical phenomena when a granular system with different GSDs and frictional properties is approaching the \"jamming\" transition. Methodology and simulation setup In this work, the classical DEM is used to simulate sheared granular systems, where the particles are spherical and follow a linear spring-dashpot contact model. The normal contact force F n ij is calculated as a Hookean contact with energy damping. The magnitude of the tangential contact force 16 Oct 2025 2 FIG. 1. Rheological behaviors: (a-c) The μeff ∼Ica and φs ∼Ica relationships plotted for systems with size distribution D1, D2, and D3, where Ica is the conventional inertial number that uses system averaged particle diameter as the characteristic length. The inset of (b) shows the relationship between two length scales, dac and daw, where dac is the average particle diameter calculated based on contact number of each particle and daw is the average particle diameter calculated based on volume of each particle. (e-f) The μeff ∼Icw and φs ∼Icw relationships plotted for systems with size distribution D1, D2, and D3, where Icw is the new dimensionless number calculated with daw. Inset of (d) presents a simulation snapshop, and the inset of (f) is the relationship between φs/φc and the proposed new inertial number Icw (see Equation 1. |ktδt ij| cannot exceed its friction limit, μp|F n ij|, where μp is the particle frictional coefficient and for most cases we set μp = 0.3. The system has no rolling resistance. The granular system is located between two sawteeth plates and is sheared with constant pressure and shear rate. For different simulations, we vary the pressure σ from 50 to 1000 Pa and the shear rate ̇γ from 0.01 to 1.0 s-1, and measure the mean value of the key variables when the sheared system is already in steady state, shown in Appendix A, where its solid fraction φs, pressure σ, shear stress τ, and velocity profile become stable. We can obtain the effective frictional coefficient μeff = τ/σ, the effective viscosity ηeff = τ/ ̇γ, and the average inertial number Ic from each simulation for further investigation. The plot of a shear cell is shown in the inset of FIG. 1(d). The particle size distributions used in this work, i.e., Fuller distribution and two power-law distributions, are elaborated in detail in Appendix B. Rheological behaviors According to the classical μ(I) rheology of dry granular systems, both μeff = τ/σ and φs should scale with the inertial number Ica = ̇γdp/ p σ/ρp, where dp is the particle diameter and also a characteristic length scale. Ge et al. [6] proposed that the length scale plays an important role in determining the rheological behavior of granular systems, and we further argue that this length scale should be adjusted to reflect changes in particle size distributions. FIGs. 1(a-b) demonstrate the μeff ∼Ica relationship for systems with GSDs of D1, D2, and D3, which are clarified in Appendix B, whereas FIG. 1(c) shows the φs ∼Ica relationship, where Ica represents the conventional inertial number that uses the average particle diameter of the system as the characteristic length. FIGs. 1(a-c) shows that, even though the μeff ∼Ica and φs ∼Ica relationships collapse for systems with any specific GSD, changing the particle gradation leads to deviations in both μeff and φs. FIGs. 1(a) and (b) show that systems with different particle gradations result in different transitional Ica, which denotes a transition from quasi-static to inertial flow. The change in this transitional Ica implies that the characteristic length of the system may also change due to changes of GSDs. Hill and Yohannes [13] reported that the average particle size should be calculated based on each particle volume. Thus, we define the characteristic length of the system as daw = [PNp i=1(dpiVpi)]/(PNp j=1 Vpj) and particles with different volumes are assumed to have different weight in calculating this \"average\" particle size. Thus, a new inertial number can be defined as Icw = ̇γdaw(ρp/σ)1/2 . (1) After adopting the new inertial number, as shown in FIGs. 1(d,e), the relationships between μeff and Icw collapse onto a single master curve, indicating that the characteristic length, daw, reflects some physical and topological nature of the sheared granular systems, and choosing an appropriate length scale helps unify the frictional rheology of dry granular systems with any size distribution. However, we admit that Icw fails to unify the φs ∼Icw relationship, although FIG. 1(f) looks much better than FIG. 1(c). For systems with different GSDs, their transitional Icw seems to be unified, and the only 3 FIG. 2. Relationship between the effective viscosity of granular systems, ηeff = τ/ ̇γ, and the solid fraction, φs. In the inset of this figure, we change the horizontal and vertical axes into |φs -φc| and ηeff -1/1.3, respectively. The dashed line represents a linear relationship. difference lies in the φs-axis, i.e., moving the φs ∼Icw relationships upward or downward can also unify this relation. For a series of simulations with the same GSD, the relationship between φs and Icw can be written as φs = φc/(1 + αφIcw), where αφ ≈0.32. Thus, if we change the vertical coordinate into φs/φc, where φc is the critical solid fraction for each GSD, we can obtain a master curve of φs/φc = 1/(1 + αφIcw), shown as the blue dashed curve in the inset of FIG. 1(f). Critical length scale and critical solid fraction Our choices of characteristic length and critical solid fraction have deep physical and geometrical reasons, even though they seem to be deliberately fitted. On one hand, daw can be related to the coordination number of each polydispersed granular system. The characteristic length should be linked to particle volumes because they are directly related to the number of contacts. Intuitively, a particle with larger size has a greater chance to contact with more particles, indicating that such a particle plays a more important role in bearing strong force networks and determining the macroscopic behavior of the whole system. To validate this, we calculated another length scale, dac, based on contact numbers of each particle, where dac = [P i(dpiNci)]/(P j Ncj) and Nci is the contact number of particle i, and plotted its relationship with daw in the inset of FIG. 1(b). This shows that daw and dac are linearly related, which verifies that daw, which can be easily obtained from GSD, has a deep root in contact statistics. φc varies with the GSD and frictional properties of particles. It is clearly neither the jamming solid fraction, φJ [10], nor the random close packing fraction, φRCP [16]. However, Zaccone [16] and his follow-up research on the random close packing of the polydisperse granular system [17] encourage us to argue that φc is a function of φRCP and μp. Since we kept μp constant, φc is predominantly controlled by φRCP , which should be related to the GSD of a granular system. Thus, we could obtain a functional relationship between φc and the GSD. We note that φc for each GSD should not be obtained from fitting of the φs ∼Icw relationship, since the functional form of the φs ∼Icw relationship is based on phenomena, not physics. Fortunately, Zaccone [16] derived the relationship between effective viscosity ηeff = τ/ ̇γ and φs from first principles, which yields ηeff ∼|φc -φs|-1.3. Thus, if we plot ηeff -1/1.3 against |φc -φs|, we should be able to obtain a linear relationship. In FIG. 2, we plot ηeff against φs for all simulations and find that ηeff always diverges at relatively high solid fractions (which are different for systems with different GSDs). The solid fraction, where a granular system is approaching infinite viscosity, indicates a critical solid fraction, φc. For each GSD, we can obtain a specific φc, and ηeff -1/1.3 and |φc -φs| are linearly related, as shown in the inset of FIG. 2. Our above analyses indicate that the length scale daw, which implies contact statistics, helps to unify the μeff ∼ Icw relationship but leaves a problem of φc for us to solve. This implies that, for systems with different GSDs but exactly the same daw, their stress-strain rate relationships behave similarly, but their solid fractions differ from each other. Since stresses in granular materials are strongly related to force networks (especially the strong force network), granular systems with different GSDs may have different amounts of rattler particles, which do not alter the strong force network, but occupy different amounts of volumes, resulting in different solid fractions. We define particles with zero contact or only one contact as rattlers, while particles with more than one contact as non-rattlers. While calculating the contact number of each particle, Nc, and the coordination number, Zc ≡⟨Nc⟩, of each simulation, we can also obtain the volumetric percentage of rattlers, Vp01/Vp, and the coordination number of non-rattlers, Zc≥2. FIG. 3(a) plots the histograms for a set of simulations with GSD of D3 and η3 = 1.0, where the legend provides the information of Icw. Increasing Icw leads to a decrease in the number of particles with large contact numbers and increases the chances of finding particles with only one contact, indicating that increasing the inertial number results in more binary collisions. With such a wide range of particle sizes, a sheared granular system naturally has many \"small\" particles, which do not play a role in the force network. FIG. 3(b) indicates that, for some cases, Zc is even below 1.0, but for each set of simulations with the same GSD, the data collapse onto a Zc ∼φs master curve. However, it is difficult to figure out how changing the GSD alters Zc. Neglecting particles with only zero and one contact number and changing the vertical axis into the average coordination number of particles with more than one contact, Zc≥2, in FIG. 3(c), the relationship Zc≥2 ∼φs shows power-law features with systematic changes introduced by changing the GSD. The inset of FIG. 3(c) shows that dividing φs by φc helps to get a unified Zc≥2 ∼φs/φc relationship, which is similar to the ηeff ∼φs/φc relationship. We argue that it is mostly particles with Nc ≤2 that determine the effective viscosity of the system. All of these imply that changing the GSD leads to a dif4 FIG. 3. (a) Histogram of the number of contact while changing Icw. This figure only plots data from the system of D3 and ηs = 1.0. (b) The Zc ∼φs relationship for systems with different GSDs, and Zc is the average coordination number. (c) The volume percentage of particles with only 0 or 1 contact is plotted against Icw. (d) Relationship between the dimensionless granular temperature, Tg/( ̇γdaw)2, and φs (inset: Tg/( ̇γdaw)2 ∼|φc -φs|). (e) The relationship between φc and the dimensionless index of the grain volume distribution, IGV D. ferent percentage of particles that participate in bearing forces in granular systems. However, FIG. 3(d) contradicts our expectation and shows that changing the GSD does not change the volume percentage of particles much, as Nc ≤1. We further plot the relationship between the normalized granular temperature Tg/( ̇γdaw)2 and φs in FIG. 3(e), where Tg ≡⟨(⃗u -⟨⃗u⟩)2⟩and ⃗u is the particle velocity. Again, the critical solid fraction φc, which is heavily influenced by the GSD, plays a key role in the scaling of Tg/( ̇γdaw)2 ∼φs, and Tg/( ̇γdaw)2 scales with ∼|φc -φs|-0.6. We argue that φc, as a critical solid fraction, marks the transition from a system with finite viscosity to that with infinite viscosity. Physically, it has similar characteristics as random close packing fractions φRCP , which are affected by both μp and GSD. Zaccone [17] and Anzivino et al. [18] suggested that φRCP can be expressed as a function of the standard deviation of GSD, while Desmond and Weeks [12] argued that they could draw a unified relation between φRCP and a combination of skewness and polydispersity of GSD with the data obtained from simulations. However, we find that both methods do not work for our system. We adopted the idea of Desmond and Weeks [12] and calculated both the skewness Sv and the polydispersity δv of the normalized grain volume distribution (GVD), instead of the GSD, so that ̃V i p = V i p -min(Vp) / [max(Vp) -min(Vp)] , (2a) Sv = ⟨∆ ̃V 3 p ⟩/ ⟨∆ ̃V 2 p ⟩ 3/2 , δv = q ⟨∆ ̃V 2 p ⟩/⟨ ̃Vp⟩, (2b) where ̃Vp is the normalized particle volume, ∆ ̃Vp = ̃Vp - ⟨ ̃Vp⟩, ⟨ ̃V n p ⟩= R ̃V n p · P( ̃Vp)d ̃Vp, and ⟨∆ ̃V (n) p ⟩= R ∆ ̃V n p · P( ̃Vp)d ̃Vp. To obtain a universal relationship, we varied both the size ranges ([1, 10] cm, [2, 10] cm, and [3.16, 10] cm) and the particle frictional coefficient (μp =0.0001, 0.1, 0.3 and 0.9). With a combined dimensionless index of the GVD, IGV D = δv +Svδ2 v, we obtained an empirical relationship between φc and IGV D so that φc =1 -(1 -φc0)/ [1 + β ln (1 + IGV D/Ig0)] , (3) where β ≈0.087 and Ig0 = 4.5 are fitting constants, φc0 is the base φc and varies with μp, and the functional form of Eq. 3 is inspired by the time evolution of the solid fraction of granular system under tapping compactions in Knight et al. [19]. FIG. 3(f) suggested that we only need to adjust φc0 to get a reasonable φc ∼IGV D relationship for systems with different μp, and Eq. 3, although being empirical, introduces a logarithmic dependence that ensures the eventual \"flattening\" of φc with increasing IGV D. Ultimately, φc should flatten out to a \"maximum\" packing fraction that is lower than or equal to 1 when IGV D →+∞, as also suggested in previous publications [12, 18, 20]. In Appendix C, we plot the φRCP ∼IGV D relationship for data extracted from previous publications to show that Eq. 3 also works for predicting the random close packing fractions. This Letter focuses on the rheology of granular systems with different grain size distributions and identifies a characteristic length scale that can be associated with the contact probability. We find that varying the GSD leads to the change in criticality behavior, i.e. the critical 5 FIG. 4. (a) The time evolution of the shear stress, τ, pressure, σn, and the solid fraction, φs. (b) the profile of particle velocities in the x-direction, Vpx. The inset of (b) shows the configuration of a simple shear simulation of the granular system. solid fraction φc that marks the transition from a finite ηeff to an infinite one. The analyses of Zc and Tg verify the importance of determining φc based on the GVD, and we have introduced a dimensionless index, IGV D, to successfully acquire a unified solution for granular systems with different particle gradations, which has the potential to be adopted to other athermal disordered systems with various polydispersities. ACKNOWLEDGMENTS Acknowledgments- We wish to acknowledge the financial support of Grant No. 12202367 from NSFC and the start-up grant from Zhejiang - nology. T.M. thanks Prof. K.M. Hill, Prof. J.-L. Le, Dr. Z. Ge, and Prof. S.A. Galindo-Torres for helpful discussions. The simulations in this work were conducted on an open-access multiphysics simulation library: MECHSYS, which can be acquired from Github (https://github.com/Axtal/mechsys.git) Appendix A: Simulations and stress calculations The discrete element method (DEM) solves the momentum balance equation for both translational and rotational motions. The governing equations of DEM can be written in the form of the Newton's second law, where the calculation of contact forces is of vital importance. In each simulation, the governing equations are integrated using a Verlet method [21], which is commonly used in molecular dynamics. Particles are spherical and follow a linear spring-dashpot contact model, where the normal contact force F n ij is calculated as a Hookean contact with energy damping, so that F n ij = -knδn ij -cn ̇δn ij, where kn is the contact stiffness, cn is the damping coefficient, and δn ij is the contact overlap. F n ij has a viscous term for energy dissipation, where cn is calculated based on the coefficient of restitution, en, so that cn = 2ξn p meffkn , ξn = -ln en/ p π2 + (ln en)2 , (A1) where meff = (1/mi + 1/mj)-1 is the reduced mass and we set en = 0.5. The magnitude of the tangential contact force |ktδt ij| cannot exceed its frictional limit, μp|F n ij|, where μp is the friction coefficient of particles. The granular system is located within and sheared by two sawteeth plates at constant pressure and shear rate, as shown in the inset of FIG. 4(b). At each timestep, the sheared granular system reaches a steady state when φs, τ, and σn become stable. Meanwhile, instead of measuring stresses from the upper and lower plates, we calculated the average stress tensor for each output timestep. The average stress tensor σij can be obtained by σij = (1/V ) Nct X c f i clj c , (A2) where V is the volume of the granular system, f i c is the ith component of a contact force vector (a summation of normal and tangential contact forces of the cth contact), and lj c is the jth component of a contact branch vector that links centroids of two contacting particles. FIG. 4(a) presents typical time-lapse evolutions of φs, τ, and σn, showing a clear steady state for determining the representative average values. Since we utilized sawteeth plates as upper and lower boundaries, the velocity profile across height is always linear [shown in FIG. 4(b)], indicating a constant shear rate. Appendix B: Particle size distributions In this letter, we use two different types of particle size distributions, i.e., Fuller distribution (D1) and power-law distributions (D2 and D3). The cumulative distribution function (CDF) of D1 can be written as C1(dp) = [(dp -dmin)/(dmax -dmin)]η1 . (B1) Then, a random particle diameter that follows this CDF can be obtained by imposing a uniformly distributed random number within [0, 1], Rand(0, 1), on the left-hand side of Eq. B1 so that dp = dmin + [Rand(0, 1)]1/η1 · (dmax -dmin). (B2) 6 FIG. 5. (a-d) show both the PDF and CDF of different GSDs for particle size ranging from 1 to 10 cm, while (c) and (f) show the PDF and CDF for systems with dp ∈[3.16, 10] cm. In Figs. (g,h), we plot the rheological properties of systems with dp ∈[3.16, 10] cm. FIG. 6. Data extracted from published papers, where the particle size in Farr and Groot [22] follows a lognormal distribution and that in Anzivino et al. [18] follows a Gamma distribution. The published data are fitted with Eq. 3. Considering the possible slopes of the PDF of power-law distributions, we take two different functional forms: C2(dp) = (dp/dmax)η2 , (B3a) dp = dmax · [Rand(0, 1)]1/η2 , (B3b) C3(dp) = 1 -(dp/dmin)-η3 , (B3c) dp = dmin · [1 -Rand(0, 1)]-1/η3 . (B3d) where we impose the dp ∈[dmin, dmax] condition every time we generate a random particle diameter. Figures 5(a-f) show both the PDF and CDF of systems with different GSDs, and FIGs. 5(g, h) show the rheological properties of granular systems with a different size range, which indicate that changing the particle size range would not change the rheological behavior, especially when we consider Icw and eliminate the effect of φc. Appendix C: φRCP extracted from previous publications We mentioned that previous research has focused on predicting the random close packing solid fraction, φRCP , which is closely related to φc in our present work. However, we make clear that φRCP is different from φc. The studies of φRCP are mostly focused on frictionless systems. Furthermore, DEM simulations, where particles are in loading conditions, allow for the existence of overlaps among contacting particles, which will result in relatively larger solid fractions. To validate the φc ∼IGV D relationship, we turn to published works to also link φRCP to IGV D. In FIG. 6, we plot the data extracted from both Farr and Groot [22] with lognormal size distribution and Anzivino et al. [18] with Γ distribution. We converted their particle size distributions into particle volume distributions and obtained the corresponding statistical parameters to calculate IGV D. 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Phys. 131, 244104 (2009)."}}},{"rowIdx":190,"cells":{"id":{"kind":"string","value":"2510.14794"},"raw":{"kind":"string","value":"1\nBridging Theory and Practice in\nReconfigurable Fluid Antenna Systems\nHalvin Yang, Member, IEEE, Yizhe Zhao, Member, IEEE, Kai-Kit Wong, Fellow, IEEE,\nHsiao-Hwa Chen, Life Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE\nAbstract—Fluid antennas, including those based on liquid, me-\nchanical, and pixel-based technologies, are poised to significantly\nenhance next-generation wireless systems by adaptively optimiz-\ning their radiation characteristics. Many theoretical analyses\nassumed near-instant reconfiguration, perfect channel knowledge,\nstatic or slowly varying propagation environments, and ideal\nmaterial properties that rarely hold in practice. In this article, we\ndissect these common assumptions and contrast them with the re-\nalities of finite actuation time, limited and imperfect channel state\ninformation, rapidly changing fading conditions, electromagnetic\ncoupling, and mechanical constraints. Through illustrative ex-\namples and simulations, we demonstrate how ignoring these\nfactors can lead to overestimated gains in capacity, coverage, etc..\nWe then propose modeling refinements, experimental validation\nmethods, and emerging control algorithms that better account\nfor real-world constraints. Our findings highlight that, while\nreconfigurable antennas remain highly promising for B5G/6G\nand Internet of things (IoT) applications, their full potential can\nonly be realized by incorporating practical considerations into\nsystem design and performance evaluation.\nIndex Terms—Fluid antenna, fluid antenna system (FAS),\nmovable antenna, modeling, resource allocation, 6G.\nI. INTRODUCTION\nW\nIRELESS networks beyond fifth-generation (5G) are\nrequired to keep up with the exponential growth in\nmobile data traffic while also providing ubiquitous connec-\ntivity. To achieve this, new wireless technologies need to\nbe developed. In recent years, multiple-input multiple-output\n(MIMO) technology has been viewed as a primary solution\nto drastically improve spectral efficiency (SE) and energy\nefficiency (EE) of wireless communications via space-time\ncoding, spatial multiplexing and beamforming. However, this\nperformance improvement comes at a high cost, namely the\naddition of more radio-frequency (RF) chains to support ad-\nditional antennas and higher computational complexity when\ntackling more complex channels. The increasing system com-\nplexity, hardware size and energy consumption pose significant\nlimitation when considering mobile or edge devices, which\nlack the computational capacity and energy supply of base\nstations (BS).\nH. Yang (email: halvin.yang@imperial.ac.uk) is with the De-\npartment of Electrical and Electronic Engineering, Imperial College Lon-\ndon, UK. Y. Zhao (email: yzzhao@uestc.edu.cn) is with the School\nof\nInformation\nand\nCommunication\nEngineering,\nUniversity\nof\nElec-\ntronic Science and Technology of China, China. K.-K. Wong (email:\nkai-kit.wong@ucl.ac.uk) is with the Department of Electronic\nand Electrical Engineering, University College London, UK, and Yon-\nsei Frontier Lab. Yonsei University, South Korea. H.-H. Chen (email:\nhshwchen@mail.ncku.edu.tw) (the corresponding author) is with the\nDepartment of Engineering Science, National Cheng Kung University, Tai-\nwan. C.-B. Chae (email: cbchae@yonsei.ac.kr) is with the School of\nIntegrated Technology, Yonsei University, South Korea.\nWith recent breakthroughs in reconfigurable antenna tech-\nnology, it is possible to design an antenna that can switch\nits position in a fixed space along predetermined points, each\nreferred to as a port [1]. Owing to this unique ability to change\nbetween multiple spatial locations, this article refers to such a\ncommunication system as a fluid antenna system (FAS). Note\nthat a physical antenna used in FAS is not necessarily liquid in\nnature; the term fluid is a description of the smooth or dynamic\nnature of the antenna rather than a physical description.\nFAS introduces the novel concept of a position-reconfigurable\nantenna that can access the null of the interference created by\nnatural fading phenomenon in a multipath-rich environment,\nsignificantly reducing the computational complexity of the\nsystem as complex beamforming and channel estimation for\nprecoding at the transmitter are no longer required [2].\nThe\nconcept\nof\nFAS\nwas\ninitially\nintroduced\nby\nWong et al. in [1], sparking further interest in utilizing re-\nconfigurable antenna technologies for wireless communication\nsystem design. Following [1], Zhu et al. introduced movable\nantenna in [3] to describe a specific class of FAS that can\nphysically relocate the antenna within a confined space using\nmechanically movable radiating elements. Recent studies even\nsuggested 3-D positioning and 3-D orientation of the antenna\nsurface for a 6-D movable antenna architecture. In this article,\nwe adopt the term FAS in a broad sense to encompass various\nforms of spatially reconfigurable antenna systems, including\nfluidic, mechanically movable, and other implementations.\nRecent advances have further envisioned to scale FAS into\nlarge reconfigurable surfaces, effectively engineering a wire-\nless environment itself [4]. This concept proposes that entire\nbuilding facades or urban infrastructure could be transformed\ninto massive FAS deployments. By dynamically shaping signal\npaths, these enormous FAS could maximize coverage and\nenhance capacity in real time, extending the principles of\nreconfigurable intelligent surfaces (RIS) into fully adaptive\ncommunication environments. While such a deployment in-\ntroduces new challenges, it represents a substantial step to-\nwards 6G-enabled smart environments, where communication\noptimization is embedded directly into physical infrastructure.\nPrevious literature has shown the viability of fluid antennas\ndeployed in communication systems with a single-antenna\nFAS, not only significantly outperforming a traditional fixed-\nposition antenna system [1], but also achieving high multi-\nplexing and diversity gains despite being a single antenna\n[2]. As FAS technology advances over the years, different\nsystems were proposed, each having different interpretation of\nfluid antennas, different terminologies, physical architecture,\nand thus different FAS technical approaches, possibly causing\nsome confusions. Readers are advised to refer to [5] for more\narXiv:2510.14794v1 [eess.SP] 16 Oct 2025\n\n2\ninformation.\nCurrent FAS research often assumes idealized material,\nchannel, and movement conditions, which do not hold in\npractice. This article will discuss those realistic conditions\nand their impact on communication systems. Corresponding\nsolutions are also recommended at the end of the article, which\nmay serve as future research directions of this emerging and\nexciting new research area.\nThe rest of the article is organized as follows. Section II\nsummarizes the key concepts, characteristics, and advantages\nand disadvantages of existing FASs, followed by a fundamen-\ntal illustration on how a typical FAS operates. Then, Section III\nexamines the theoretical assumptions made in FAS research\nand compares them with practical realities, highlighting their\nimplications on system performance. Section IV discusses\nthe broader impact of these assumptions on communication\nsystems. After that, Section V proposes solutions and outlines\nfuture research directions. Finally, Section VI concludes the\narticle.\nII. OVERVIEW ON FAS TECHNOLOGIES\nThis section provides an overview on how FASs are im-\nplemented practically by looking at different fluid antenna\ndesigns, and then discusses about the fundamental operating\nprinciples of FAS and related enabling technologies.\nA. Physical Design\nA fluid antenna refers to any radiating structure in which\nsoftware-controlled conductive or dielectric elements dynami-\ncally alter their shape, position and/or geometry to reconfigure\nkey metrics such as operating frequency, polarization, or ra-\ndiation pattern. These structures may involve fluidic elements,\nsuch as conductive liquids namely, Eutectic Gallium-Indium,\nGalinstan, or ionized solutions (e.g., NaCl) [6], [7]. Depending\non the design, such fluid antennas can enable frequency or\nspatial reconfigurability, but exhibit slow switching speeds due\nto fluid inertia and often require bulky pumps or reservoirs.\nIn contrast, pixel-based antennas use electronically switched\nstatic radiating elements, e.g., PIN diodes or MEMS, to form\nreconfigurable patterns without mechanical movement [8], en-\nabling microsecond- or even nanosecond-level switching and\nsupporting both shape and position reconfiguration. Similarly,\nmetamaterial-based antennas employ tunable electromagnetic\nsurfaces composed of reconfigurable unit cells, thus achieving\nagile and fully electronic beam control without moving parts\nand offering comparable switching speeds [9].\nBy comparison, mechanical antennas physically reposition\nradiating elements, often in 2D or 6D space (3D transla-\ntion and 3D rotation), using motors along predefined tracks.\nWhile they may offer fine spatial resolution, they often suffer\nfrom slow actuation, big energy overhead, and wear-and-\ntear. Contrary to some claims, mechanical antennas are not\nsimple to implement, requiring structural supports, motion\ncontrol, and synchronization. Moreover, antenna size is not\narchitecture-dependent, and large arrays can be realized using\nany technology if deployment constraints permit. Table I\nsummarizes the main fluid antenna architectures, highlighting\ntheir mechanisms, benefits, and limitations.\nFig. 1. An illustration of a FAS’s browsing capabilities [1].\nB. Operating Principles\nFig. 1 exemplifies a FAS. By enabling radiating elements\nto change positions in space, a fluid antenna allows users\nto browse through different fading envelopes located at each\ndifferent spatial location. Not only does this enable benefits to\ndiversity and capacity, but also can users select the position\nin space, where the signal power is the highest and the noise\nor interference is suppressed by a deep fade.\nBy exploiting the capability of FAS to resolve interference\nby selecting the optimal port (location in space), multiuser\ncommunication can be achieved without any complex beam-\nforming or channel feedback. In a traditional MIMO system,\nwhere time-frequency resources are shared, signal processing\nand beamforming were required to create peaks and nulls\nof the fading envelopes with channel state information (CSI)\nbeforehand. The interference at each user in a FAS is expected\nto be mitigated by the fluid antenna via a process known as\nport selection, in which the port with the highest signal-to-\ninterference plus noise ratio (SINR) can be selected. A more\nsophisticated scheme that selects more than one ports is also\npossible to enhance interference immunity [5, Section V-D].\nC. State of the Art\n• Channel Characteristics: Due to the proximity of ports,\nwhich can be arbitrarily close, correlation between ports\nis a key consideration in FAS. Thus, to get an accurate\nrepresentation of a FAS channel, the correlation needs\nto be modeled accurately because an underestimation of\nthe correlation would result in unrealistic performance\nevaluation. There are already several correlation models\nproposed recently, including the block correlation model,\nwhich assumes a large number of correlated random vari-\nables expressed in a matrix, and the single port correlation\nmodel, in which it is assumed each port is correlated with\na single reference port and therefore there is one mutual\nrandom variable at each port. The block correlation model\nexhibits the highest accuracy, while single port correlation\nmodel has a lower analysis complexity [5, Section II].\n• Modeling and Performance Analysis: Performance\nanalysis is essential in understanding the viability of FAS\n\n3\nTABLE I\nSUMMARY OF DIFFERENT FLUID ANTENNA ARCHITECTURES\nTechnology Type\nArchitecture\nOperating Principle\nAdvantages/Disadvantages\nPixel-Based Antenna\n[8]\nA grid of static radiating elements is\nreconfigured electronically by toggling\nintegrated switches (e.g., PIN diodes or\nMEMS) on/off to form different radiation\npatterns. No physical movement is involved,\nallowing ultra-fast and reliable switching.\nScalability depends on the number and\nlayout of switches.\n• Extremely high-speed\nswitching (in 𝜇s range, even\nreaching 𝑛s)\n• Limited by the number of\nswitches\n• Large unused sections\n• Design complexity increases\nwith reconfiguration capability\nMechanical Antenna\n[3]\nAntenna element is repositioned\nmechanically in 2D or 3D space using\nmotors and a structural frame with\npredefined paths or tracks. Advanced\nvariants (e.g., 6DMA) support simultaneous\ntranslation and rotation. It requires physical\nsupport infrastructure, power control, and\nreal-time coordination. Signal is routed\nthrough an RF chain for processing.\n• 2D or 6D spatial coverage\ndepending on configuration\n• Slow switching response\n(similar to liquid-based fluid\nantennas)\n• Requires structural frame,\ncontrol logic, and mechanical\nactuation\n• Power consumption and\nmechanical wear were often\noverlooked in analysis\n• Weight and size limitations\ndepend on deployment\nSurface-Wave\nPosition-Flexible\nAntenna [6], [7]\nIt combines fluid monopole and\nsurface-wave designs under a shared\ncategory. Liquid-based antennas control RF\nbehavior by relocating conductive fluid\ninside a channel. Frequency-reconfigurable\ndesigns (monopole type) vary the fluid’s\nlength and surface-wave types adjust fluid\nposition to steer or receive guided waves. It\nis typically actuated using pumps or pressure\ngradients.\n• Enables frequency or position\nreconfigurability depending on\ndesign\n• Moderate spatial control\nprecision\n• Slow switching speed due to\nfluid inertia\n• Design may be bulky due to\npumps and reservoirs\n• No solid-state switching, with\nlimited responsiveness\ncompared to pixel or\nmetamaterial designs\nMetamaterial-Based\nAntenna [9]\nIt utilizes programmable metasurfaces or\ntunable unit cells to manipulate\nelectromagnetic wave propagation without\nmechanical movement. These structures can\ndynamically adjust their properties to control\nbeam direction, shape and polarization,\nenabling agile and efficient beamforming.\n• No mechanical movement,\nand fully electronic\nreconfiguration\n• Ultra-fast switching speeds\n• Compact and integrable into\nvarious form factors\n• Requires complex control\ncircuitry\n• High design and fabrication\ncomplexity\nin a communications environment. To this end, a majority\nof previous works was focused on the derivation of such\nperformance indicators as outage probability, multiplex-\ning gain, bit error rate, and data rate. The accuracy of\nthe channel model will impact the performance analysis\non different indicators in different scenarios, while more\naccurate models tend to be significantly more complex.\n• Resource Allocation and Optimization: Resource al-\nlocation and optimization strategies also depend on the\nunderlying FAS architecture. Pixel-based and metama-\nterial antennas, due to their ultra-fast switching, are\nparticularly suited for optimization problems that require\nfrequent reconfiguration, such as fast port selection, user\nscheduling, and multiuser interference suppression. By\ncontrast, mechanical and liquid-based designs face slower\nactuation and higher energy costs, making them less\neffective for real-time optimization but still valuable in\nlong-term tasks such as static port assignment, spectrum\nplanning, or energy efficiency optimization. Liquid and\nsurface-wave designs further enable frequency and posi-\ntion agility, which can be exploited in spectrum allocation\nand interference avoidance, albeit with slower responsive-\nness. Hence, the choice of optimization strategy should\nreflect both the communication objective and the physical\nlimitations of the chosen architecture.\nIII. THEORETICAL ASSUMPTIONS VS REALITY\nFluid antennas, often described as reconfigurable, mechan-\nical, or pixel-based solutions, promise to dynamically modify\nantenna parameters to maximize signal-to-noise ratio (SNR),\ndata rates, or coverage. Despite this potential, many studies\nrelied on simplified assumptions that may lead to inflated\nexpectations or overly idealized system models. In the text\nfollowed, we examine these assumptions and contrast them\nwith practical constraints. We also discuss the implications\nof these idealized assumptions on communication system\nperformance.\n\n4\n0\n2\n4\n6\n8\n10\nTime (Years)\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\nOutage Probability\nOutage Probability of Different Antenna Types Over Time (Degradation)\nWear-Based Mechanical Antenna\nFluid Exposure Antenna\nControlled Precision Fluid Antenna\nFig. 2.\nPerformance degradation of three antenna types over time. Outage probability trends were simulated with gradual signal loss and small random\nvariations to reflect environmental impacts such as wear-and-tear, fluid exposure, and temperature changes.\nA. Ideal Material Properties\nAssumption: Existing fluid-based antenna models treat the\nfluid as having constant, frequency-independent conductivity\nand negligible temperature dependence. Similarly, structural\nmaterials may be taken as perfect dielectrics or perfectly flex-\niblity. Analytical models often assumed that once an optimal\nconfiguration is found, the antenna can reliably return to that\nexact state with minimal error each time.\nReality: Two issues need to be considered carefully:\n1) Frequency Dependence: Conductive liquids, gels, and\nalloys often exhibit dispersive properties, where conduc-\ntivity and permittivity vary with frequency.\n2) Thermal and Aging Effects: Temperature shifts can alter\nfluid viscosity or cause metal oxidation, leading to\nperformance drifts. Over time, repeated mechanical or\nchemical stress can degrade materials as well [10].\nImpact: Ignoring real materials’ frequency dependence and\naging effect leads to overly optimistic predictions of antenna\ngain and efficiency. Designers may find their prototypes un-\nderperforming in practical temperature ranges or long-term\ndeployment. More experimental data on these issues can be\nfound in [10]. In practice, a ‘best configuration’ might vary\nslightly each time. This inconsistency impacts link reliability\nand may require additional measurements or dynamic cal-\nibration, increasing overhead. This may require calibration\nloops to ensure performance reliability [10]. Fig. 2 shows\na rough indication of the impact of wear-and-tear on the\nperformance of antennas, showing the impact of antenna wear\ndue to mechanical movement, antenna fluid degradation from\nenvironmental exposure, and the impact of degradation of\ncontrol mechanisms.\nFig. 2 illustrates how three different antenna types degrade\nin performance over a ten-year period, expressed in terms\nof an increasing outage probability, the likelihood that the\nantenna fails to maintain a reliable connection. Each curve\nwas generated by simulating a gradual decline in signal quality\nover time, with added random variation to reflect environ-\nmental impacts like weather, dust, or material inconsistencies.\nThe mechanical antenna assumes a steady performance loss\nfrom physical wear, corrosion, and moving-part fatigue. The\nfluid exposure antenna degrades more slowly, capturing the\neffects of fluid leakage or contamination over extended use.\nThe directional fluid antenna shows the most stable behavior,\nthereby reflecting its reliance on solid-state control compo-\nnents rather than moving parts or exposed fluids. The rate\nat which performance degrades varies across the antennas,\ndepending on how sensitive each design is to material or\ncontrol deterioration.\nB. No Impact of Packaging or Enclosures\nAssumption: Antenna performance simulations were usu-\nally performed in free space, assuming no physical constraints\non movement or additional losses from device enclosures.\nReality: The actual performance does depend on the fol-\nlowing factors.\n1) Space Constraints: Mobile handsets, IoT devices, or\nunmanned aerial vehicles (UAVs) have different limiting\nfactors that restrict antenna motion or fluid distribution\npathways [6], [7].\n2) Interaction with Enclosures and Users: Plastics, metals,\nand even user’s hands can detune the antennas, causing\nmodeled reflections and absorption.\nImpact: In practice, antenna elements often cannot be\nphysically moved into theoretically ‘optimal’ positions. Real\npackaging might reduce the reconfigurable range, adding new\nlosses or scattering effects that degrade performance gains.\nC. Negligible Mechanical and Electromagnetic Coupling\nAssumption: Some studies analyzed a single ‘element’ or\nassumed perfect isolation between antenna elements, ignoring\nthe mechanical constraints or electromagnetic interactions that\nnaturally arise in closely packed arrays. The change in antenna\ngeometry in any form can have a significant impact on the RF\nperformance of the whole system, or in the case of multiple\nantennas affect adjacent radiating elements [11].\n\n5\n0\n100\n200\n300\n400\n500\nTime Steps\n10\n5\n0\n5\n10\nChannel Gain (dB)\nImpact of Incorrect Slow Fading Assumption on FAS Performance\nFast Fading (Frequent Switching)\nSlow Fading (Slow Switching)\n(a)\nFast Fading\nSlow Fading\nFading Assumption\n0\n2\n4\n6\n8\n10\n12\n14\nAverage SNR (dB)\n0\n1\n2\n3\n4\n5\n6\n7\n8\nOutage Probability (%)\nComparison of Average SNR and Outage Probability \n for Fading Assumptions\nAverage SNR (dB)\nOutage Probability (%)\n(b)\nFig. 3.\nThe effect of an inaccurate fading assumption on the performance of the system: (a) the channel gain of slow and fast fading assumptions in a\nfast fading environment, and (b) the SNR and outage performance of fading assumptions in a fast fading channel. These results illustrate the importance of\nmodeling realistic channel dynamics for FAS.\nReality: The following effects matter.\n1) Mutual Coupling: In multi-element reconfigurable sys-\ntems, shifting or resizing one antenna can detune its\nneighbors, altering the collective radiation pattern in\nunpredictable ways [11].\n2) Mechanical Friction and Alignment: Real mechanical\nactuators encounter gear backlash, friction, and align-\nment tolerances, meaning that the intended final position\noften differs from the actual one.\n3) Actuator Precision: Mechanical and fluidic systems each\nhave tolerances, and small deviations may accumulate\nover multiple cycles, which could become unacceptable.\nImpact: Overlooking these coupling effects can invalidate\nthe assumed radiation pattern, mismatch, and overall system\nperformance. Reconfiguration might improve one metric (e.g.,\ngain in a certain direction) but degrade the others (e.g.,\ninterference to neighboring elements).\nD. Instantaneous Reconfiguration\nAssumption: Many analyses assumed that antennas can be\nreconfigured with negligible latency and without additional\nenergy overhead. Under such assumptions, the system can\nalways ‘hop’ to the best configuration whenever channel\nconditions change. However, this is often unrealistic across\nmultiple antenna types. For instance, [6], [7] showed that\nliquid-based antennas require pumps or valves and involve\nnon-negligible delays. Similarly, mechanical antennas must\nphysically reposition the radiating elements using motors,\nwhich also incurs latency and energy cost. Even electronically\nreconfigurable designs, such as pixel or metamaterial-based\nantennas, can involve control signaling, timing constraints, or\ndriver delays.\nReality: The following factors must be considered.\n1) Latency and Energy: Actuation hardware, no matter\nwhether involving fluid redistribution, motorized move-\nment, or complex switching logic, consumes both time\nand power. Frequent reconfiguration can drain battery\n\n6\nlife in portable devices and create communication gaps\nduring the transition.\n2) Control Overhead: Synchronizing reconfiguration com-\nmands with fast-changing channels requires additional\nprocessing and signaling, introducing further delays in\nsystem responsiveness.\nImpact: Ignoring reconfiguration latency risks exaggerating\ntheoretical throughput or coverage gains. In practice, systems\nmay lose critical time adapting to channel conditions that\nmight have already changed. Scheduling algorithms that as-\nsume instantaneous reconfiguration often overlook switching\ndelays or transitional downtimes, leading to overly optimistic\nperformance predictions that are difficult to reproduce in\nreal deployments. Low-latency or mission-critical applications,\nsuch as industrial IoT and autonomous systems, are par-\nticularly sensitive to such overhead. Latency introduced by\nantenna switching may violate strict quality-of-service (QoS)\nrequirements. Moreover, misaligned timing or rapid channel\nvariation can degrade SNR and compromise system reliability\nif reconfiguration is not executed with a sufficient precision.\nE. Static or Slowly Varying Channels\nAssumption: Another common assumption is that the chan-\nnel remains largely unchanged during the reconfiguration\nprocess, or it only changes really slowly that once the ‘best’\nantenna state is found, or it remains optimal for long periods\n[12]. This assumption may not be realistic, as [13] showed\nthat millimeter-wave channels can fluctuate rapidly.\nReality: In real-world conditions, we have to take into\naccount the following factors.\n1) Mobility and Fast Fading: In urban or vehicular settings,\nmultipath components vary rapidly. By the time an\nantenna arrives at its new state, the optimal direction\nor frequency response may have shifted.\n2) Short Coherence Times: In high-frequency bands (e.g.,\nmillimeter-wave or terahertz bands), coherence time\ncan be on the order of milliseconds, challenging any\nreconfiguration that takes longer time than that.\n3) Channel Dynamics: Mobility and environmental changes\ncan cause rapid channel fluctuations.\nImpact: Models assuming quasi-static channels can largely\noverestimate achievable performance in real dynamic sce-\nnarios. Protocols that rely on repeated reconfiguration may\nstruggle to keep pace with channel variations, leading to under-\nutilized potential or wasted energy. The work in [13] offered\nfurther insight into the performance of FAS specifically under\nthese fast channel variations and the detriment to performance\nfrom assuming a slow fading channel. Fig. 3 illustrates the\nimpact of inaccurate fading assumptions. These inaccuracies\nand a larger channel frequency fluctuations may result in the\noptimal port having to change more frequently, requiring more\nprocessing power for port selection. It is worth noting that\nthe results in Fig. 3 are based on a generic FAS channel\nmodel rather than a specific hardware implementation. The\nintention is to illustrate the universal impact of inaccurate\nfading assumptions, which all FAS architectures are subject\nto. Nevertheless, the severity of degradation can vary: pixel-\nbased and metamaterial antennas, with their fast electronic\nswitching, can adapt more effectively to fast-fading environ-\nments, whereas mechanical and liquid-based designs are more\nsusceptible to performance loss due to their slower actuation.\nF. Perfect Knowledge of Channel\nAssumption: It is often assumed that the transmitter (or a\ncentral controller) has perfect knowledge of the instantaneous\nCSI for all possible antenna states, enabling optimal selection\namong them. This is a very strong assumption, as shown\nin [14], which highlighted the measurement complexity in\ncapturing channel responses for numerous possible antenna\nstates.\nReality: Acquiring CSI comes with practical issues.\n1) Measurement Overhead: Probing all candidate antenna\nstates to obtain CSI can be prohibitively time-consuming\nand complex, especially in dense multipath or high-\nfrequency (millimeter-wave) bands.\n2) Estimation Errors: Noise, limited training symbols, and\npilot contamination (in multiuser scenarios) may degrade\nCSI estimation accuracy.\nImpact: The mismatch between assumed perfect CSI and\nreal, noisy, partial, or outdated CSI can greatly reduce the\nsystem’s ability to consistently select the best antenna con-\nfiguration. Ultimately, capacity gains predicted by idealized\nmodels often shrink significantly when estimation errors or\nfeedback delays are included. The challenges of CSI overhead\nwere further discussed in [14], which also explored the concept\nof adaptive selection from multiple antenna states. The impact\nof CSI errors on FAS can be observed in Fig. 4.\nFig. 4 shows the impact of imperfect CSI on wireless\nlink capacity across an SNR range from −10 dB to 30 dB.\nThe simulation assumed a flat Rayleigh fading channel with\nGaussian signaling and no feedback. CSI errors were modeled\nas zero-mean complex Gaussian noise with variances ranging\nfrom 0 (perfect CSI) to an error of 40%. Each curve represents\nthe average of 1, 000 Monte Carlo trials. The results reveal\nthat while CSI errors have minimal effect at a low SNR, their\nimpact grows significantly in a high-SNR region, in which\neven a modest estimation error can severely limit achievable\ncapacity. This underscores the critical role of accurate channel\nestimation in high-performance communication systems.\nIV. IMPLICATIONS IN COMMUNICATION\nSYSTEMS\nThe ideal assumptions discussed above can collectively\nlead to multiple implications on communication systems in\nreal-world scenarios, as illustrated in Fig. 5. Some important\nimplications are discussed below. These physical layer impacts\nunderscore the need for joint mechanical-electrical modeling,\nensuring the dynamic antenna behavior is captured in link-\nlevel simulations. By anticipating the shifts in resonance, plan-\nning for demodulation disruptions, and incorporating offset-\ntracking loops, designers can mitigate transient degradations\nand preserve robust physical-layer performance.\n\n7\n10\n5\n0\n5\n10\n15\n20\n25\n30\nSNR (dB)\n0\n2\n4\n6\n8\nCapacity (bps/Hz)\nCapacity vs SNR on Different CSI Variances\nCSI Error Variance: 0.0\nCSI Error Variance: 0.05\nCSI Error Variance: 0.1\nCSI Error Variance: 0.2\nCSI Error Variance: 0.3\nCSI Error Variance: 0.4\nFig. 4.\nImpact of imperfect CSI on FAS capacity. Simulations used a flat Rayleigh fading channel with Gaussian signaling and CSI errors modeled as\nGaussian noise. Higher error variance significantly reduces capacity at a high SNR.\nFig. 5. Impacts of a non-ideal fluid antenna on communication systems.\nA. Impedance and Matching Networks\nFluid movement can change antenna’s impedance profile,\nimpacting the match to transmission line or power amplifier\n[15]. A mismatch leads to reflected power and potential\namplifier distortion, necessitating either real-time tuning (e.g.,\nvia variable matching networks) or the use of conservative\ndesign margins that reduce the benefits from reconfiguration.\nB. Frequency Shift and Resonance Stability\nFluidic antennas typically rely on shape or volume changes\nto alter their resonance. If the antenna’s effective electrical\nlength shifts unpredictably during or after reconfiguration, then\nthe system can experience frequency offset and mismatches\nin the expected resonant band [15]. This may cause unex-\npected errors for RF-to-baseband and orthogonal frequency\ndivision multiplexing (OFDM) demodulation modules, which\nis particularly significant for higher-frequency operation (e.g.,\nin millimeter-wave bands), where small dimensional changes\ncan result in large resonance shifts.\nC. Doppler and Mobility Considerations\nAlthough fluidic antennas are not necessarily moving at\nhigh speeds themselves, rapid mechanical actuation or fluid\nflow can momentarily mimic Doppler-like effects in the air\ninterface. This can lead to apparent frequency offsets during\nreconfiguration, requiring updated CFO (i.e., carrier frequency\noffset) tracking at the receiver. In high-mobility scenarios, such\nas vehicular communications, fluid antennas must not only\ncope with channel fading but also ensure that any antenna\nreconfiguration does not significantly worsen the already chal-\nlenging Doppler spread. This may have similar impacts as the\naforementioned frequency shift on a communication system.\nD. Inaccurate and Outdated CSI\nThe massive antenna positions make it impractical to esti-\nmate a wireless channel of all antenna states, which results\nin imperfect CSI at a transmitter. Meanwhile, fast fading in\nrapidly varying channels may occasionally exceed channel\nestimation frequency, which makes the acquired CSI out-\ndated. Therefore, the beamforming efficiency can be readily\ndegraded, since transmit precoding and receive combining both\nhave strict requirements of perfect CSI.\nE. Demodulation Complexity and Synchronization\nReal-time changes in the antenna gain pattern can introduce\nadditional phase noise and time-varying channel responses,\ncomplicating receiver synchronization and channel estimation.\nFor example, as the antenna reorients or reshapes, partial CSI\n\n8\ncan become obsolete much more quickly [6]. Consequently,\ndemodulation algorithms may need frequent pilot re-insertion\nor adaptive equalization to handle dynamic interference or\nvarying signal strengths during reconfiguration events.\nF. QoS Degradation\nDynamic FAS positioning inevitably incurs additional la-\ntency and energy consumption, which further degrades the\nenergy efficiency of a communication link. Moreover, the\nwasted time for antenna re-positioning also contributes to the\nloss of throughput. These factors altogether result in the QoS\ndegradation of the end-to-end communication link.\nV. RECOMMENDATIONS AND FUTURE\nDIRECTIONS\nBridging the gap between theoretical promise and commer-\ncial deployment of reconfigurable antenna systems requires\nfresh perspectives and rigorous research efforts considering\ndifferent aspects. Below are several key directions from the\nperspective of wireless communications. Innovations in other\nfields such as materials can also benefit FAS. For instance,\nthere is a need for stable high-performance fluids. New con-\nductive fluids with a lower viscosity and reduced temperature\nsensitivity can enhance reliability. Metallurgical advances in\ngallium-based alloys or ion-rich solutions can improve con-\nductivity and longevity. Durable mechanical components are\nalso necessary. Shape-memory alloys, flexible plastics, or 3D-\nprinted waveguides may enable smoother and more repeatable\nmotion, mitigating mechanical wear-and-tear issues. Apart\nfrom these, there are some promising directions that deserve\nmore investigations, which we briefly discuss as follows.\nA. Stochastic or Hybrid Channel Models\nBecause channels and reconfiguration processes are inher-\nently uncertain and time-varying, purely deterministic models\nare insufficient. Researchers are beginning to adopt stochastic\napproaches, where each antenna configuration transition is\nassigned a probability of success, a mean delay, and an\nassociated CSI uncertainty distribution. Hybrid models that\nblend measurement-based data with theoretical frameworks\ncan more accurately reflect practical performance envelopes,\nguiding realistic system-level optimizations.\nB. Medium Access Control Protocols\nMedium access control (MAC), which relies on fine-grained\non-the-fly antenna configuration changes, must incorporate\nlatency and channel-estimation overhead into their scheduling\nmechanisms. Slot-based protocols, for example, may require\nlonger guard intervals or advanced predictive algorithms if\nthe antenna reconfiguration time is not negligible. In addition,\nresource allocation strategies that assume perfect CSI for all\nantenna states need to incorporate robust feedback loops,\nreducing effective spectral efficiency. In multiuser scenarios\n(e.g., MIMO networks), fluid antennas might coordinate with\neach other to conduct cooperative positioning in order to avoid\ninterference or optimize coverage collectively.\nC. Cross-Layer Design\nReconfigurable antennas can no longer be viewed solely as\na physical-layer phenomenon. Cross-layer frameworks, where\nchanges at the physical layer interact with MAC scheduling\nand even application-layer demands, will be essential. For in-\nstance, when reconfiguration overhead is high, an application-\nlayer decision to buffer data during reconfiguration intervals\nmay improve overall system efficiency.\nD. Advanced Control and Optimization Methods\nMore creative and practical approaches need to be sought\nout in order to make FAS mature for a large scale deployment.\nSome of the thoughts are listed as follows.\n• Predictive and Machine Learning Techniques: Instead of\nreacting to instantaneous channel measurements, systems\ncan utilize machine learning models to anticipate channel\nvariations based on historical data, reducing unnecessary\nreconfigurations.\n• Limited-Codebook Approaches: To reduce the overhead\nof searching a continuum of possible antenna states, de-\nsigners can adopt a finite codebook of well-chosen config-\nurations. This strategy reduces complexity and channel-\nmeasurement overhead.\n• Energy-Efficient and Time-Limited Port Selection: Prac-\ntical fluid or reconfigurable antennas often have multiple\nports or states, each with different power-consumption\nand performance trade-offs. To address energy and la-\ntency constraints, researchers have proposed time-limited\nport selection algorithms, which optimized port usage\nonly within pre-defined time windows. By scheduling\nreconfiguration intervals more judiciously, the system\navoids excessive energy draw from continuous scanning\nof all possible antenna states. In tandem, energy-efficient\nmetrics (e.g., bits per joule or energy per reconfiguration)\ncan guide selection policies, ensuring that any perfor-\nmance gain from reconfiguration outweighs its associated\ncost.\nE. Hardware-In-the-Loop and Field Trials\nThese include:\n• Realistic Prototyping: Small-scale prototypes, complete\nsetup with actuation hardware, fluid channels or mechani-\ncal pivot arms, should be tested in realistic indoor/outdoor\nenvironments.\n• Extended Network Trials: Multi-node testbeds (e.g., in\nan anechoic chamber or urban corridor) can expose the\ninterplay of mobility, interference, and real packing con-\nstraints, the factors often absent in lab-based experiments.\nF. Standardization Efforts\nStandardization efforts are also essential. As reconfigurable\nantennas mature, industry and regulatory bodies may need\nnew guidelines for testing, compliance, and performance eval-\nuation, similar to the existing antenna standards (e.g., ETSI,\n3GPP).\n\n9\nVI. CONCLUSION\nWhile FAS hold great promise for dynamically optimiz-\ning performance, the assumptions listed above, often made\nfor analytical tractability or simplicity, can lead to inflated\ntheoretical gains. In practice, real-world constraints such as\ntime and energy costs, material properties, limited resolution,\nmeasurement and alignment errors, dynamic channels, and\nmultiuser interference can significantly diminish these benefits.\nWhen models overlook the factors such as reconfiguration\nlatency and energy consumption, imperfect or limited channel\nknowledge, mechanical and electromagnetic coupling, non-\nideal materials, packaging limitations, and misalignment is-\nsues, they risk being overly optimistic. To better predict actual\nperformance in real-world deployments, future studies should\nincorporate more realistic models that account for hardware\nconstraints, reconfiguration overhead, channel dynamics, and\nnetwork-level interactions.\nKey Takeaways\n• Many FAS studies relied on overly ideal assumptions\n(e.g., perfect CSI, instantaneous switching, ideal mate-\nrials).\n• Practical constraints such as material aging, packaging\nlimits, and switching delays can significantly reduce the\nexpected gains.\n• Realistic stochastic and hardware-aware models are\nneeded to evaluate system performance accurately.\n• Cross-layer design is essential to manage reconfiguration\noverheads and fast-changing channels.\n• Prototyping, field trials, and standardization are critical\nsteps to translate FAS theory into practical applications.\nREFERENCES\n[1] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna\nsystems,” IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950–1962,\nMar. 2021.\n[2] K. K. Wong and K. F. Tong, “Fluid antenna multiple access,” IEEE\nTrans. Wireless Commun., vol. 21, no. 7, pp. 4801–4815, Jul. 2022.\n[3] L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless commu-\nnication: Opportunities and challenges,” IEEE Commun. Mag., vol. 62,\nno. 6, pp. 114–120, Jun. 2024.\n[4] K. K. Wong, K. F. Tong, Z. Chu, and Y. Zhang, “A vision to smart\nradio environment: Surface wave communication superhighways,” IEEE\nWireless Commun., vol. 28, no. 1, pp. 112–119, Feb. 2021.\n[5] W. K. New et al., “A tutorial on fluid antenna system for 6G networks:\nEncompassing communication theory, optimization methods and hard-\nware designs,” IEEE Commun. Surv. & Tut., doi:10.1109/COMST.2024.\n3498855, 2024.\n[6] H. Abu Bakar, R. A. Rahim, P. J. Soh, and P. Akkaraekthalin, “Liquid-\nbased reconfigurable antenna technology: Recent developments, chal-\nlenges and future,” Sensors, vol. 21, no. 3, p. 827, Jan. 2021.\n[7] K.-F. Tong, B. Liu, and K.-K. Wong, “Designs and challenges in\nfluid antenna system hardware,” Electronics, Special Issue on Futuristic\nAntennas: Sustainable, Efficient, Reconfigurable, and Intelligent Design,\nvol. 14, no. 7, pp. 1458, Apr. 2025.\n[8] S. Song and R. D. Murch, “An efficient approach for optimizing\nfrequency reconfigurable pixel antennas using genetic algorithms,” IEEE\nTrans. Antennas Propag., vol. 62, no. 2, pp. 609–620, Feb. 2014.\n[9] B. Liu, K.-F. Tong, K.-K. Wong, C.-B. Chae, and H. Wong, “Be water,\nmy antennas: Riding on radio wave fluctuation in nature for spatial\nmultiplexing using programmable meta-fluid antenna,” arXiv preprint,\narXiv:2502.04693, 2025.\n[10] M. Kubo et al., “Stretchable microfluidic radiofrequency antennas,” Adv.\nMater., vol. 22, no. 25, pp. 2749–2752, Jul. 2010.\n[11] J. Zhang, M. O. Akinsolu, B. Liu, and G. A. E. Vandenbosch, “Automatic\nAI-driven design of mutual coupling reducing topologies for frequency\nreconfigurable antenna arrays,” IEEE Trans. Antennas Propag., vol. 69,\nno. 3, pp. 1831–1836, Mar. 2021.\n[12] J. Zou, S. Sun, and C. Wang, “Online learning-induced port selection for\nfluid antenna in dynamic channel environment,” IEEE Wireless Commun.\nLett., vol. 13, no. 2, pp. 313–317, Feb. 2024.\n[13] C. Gupta et al., “Accurate and computationally efficient modeling of\nnonquasi static effects in MOSFETs for millimeter-wave applications,”\nIEEE Trans. Electron Devices, vol. 66, no. 1, pp. 44–51, Jan. 2019\n[14] Y. Wang, H. Shen, C. Han, and M. Tao, “Movable antennas: Channel\nmeasurement, modeling, and performance evaluation,” arXiv preprint ,\narXiv:2409.03386, 2024.\n[15] T. Jang, C. Zhang, H. Youn, J. Zhou, and L. J. Guo, “Semitransparent\nand flexible mechanically reconfigurable electrically small antennas\nbased on tortuous metallic micromesh,” IEEE Trans. Antennas Propag.,\nvol. 65, no. 1, pp. 150–158, Jan. 2017.\n"},"cleaned":{"kind":"string","value":"1 Bridging Theory and Practice in Reconfigurable Fluid Antenna Systems Halvin Yang, Member, IEEE, Yizhe Zhao, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Hsiao-Hwa Chen, Life Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE Abstract-Fluid antennas, including those based on liquid, mechanical, and pixel-based technologies, are poised to significantly enhance next-generation wireless systems by adaptively optimizing their radiation characteristics. Many theoretical analyses assumed near-instant reconfiguration, perfect channel knowledge, static or slowly varying propagation environments, and ideal material properties that rarely hold in practice. In this article, we dissect these common assumptions and contrast them with the realities of finite actuation time, limited and imperfect channel state information, rapidly changing fading conditions, electromagnetic coupling, and mechanical constraints. Through illustrative examples and simulations, we demonstrate how ignoring these factors can lead to overestimated gains in capacity, coverage, etc.. We then propose modeling refinements, experimental validation methods, and emerging control algorithms that better account for real-world constraints. Our findings highlight that, while reconfigurable antennas remain highly promising for B5G/6G and Internet of things (IoT) applications, their full potential can only be realized by incorporating practical considerations into system design and performance evaluation. Index Terms-Fluid antenna, fluid antenna system (FAS), movable antenna, modeling, resource allocation, 6G. I. INTRODUCTION W IRELESS networks beyond fifth-generation (5G) are required to keep up with the exponential growth in mobile data traffic while also providing ubiquitous connectivity. To achieve this, new wireless technologies need to be developed. In recent years, multiple-input multiple-output (MIMO) technology has been viewed as a primary solution to drastically improve spectral efficiency (SE) and energy efficiency (EE) of wireless communications via space-time coding, spatial multiplexing and beamforming. However, this performance improvement comes at a high cost, namely the addition of more radio-frequency (RF) chains to support additional antennas and higher computational complexity when tackling more complex channels. The increasing system complexity, hardware size and energy consumption pose significant limitation when considering mobile or edge devices, which lack the computational capacity and energy supply of base stations (BS). H. Yang (email: ) is with the Department of Electrical and Electronic Engineering, Imperial College London, UK. Y. Zhao (email: ) is with the - tronic Science and Technology of China, China. K.-K. Wong (email: ) is with the - sei Frontier Lab. Yonsei University, South Korea. H.-H. Chen (email: ) (the corresponding author) is with the - wan. C.-B. Chae (email: ) is with the . With recent breakthroughs in reconfigurable antenna technology, it is possible to design an antenna that can switch its position in a fixed space along predetermined points, each referred to as a port [1]. Owing to this unique ability to change between multiple spatial locations, this article refers to such a communication system as a fluid antenna system (FAS). Note that a physical antenna used in FAS is not necessarily liquid in nature; the term fluid is a description of the smooth or dynamic nature of the antenna rather than a physical description. FAS introduces the novel concept of a position-reconfigurable antenna that can access the null of the interference created by natural fading phenomenon in a multipath-rich environment, significantly reducing the computational complexity of the system as complex beamforming and channel estimation for precoding at the transmitter are no longer required [2]. The concept of FAS was initially introduced by Wong et al. in [1], sparking further interest in utilizing reconfigurable antenna technologies for wireless communication system design. Following [1], Zhu et al. introduced movable antenna in [3] to describe a specific class of FAS that can physically relocate the antenna within a confined space using mechanically movable radiating elements. Recent studies even suggested 3-D positioning and 3-D orientation of the antenna surface for a 6-D movable antenna architecture. In this article, we adopt the term FAS in a broad sense to encompass various forms of spatially reconfigurable antenna systems, including fluidic, mechanically movable, and other implementations. Recent advances have further envisioned to scale FAS into large reconfigurable surfaces, effectively engineering a wireless environment itself [4]. This concept proposes that entire building facades or urban infrastructure could be transformed into massive FAS deployments. By dynamically shaping signal paths, these enormous FAS could maximize coverage and enhance capacity in real time, extending the principles of reconfigurable intelligent surfaces (RIS) into fully adaptive communication environments. While such a deployment introduces new challenges, it represents a substantial step towards 6G-enabled smart environments, where communication optimization is embedded directly into physical infrastructure. Previous literature has shown the viability of fluid antennas deployed in communication systems with a single-antenna FAS, not only significantly outperforming a traditional fixedposition antenna system [1], but also achieving high multiplexing and diversity gains despite being a single antenna [2]. As FAS technology advances over the years, different systems were proposed, each having different interpretation of fluid antennas, different terminologies, physical architecture, and thus different FAS technical approaches, possibly causing some confusions. Readers are advised to refer to [5] for more 16 Oct 2025 2 information. Current FAS research often assumes idealized material, channel, and movement conditions, which do not hold in practice. This article will discuss those realistic conditions and their impact on communication systems. Corresponding solutions are also recommended at the end of the article, which may serve as future research directions of this emerging and exciting new research area. The rest of the article is organized as follows. Section II summarizes the key concepts, characteristics, and advantages and disadvantages of existing FASs, followed by a fundamental illustration on how a typical FAS operates. Then, Section III examines the theoretical assumptions made in FAS research and compares them with practical realities, highlighting their implications on system performance. Section IV discusses the broader impact of these assumptions on communication systems. After that, Section V proposes solutions and outlines future research directions. Finally, Section VI concludes the article. II. OVERVIEW ON FAS TECHNOLOGIES This section provides an overview on how FASs are implemented practically by looking at different fluid antenna designs, and then discusses about the fundamental operating principles of FAS and related enabling technologies. A. Physical Design A fluid antenna refers to any radiating structure in which software-controlled conductive or dielectric elements dynamically alter their shape, position and/or geometry to reconfigure key metrics such as operating frequency, polarization, or radiation pattern. These structures may involve fluidic elements, such as conductive liquids namely, Eutectic Gallium-Indium, Galinstan, or ionized solutions (e.g., NaCl) [6], [7]. Depending on the design, such fluid antennas can enable frequency or spatial reconfigurability, but exhibit slow switching speeds due to fluid inertia and often require bulky pumps or reservoirs. In contrast, pixel-based antennas use electronically switched static radiating elements, e.g., PIN diodes or MEMS, to form reconfigurable patterns without mechanical movement [8], enabling microsecond- or even nanosecond-level switching and supporting both shape and position reconfiguration. Similarly, metamaterial-based antennas employ tunable electromagnetic surfaces composed of reconfigurable unit cells, thus achieving agile and fully electronic beam control without moving parts and offering comparable switching speeds [9]. By comparison, mechanical antennas physically reposition radiating elements, often in 2D or 6D space (3D translation and 3D rotation), using motors along predefined tracks. While they may offer fine spatial resolution, they often suffer from slow actuation, big energy overhead, and wear-andtear. Contrary to some claims, mechanical antennas are not simple to implement, requiring structural supports, motion control, and synchronization. Moreover, antenna size is not architecture-dependent, and large arrays can be realized using any technology if deployment constraints permit. Table I summarizes the main fluid antenna architectures, highlighting their mechanisms, benefits, and limitations. Fig. 1. An illustration of a FAS's browsing capabilities [1]. B. Operating Principles Fig. 1 exemplifies a FAS. By enabling radiating elements to change positions in space, a fluid antenna allows users to browse through different fading envelopes located at each different spatial location. Not only does this enable benefits to diversity and capacity, but also can users select the position in space, where the signal power is the highest and the noise or interference is suppressed by a deep fade. By exploiting the capability of FAS to resolve interference by selecting the optimal port (location in space), multiuser communication can be achieved without any complex beamforming or channel feedback. In a traditional MIMO system, where time-frequency resources are shared, signal processing and beamforming were required to create peaks and nulls of the fading envelopes with channel state information (CSI) beforehand. The interference at each user in a FAS is expected to be mitigated by the fluid antenna via a process known as port selection, in which the port with the highest signal-tointerference plus noise ratio (SINR) can be selected. A more sophisticated scheme that selects more than one ports is also possible to enhance interference immunity [5, Section V-D]. C. State of the Art • Channel Characteristics: Due to the proximity of ports, which can be arbitrarily close, correlation between ports is a key consideration in FAS. Thus, to get an accurate representation of a FAS channel, the correlation needs to be modeled accurately because an underestimation of the correlation would result in unrealistic performance evaluation. There are already several correlation models proposed recently, including the block correlation model, which assumes a large number of correlated random variables expressed in a matrix, and the single port correlation model, in which it is assumed each port is correlated with a single reference port and therefore there is one mutual random variable at each port. The block correlation model exhibits the highest accuracy, while single port correlation model has a lower analysis complexity [5, Section II]. • Modeling and Performance Analysis: Performance analysis is essential in understanding the viability of FAS 3 TABLE I SUMMARY OF DIFFERENT FLUID ANTENNA ARCHITECTURES Technology Type Architecture Operating Principle Advantages/Disadvantages Pixel-Based Antenna [8] A grid of static radiating elements is reconfigured electronically by toggling integrated switches (e.g., PIN diodes or MEMS) on/off to form different radiation patterns. No physical movement is involved, allowing ultra-fast and reliable switching. Scalability depends on the number and layout of switches. • Extremely high-speed switching (in μs range, even reaching ns) • Limited by the number of switches • Large unused sections • Design complexity increases with reconfiguration capability Mechanical Antenna [3] Antenna element is repositioned mechanically in 2D or 3D space using motors and a structural frame with predefined paths or tracks. Advanced variants (e.g., 6DMA) support simultaneous translation and rotation. It requires physical support infrastructure, power control, and real-time coordination. Signal is routed through an RF chain for processing. • 2D or 6D spatial coverage depending on configuration • Slow switching response (similar to liquid-based fluid antennas) • Requires structural frame, control logic, and mechanical actuation • Power consumption and mechanical wear were often overlooked in analysis • Weight and size limitations depend on deployment Surface-Wave Position-Flexible Antenna [6], [7] It combines fluid monopole and surface-wave designs under a shared category. Liquid-based antennas control RF behavior by relocating conductive fluid inside a channel. Frequency-reconfigurable designs (monopole type) vary the fluid's length and surface-wave types adjust fluid position to steer or receive guided waves. It is typically actuated using pumps or pressure gradients. • Enables frequency or position reconfigurability depending on design • Moderate spatial control precision • Slow switching speed due to fluid inertia • Design may be bulky due to pumps and reservoirs • No solid-state switching, with limited responsiveness compared to pixel or metamaterial designs Metamaterial-Based Antenna [9] It utilizes programmable metasurfaces or tunable unit cells to manipulate electromagnetic wave propagation without mechanical movement. These structures can dynamically adjust their properties to control beam direction, shape and polarization, enabling agile and efficient beamforming. • No mechanical movement, and fully electronic reconfiguration • Ultra-fast switching speeds • Compact and integrable into various form factors • Requires complex control circuitry • High design and fabrication complexity in a communications environment. To this end, a majority of previous works was focused on the derivation of such performance indicators as outage probability, multiplexing gain, bit error rate, and data rate. The accuracy of the channel model will impact the performance analysis on different indicators in different scenarios, while more accurate models tend to be significantly more complex. • Resource Allocation and Optimization: Resource allocation and optimization strategies also depend on the underlying FAS architecture. Pixel-based and metamaterial antennas, due to their ultra-fast switching, are particularly suited for optimization problems that require frequent reconfiguration, such as fast port selection, user scheduling, and multiuser interference suppression. By contrast, mechanical and liquid-based designs face slower actuation and higher energy costs, making them less effective for real-time optimization but still valuable in long-term tasks such as static port assignment, spectrum planning, or energy efficiency optimization. Liquid and surface-wave designs further enable frequency and position agility, which can be exploited in spectrum allocation and interference avoidance, albeit with slower responsiveness. Hence, the choice of optimization strategy should reflect both the communication objective and the physical limitations of the chosen architecture. III. THEORETICAL ASSUMPTIONS VS REALITY Fluid antennas, often described as reconfigurable, mechanical, or pixel-based solutions, promise to dynamically modify antenna parameters to maximize signal-to-noise ratio (SNR), data rates, or coverage. Despite this potential, many studies relied on simplified assumptions that may lead to inflated expectations or overly idealized system models. In the text followed, we examine these assumptions and contrast them with practical constraints. We also discuss the implications of these idealized assumptions on communication system performance. 4 0 2 4 6 8 10 Time (Years) 0.0 0.1 0.2 0.3 0.4 0.5 Outage Probability Outage Probability of Different Antenna Types Over Time (Degradation) Wear-Based Mechanical Antenna Fluid Exposure Antenna Controlled Precision Fluid Antenna Fig. 2. Performance degradation of three antenna types over time. Outage probability trends were simulated with gradual signal loss and small random variations to reflect environmental impacts such as wear-and-tear, fluid exposure, and temperature changes. A. Ideal Material Properties Assumption: Existing fluid-based antenna models treat the fluid as having constant, frequency-independent conductivity and negligible temperature dependence. Similarly, structural materials may be taken as perfect dielectrics or perfectly flexiblity. Analytical models often assumed that once an optimal configuration is found, the antenna can reliably return to that exact state with minimal error each time. Reality: Two issues need to be considered carefully: 1) Frequency Dependence: Conductive liquids, gels, and alloys often exhibit dispersive properties, where conductivity and permittivity vary with frequency. 2) Thermal and Aging Effects: Temperature shifts can alter fluid viscosity or cause metal oxidation, leading to performance drifts. Over time, repeated mechanical or chemical stress can degrade materials as well [10]. Impact: Ignoring real materials' frequency dependence and aging effect leads to overly optimistic predictions of antenna gain and efficiency. Designers may find their prototypes underperforming in practical temperature ranges or long-term deployment. More experimental data on these issues can be found in [10]. In practice, a 'best configuration' might vary slightly each time. This inconsistency impacts link reliability and may require additional measurements or dynamic calibration, increasing overhead. This may require calibration loops to ensure performance reliability [10]. Fig. 2 shows a rough indication of the impact of wear-and-tear on the performance of antennas, showing the impact of antenna wear due to mechanical movement, antenna fluid degradation from environmental exposure, and the impact of degradation of control mechanisms. Fig. 2 illustrates how three different antenna types degrade in performance over a ten-year period, expressed in terms of an increasing outage probability, the likelihood that the antenna fails to maintain a reliable connection. Each curve was generated by simulating a gradual decline in signal quality over time, with added random variation to reflect environmental impacts like weather, dust, or material inconsistencies. The mechanical antenna assumes a steady performance loss from physical wear, corrosion, and moving-part fatigue. The fluid exposure antenna degrades more slowly, capturing the effects of fluid leakage or contamination over extended use. The directional fluid antenna shows the most stable behavior, thereby reflecting its reliance on solid-state control components rather than moving parts or exposed fluids. The rate at which performance degrades varies across the antennas, depending on how sensitive each design is to material or control deterioration. B. No Impact of Packaging or Enclosures Assumption: Antenna performance simulations were usually performed in free space, assuming no physical constraints on movement or additional losses from device enclosures. Reality: The actual performance does depend on the following factors. 1) Space Constraints: Mobile handsets, IoT devices, or unmanned aerial vehicles (UAVs) have different limiting factors that restrict antenna motion or fluid distribution pathways [6], [7]. 2) Interaction with Enclosures and Users: Plastics, metals, and even user's hands can detune the antennas, causing modeled reflections and absorption. Impact: In practice, antenna elements often cannot be physically moved into theoretically 'optimal' positions. Real packaging might reduce the reconfigurable range, adding new losses or scattering effects that degrade performance gains. C. Negligible Mechanical and Electromagnetic Coupling Assumption: Some studies analyzed a single 'element' or assumed perfect isolation between antenna elements, ignoring the mechanical constraints or electromagnetic interactions that naturally arise in closely packed arrays. The change in antenna geometry in any form can have a significant impact on the RF performance of the whole system, or in the case of multiple antennas affect adjacent radiating elements [11]. 5 0 100 200 300 400 500 Time Steps 10 5 0 5 10 Channel Gain (dB) Impact of Incorrect Slow Fading Assumption on FAS Performance Fast Fading (Frequent Switching) Slow Fading (Slow Switching) (a) Fast Fading Slow Fading Fading Assumption 0 2 4 6 8 10 12 14 Average SNR (dB) 0 1 2 3 4 5 6 7 8 Outage Probability (%) Comparison of Average SNR and Outage Probability for Fading Assumptions Average SNR (dB) Outage Probability (%) (b) Fig. 3. The effect of an inaccurate fading assumption on the performance of the system: (a) the channel gain of slow and fast fading assumptions in a fast fading environment, and (b) the SNR and outage performance of fading assumptions in a fast fading channel. These results illustrate the importance of modeling realistic channel dynamics for FAS. Reality: The following effects matter. 1) Mutual Coupling: In multi-element reconfigurable systems, shifting or resizing one antenna can detune its neighbors, altering the collective radiation pattern in unpredictable ways [11]. 2) Mechanical Friction and Alignment: Real mechanical actuators encounter gear backlash, friction, and alignment tolerances, meaning that the intended final position often differs from the actual one. 3) Actuator Precision: Mechanical and fluidic systems each have tolerances, and small deviations may accumulate over multiple cycles, which could become unacceptable. Impact: Overlooking these coupling effects can invalidate the assumed radiation pattern, mismatch, and overall system performance. Reconfiguration might improve one metric (e.g., gain in a certain direction) but degrade the others (e.g., interference to neighboring elements). D. Instantaneous Reconfiguration Assumption: Many analyses assumed that antennas can be reconfigured with negligible latency and without additional energy overhead. Under such assumptions, the system can always 'hop' to the best configuration whenever channel conditions change. However, this is often unrealistic across multiple antenna types. For instance, [6], [7] showed that liquid-based antennas require pumps or valves and involve non-negligible delays. Similarly, mechanical antennas must physically reposition the radiating elements using motors, which also incurs latency and energy cost. Even electronically reconfigurable designs, such as pixel or metamaterial-based antennas, can involve control signaling, timing constraints, or driver delays. Reality: The following factors must be considered. 1) Latency and Energy: Actuation hardware, no matter whether involving fluid redistribution, motorized movement, or complex switching logic, consumes both time and power. Frequent reconfiguration can drain battery 6 life in portable devices and create communication gaps during the transition. 2) Control Overhead: Synchronizing reconfiguration commands with fast-changing channels requires additional processing and signaling, introducing further delays in system responsiveness. Impact: Ignoring reconfiguration latency risks exaggerating theoretical throughput or coverage gains. In practice, systems may lose critical time adapting to channel conditions that might have already changed. Scheduling algorithms that assume instantaneous reconfiguration often overlook switching delays or transitional downtimes, leading to overly optimistic performance predictions that are difficult to reproduce in real deployments. Low-latency or mission-critical applications, such as industrial IoT and autonomous systems, are particularly sensitive to such overhead. Latency introduced by antenna switching may violate strict quality-of-service (QoS) requirements. Moreover, misaligned timing or rapid channel variation can degrade SNR and compromise system reliability if reconfiguration is not executed with a sufficient precision. E. Static or Slowly Varying Channels Assumption: Another common assumption is that the channel remains largely unchanged during the reconfiguration process, or it only changes really slowly that once the 'best' antenna state is found, or it remains optimal for long periods [12]. This assumption may not be realistic, as [13] showed that millimeter-wave channels can fluctuate rapidly. Reality: In real-world conditions, we have to take into account the following factors. 1) Mobility and Fast Fading: In urban or vehicular settings, multipath components vary rapidly. By the time an antenna arrives at its new state, the optimal direction or frequency response may have shifted. 2) Short Coherence Times: In high-frequency bands (e.g., millimeter-wave or terahertz bands), coherence time can be on the order of milliseconds, challenging any reconfiguration that takes longer time than that. 3) Channel Dynamics: Mobility and environmental changes can cause rapid channel fluctuations. Impact: Models assuming quasi-static channels can largely overestimate achievable performance in real dynamic scenarios. Protocols that rely on repeated reconfiguration may struggle to keep pace with channel variations, leading to underutilized potential or wasted energy. The work in [13] offered further insight into the performance of FAS specifically under these fast channel variations and the detriment to performance from assuming a slow fading channel. Fig. 3 illustrates the impact of inaccurate fading assumptions. These inaccuracies and a larger channel frequency fluctuations may result in the optimal port having to change more frequently, requiring more processing power for port selection. It is worth noting that the results in Fig. 3 are based on a generic FAS channel model rather than a specific hardware implementation. The intention is to illustrate the universal impact of inaccurate fading assumptions, which all FAS architectures are subject to. Nevertheless, the severity of degradation can vary: pixelbased and metamaterial antennas, with their fast electronic switching, can adapt more effectively to fast-fading environments, whereas mechanical and liquid-based designs are more susceptible to performance loss due to their slower actuation. F. Perfect Knowledge of Channel Assumption: It is often assumed that the transmitter (or a central controller) has perfect knowledge of the instantaneous CSI for all possible antenna states, enabling optimal selection among them. This is a very strong assumption, as shown in [14], which highlighted the measurement complexity in capturing channel responses for numerous possible antenna states. Reality: Acquiring CSI comes with practical issues. 1) Measurement Overhead: Probing all candidate antenna states to obtain CSI can be prohibitively time-consuming and complex, especially in dense multipath or highfrequency (millimeter-wave) bands. 2) Estimation Errors: Noise, limited training symbols, and pilot contamination (in multiuser scenarios) may degrade CSI estimation accuracy. Impact: The mismatch between assumed perfect CSI and real, noisy, partial, or outdated CSI can greatly reduce the system's ability to consistently select the best antenna configuration. Ultimately, capacity gains predicted by idealized models often shrink significantly when estimation errors or feedback delays are included. The challenges of CSI overhead were further discussed in [14], which also explored the concept of adaptive selection from multiple antenna states. The impact of CSI errors on FAS can be observed in Fig. 4. Fig. 4 shows the impact of imperfect CSI on wireless link capacity across an SNR range from -10 dB to 30 dB. The simulation assumed a flat Rayleigh fading channel with Gaussian signaling and no feedback. CSI errors were modeled as zero-mean complex Gaussian noise with variances ranging from 0 (perfect CSI) to an error of 40%. Each curve represents the average of 1, 000 Monte Carlo trials. The results reveal that while CSI errors have minimal effect at a low SNR, their impact grows significantly in a high-SNR region, in which even a modest estimation error can severely limit achievable capacity. This underscores the critical role of accurate channel estimation in high-performance communication systems. IV. IMPLICATIONS IN COMMUNICATION SYSTEMS The ideal assumptions discussed above can collectively lead to multiple implications on communication systems in real-world scenarios, as illustrated in Fig. 5. Some important implications are discussed below. These physical layer impacts underscore the need for joint mechanical-electrical modeling, ensuring the dynamic antenna behavior is captured in linklevel simulations. By anticipating the shifts in resonance, planning for demodulation disruptions, and incorporating offsettracking loops, designers can mitigate transient degradations and preserve robust physical-layer performance. 7 10 5 0 5 10 15 20 25 30 SNR (dB) 0 2 4 6 8 Capacity (bps/Hz) Capacity vs SNR on Different CSI Variances CSI Error Variance: 0.0 CSI Error Variance: 0.05 CSI Error Variance: 0.1 CSI Error Variance: 0.2 CSI Error Variance: 0.3 CSI Error Variance: 0.4 Fig. 4. Impact of imperfect CSI on FAS capacity. Simulations used a flat Rayleigh fading channel with Gaussian signaling and CSI errors modeled as Gaussian noise. Higher error variance significantly reduces capacity at a high SNR. Fig. 5. Impacts of a non-ideal fluid antenna on communication systems. A. Impedance and Matching Networks Fluid movement can change antenna's impedance profile, impacting the match to transmission line or power amplifier [15]. A mismatch leads to reflected power and potential amplifier distortion, necessitating either real-time tuning (e.g., via variable matching networks) or the use of conservative design margins that reduce the benefits from reconfiguration. B. Frequency Shift and Resonance Stability Fluidic antennas typically rely on shape or volume changes to alter their resonance. If the antenna's effective electrical length shifts unpredictably during or after reconfiguration, then the system can experience frequency offset and mismatches in the expected resonant band [15]. This may cause unexpected errors for RF-to-baseband and orthogonal frequency division multiplexing (OFDM) demodulation modules, which is particularly significant for higher-frequency operation (e.g., in millimeter-wave bands), where small dimensional changes can result in large resonance shifts. C. Doppler and Mobility Considerations Although fluidic antennas are not necessarily moving at high speeds themselves, rapid mechanical actuation or fluid flow can momentarily mimic Doppler-like effects in the air interface. This can lead to apparent frequency offsets during reconfiguration, requiring updated CFO (i.e., carrier frequency offset) tracking at the receiver. In high-mobility scenarios, such as vehicular communications, fluid antennas must not only cope with channel fading but also ensure that any antenna reconfiguration does not significantly worsen the already challenging Doppler spread. This may have similar impacts as the aforementioned frequency shift on a communication system. D. Inaccurate and Outdated CSI The massive antenna positions make it impractical to estimate a wireless channel of all antenna states, which results in imperfect CSI at a transmitter. Meanwhile, fast fading in rapidly varying channels may occasionally exceed channel estimation frequency, which makes the acquired CSI outdated. Therefore, the beamforming efficiency can be readily degraded, since transmit precoding and receive combining both have strict requirements of perfect CSI. E. Demodulation Complexity and Synchronization Real-time changes in the antenna gain pattern can introduce additional phase noise and time-varying channel responses, complicating receiver synchronization and channel estimation. For example, as the antenna reorients or reshapes, partial CSI 8 can become obsolete much more quickly [6]. Consequently, demodulation algorithms may need frequent pilot re-insertion or adaptive equalization to handle dynamic interference or varying signal strengths during reconfiguration events. F. QoS Degradation Dynamic FAS positioning inevitably incurs additional latency and energy consumption, which further degrades the energy efficiency of a communication link. Moreover, the wasted time for antenna re-positioning also contributes to the loss of throughput. These factors altogether result in the QoS degradation of the end-to-end communication link. V. RECOMMENDATIONS AND FUTURE DIRECTIONS Bridging the gap between theoretical promise and commercial deployment of reconfigurable antenna systems requires fresh perspectives and rigorous research efforts considering different aspects. Below are several key directions from the perspective of wireless communications. Innovations in other fields such as materials can also benefit FAS. For instance, there is a need for stable high-performance fluids. New conductive fluids with a lower viscosity and reduced temperature sensitivity can enhance reliability. Metallurgical advances in gallium-based alloys or ion-rich solutions can improve conductivity and longevity. Durable mechanical components are also necessary. Shape-memory alloys, flexible plastics, or 3Dprinted waveguides may enable smoother and more repeatable motion, mitigating mechanical wear-and-tear issues. Apart from these, there are some promising directions that deserve more investigations, which we briefly discuss as follows. A. Stochastic or Hybrid Channel Models Because channels and reconfiguration processes are inherently uncertain and time-varying, purely deterministic models are insufficient. Researchers are beginning to adopt stochastic approaches, where each antenna configuration transition is assigned a probability of success, a mean delay, and an associated CSI uncertainty distribution. Hybrid models that blend measurement-based data with theoretical frameworks can more accurately reflect practical performance envelopes, guiding realistic system-level optimizations. B. Medium Access Control Protocols Medium access control (MAC), which relies on fine-grained on-the-fly antenna configuration changes, must incorporate latency and channel-estimation overhead into their scheduling mechanisms. Slot-based protocols, for example, may require longer guard intervals or advanced predictive algorithms if the antenna reconfiguration time is not negligible. In addition, resource allocation strategies that assume perfect CSI for all antenna states need to incorporate robust feedback loops, reducing effective spectral efficiency. In multiuser scenarios (e.g., MIMO networks), fluid antennas might coordinate with each other to conduct cooperative positioning in order to avoid interference or optimize coverage collectively. C. Cross-Layer Design Reconfigurable antennas can no longer be viewed solely as a physical-layer phenomenon. Cross-layer frameworks, where changes at the physical layer interact with MAC scheduling and even application-layer demands, will be essential. For instance, when reconfiguration overhead is high, an applicationlayer decision to buffer data during reconfiguration intervals may improve overall system efficiency. D. Advanced Control and Optimization Methods More creative and practical approaches need to be sought out in order to make FAS mature for a large scale deployment. Some of the thoughts are listed as follows. • Predictive and Machine Learning Techniques: Instead of reacting to instantaneous channel measurements, systems can utilize machine learning models to anticipate channel variations based on historical data, reducing unnecessary reconfigurations. • Limited-Codebook Approaches: To reduce the overhead of searching a continuum of possible antenna states, designers can adopt a finite codebook of well-chosen configurations. This strategy reduces complexity and channelmeasurement overhead. • Energy-Efficient and Time-Limited Port Selection: Practical fluid or reconfigurable antennas often have multiple ports or states, each with different power-consumption and performance trade-offs. To address energy and latency constraints, researchers have proposed time-limited port selection algorithms, which optimized port usage only within pre-defined time windows. By scheduling reconfiguration intervals more judiciously, the system avoids excessive energy draw from continuous scanning of all possible antenna states. In tandem, energy-efficient metrics (e.g., bits per joule or energy per reconfiguration) can guide selection policies, ensuring that any performance gain from reconfiguration outweighs its associated cost. E. Hardware-In-the-Loop and Field Trials These include: • Realistic Prototyping: Small-scale prototypes, complete setup with actuation hardware, fluid channels or mechanical pivot arms, should be tested in realistic indoor/outdoor environments. • Extended Network Trials: Multi-node testbeds (e.g., in an anechoic chamber or urban corridor) can expose the interplay of mobility, interference, and real packing constraints, the factors often absent in lab-based experiments. F. Standardization Efforts Standardization efforts are also essential. As reconfigurable antennas mature, industry and regulatory bodies may need new guidelines for testing, compliance, and performance evaluation, similar to the existing antenna standards (e.g., ETSI, 3GPP). 9 VI. CONCLUSION While FAS hold great promise for dynamically optimizing performance, the assumptions listed above, often made for analytical tractability or simplicity, can lead to inflated theoretical gains. In practice, real-world constraints such as time and energy costs, material properties, limited resolution, measurement and alignment errors, dynamic channels, and multiuser interference can significantly diminish these benefits. When models overlook the factors such as reconfiguration latency and energy consumption, imperfect or limited channel knowledge, mechanical and electromagnetic coupling, nonideal materials, packaging limitations, and misalignment issues, they risk being overly optimistic. To better predict actual performance in real-world deployments, future studies should incorporate more realistic models that account for hardware constraints, reconfiguration overhead, channel dynamics, and network-level interactions. Key Takeaways • Many FAS studies relied on overly ideal assumptions (e.g., perfect CSI, instantaneous switching, ideal materials). • Practical constraints such as material aging, packaging limits, and switching delays can significantly reduce the expected gains. • Realistic stochastic and hardware-aware models are needed to evaluate system performance accurately. • Cross-layer design is essential to manage reconfiguration overheads and fast-changing channels. • Prototyping, field trials, and standardization are critical steps to translate FAS theory into practical applications. REFERENCES [1] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, \"Fluid antenna systems,\" IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950-1962, Mar. 2021. [2] K. K. Wong and K. F. Tong, \"Fluid antenna multiple access,\" IEEE Trans. Wireless Commun., vol. 21, no. 7, pp. 4801-4815, Jul. 2022. [3] L. Zhu, W. Ma, and R. Zhang, \"Movable antennas for wireless communication: Opportunities and challenges,\" IEEE Commun. Mag., vol. 62, no. 6, pp. 114-120, Jun. 2024. [4] K. K. Wong, K. F. Tong, Z. Chu, and Y. Zhang, \"A vision to smart radio environment: Surface wave communication superhighways,\" IEEE Wireless Commun., vol. 28, no. 1, pp. 112-119, Feb. 2021. [5] W. K. New et al., \"A tutorial on fluid antenna system for 6G networks: Encompassing communication theory, optimization methods and hardware designs,\" IEEE Commun. Surv. & Tut., 3498855, 2024. [6] H. Abu Bakar, R. A. Rahim, P. J. Soh, and P. Akkaraekthalin, \"Liquidbased reconfigurable antenna technology: Recent developments, challenges and future,\" Sensors, vol. 21, no. 3, p. 827, Jan. 2021. [7] K.-F. Tong, B. Liu, and K.-K. Wong, \"Designs and challenges in fluid antenna system hardware,\" Electronics, Special Issue on Futuristic Antennas: Sustainable, Efficient, Reconfigurable, and Intelligent Design, vol. 14, no. 7, pp. 1458, Apr. 2025. [8] S. Song and R. D. Murch, \"An efficient approach for optimizing frequency reconfigurable pixel antennas using genetic algorithms,\" IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 609-620, Feb. 2014. [9] B. Liu, K.-F. Tong, K.-K. Wong, C.-B. Chae, and H. Wong, \"Be water, my antennas: Riding on radio wave fluctuation in nature for spatial multiplexing using programmable meta-fluid antenna,\" arXiv preprint, , 2025. [10] M. Kubo et al., \"Stretchable microfluidic radiofrequency antennas,\" Adv. Mater., vol. 22, no. 25, pp. 2749-2752, Jul. 2010. [11] J. Zhang, M. O. Akinsolu, B. Liu, and G. A. E. Vandenbosch, \"Automatic AI-driven design of mutual coupling reducing topologies for frequency reconfigurable antenna arrays,\" IEEE Trans. Antennas Propag., vol. 69, no. 3, pp. 1831-1836, Mar. 2021. [12] J. Zou, S. Sun, and C. Wang, \"Online learning-induced port selection for fluid antenna in dynamic channel environment,\" IEEE Wireless Commun. Lett., vol. 13, no. 2, pp. 313-317, Feb. 2024. [13] C. Gupta et al., \"Accurate and computationally efficient modeling of nonquasi static effects in MOSFETs for millimeter-wave applications,\" IEEE Trans. Electron Devices, vol. 66, no. 1, pp. 44-51, Jan. 2019 [14] Y. Wang, H. Shen, C. Han, and M. Tao, \"Movable antennas: Channel measurement, modeling, and performance evaluation,\" arXiv preprint , , 2024. [15] T. Jang, C. Zhang, H. Youn, J. Zhou, and L. J. Guo, \"Semitransparent and flexible mechanically reconfigurable electrically small antennas based on tortuous metallic micromesh,\" IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 150-158, Jan. 2017."}}},{"rowIdx":191,"cells":{"id":{"kind":"string","value":"2510.14790"},"raw":{"kind":"string","value":"ACTIVE JAMMER LOCALIZATION VIA ACQUISITION-AWARE PATH PLANNING\nLuis Gonz´alez-Gudi˜no1, Mariona Jaramillo-Civill2, Pau Closas2, Tales Imbiriba1\n1Dept. of Computer Science, University of Massachusetts Boston, Boston, MA, USA\n2Dept. of Electrical & Computer Engineering, Northeastern University, Boston, MA, USA\nABSTRACT\nWe propose an active jammer localization framework that combines\nBayesian optimization with acquisition-aware path planning. Un-\nlike passive crowdsourced methods, our approach adaptively guides\na mobile agent to collect high-utility Received Signal Strength\nmeasurements while accounting for urban obstacles and mobil-\nity constraints. For this, we modified the A* algorithm, A-UCB*,\nby incorporating acquisition values into trajectory costs, leading\nto high-acquisition planned paths. Simulations on realistic urban\nscenarios show that the proposed method achieves accurate localiza-\ntion with fewer measurements compared to uninformed baselines,\ndemonstrating consistent performance under different environments.\nIndex Terms— Jammer localization, GNSS interference, Bayesian\noptimization, Gaussian processes, Path planning\n1. INTRODUCTION\nGlobal Navigation Satellite Systems (GNSS) such as GPS, Galileo,\nGLONASS and BeiDou provide critical position, navigation, and\ntiming (PNT) services for a wide array of applications, from intel-\nligent transportation and precision agriculture to timing-dependent\ninfrastructures like banking systems and cellular networks [1]. How-\never, the strong dependence on GNSS makes these systems vulner-\nable to both unintentional and intentional interference [2, 3]. Unin-\ntentional interference may originate from out-of-band sources such\nas terrestrial digital video broadcasting (DVB-T) or amateur radios,\nas well as from in-band sources like distance measuring equipment\n(DME) or civilian radars. Additionally, GNSS signals are suscepti-\nble to jamming by personal privacy devices (PPDs). These devices,\nwhich are inexpensive and readily available online despite being ille-\ngal, emit high-power signals in the L-band (the frequency band used\nby GNSS) and can disrupt reception over distances ranging from tens\nof meters to several kilometers. These interferences can overwhelm\nreceiver front-ends and result in service denial, with incidents re-\nported in ports and air traffic control zones [4].\nDetecting and localizing such interferers is essential for resilient\nPNT operations. A cost-effective and scalable solution is crowd-\nsourced data [5, 6, 7], especially in densely populated or high-traffic\nareas with many GNSS users. Crowdsourcing leverages existing\nGNSS-enabled devices (e.g., smartphones) to collect environmen-\ntal data in a distributed way. A key measurement here is Received\nSignal Strength (RSS), which quantifies signal power at a given\nlocation and frequency. Under jamming, RSS can deviate markedly\nfrom expected GNSS levels, indicating abnormal activity. By an-\nalyzing spatially distributed RSS readings, it becomes possible to\ninfer the presence and even approximate location of interference\nThis work was partially supported by the National Science Foundation\nunder Awards 1845833, 2326559 and 2530870.\nsources without requiring a dense deployment of dedicated monitor-\ning stations.\nThis principle has given rise to a variety of localization tech-\nniques. Many approaches in crowdsourced jammer localization\ntypically rely on fitting power measurements, such as Carrier-to-\nNoise-density ratio C/N0 or Automatic Gain Control (AGC) values,\nto a simple physical model [7, 8, 9, 10, 11]. These methods often\nassume a known path-loss propagation model that, while effective in\nopen-sky scenarios, struggles in complex urban environments where\nmultipath, shadowing and occlusions introduce significant devia-\ntions from the ideal path-loss function. To address these limitations,\nmore recent work has explored data-driven approaches. Instead of\nassuming a fixed physical model, these methods use tools like neu-\nral networks to learn the complex, non-linear relationship between\nlocation and RSS directly from the data [6, 12].\nA common thread in the aforementioned research is its pas-\nsive nature. These methods rely on data collected by users pursu-\ning their own objectives, meaning samples are gathered inciden-\ntally. This leads to inefficient localization, as measurements may\nbe sparse, clustered in redundant areas, or fail to cover regions of\nhighest uncertainty. As a result, many samples may be needed for a\nconfident jammer estimate. Several recent works have explored alter-\nnative approaches to overcome these limitations. For instance, [13]\npropose a UAV-based system that scans the environment by hover-\ning at preplanned waypoints and rotating a directional antenna to\nmeasure signal strength. However, the UAV follows a static plan and\ndoes not adapt its path to the data, limiting efficiency. In contrast,\n[14] deploy a UAV tailored for jamming scenarios that performs\ngreedy bearing-based triangulation. This adds partial adaptivity by\niteratively steering toward estimated jammer directions, but remains\nlimited to short-horizon heuristics without global reasoning about\npromising regions.\nThese limitations reveal a gap in the literature: the absence of\nadaptive strategies that guide data collection in a sample-efficient\nand environment-aware manner. To address this, we propose an\nactive localization framework that combines Bayesian optimiza-\ntion with acquisition-aware path planning, enabling an autonomous\nagent to navigate complex environments while wisely selecting\nhigh-value measurements. To isolate the performance of our pro-\nposed framework, we focus on the localization of a single static\njammer. However, with an appropriately designed multimodal sur-\nrogate model, the framework could be extended to handle multiple\njammers. In summary, our main contributions are:\n• A novel Bayesian optimization framework for active jammer\nlocalization.\n• An acquisition-aware path planning strategy that balances\nmovement cost and acquisition gain.\n• A sample-efficient strategy that accurately localizes the jam-\nmer with minimal measurements.\narXiv:2510.14790v1 [cs.LG] 16 Oct 2025\n\n2. PROBLEM FORMULATION\nIn this work, we address the problem of localizing a single stationary\njamming source in urban environments, using crowdsourced RSS\nmeasurements from static agents together with adaptive active sens-\ning by an autonomous mobile agent. Our key assumption is that\nthe jammer’s true position xJ lies in a bounded two-dimensional\narea X ⊂R2 and induces an (unknown) interference-power field\nftrue(x; xJ), whose global maximum corresponds to the location\nwhere the RSS from the jammer is strongest. Thus, our goal is to\nestimate it by finding the global maximizer of this field:\nˆxJ = arg max\nx∈X ftrue(x; xJ)\n(1)\nObtaining a closed-form analytical model for ftrue in urban sce-\nnarios is extremely challenging due to multipath, non-line-of-sight\nconditions, and shadowing effects that make the field highly non-\nconvex and dependent on city-specific characteristics like building\nlayout and materials. As a consequence, ftrue is treated as a black-\nbox function. In this context, an autonomous agent is deployed to\nsequentially visit probing locations can collect data with the ulti-\nmate goal of solving (1). At each location x ∈X, the agent records\na noisy RSS measurement:\nyn = ftrue(x ; xJ) + ξn,\nn = 1, . . . , N\n(2)\nwith ξn representing additive measurement noise. Importantly, the\nagent operates under navigation constraints: some regions of X are\ninaccessible due to static obstacles such as buildings, walls, or re-\nstricted zones. We assume that these obstacles (and hence the feasi-\nble subset of X) are known a priori. Therefore, the challenge is not\nonly to search for the global maximizer, but to decide where to mea-\nsure next so that the maximizer can be identified with as few samples\nas possible. This motivates an approach that smartly balances explo-\nration and exploitation.\n3. FRAMEWORK FOR JAMMER LOCALIZATION\nTo tackle this problem effectively, we require a strategy capable of\nnavigating complex, noisy, and partially observable environments in\na data-efficient manner. Bayesian optimization (BO) provides a nat-\nural fit for this setting: it is specifically designed for optimizing ex-\npensive, black-box functions with limited samples, while explicitly\nmodeling uncertainty. In our context, BO enables the agent to rea-\nson about both the expected interference power and the confidence\nof that estimate across the environment, allowing it to actively trade\noff, through an acquisition function, between exploring uncertain re-\ngions and exploiting promising ones. For a detailed overview of the\nBO framework, we refer the reader to [15].\nInspired by BO, we adopt an iterative framework (see Fig.1)\nwhere at each iteration, the agent performs four main steps: (i) ac-\nquires a new measurement at a selected location, (ii) updates a prob-\nabilistic model of the interference field, (iii) computes an acquisition\nfunction that determines the next sensing location, and (iv) plans an\nacquisition-guided path to this location. This loop continues until\nconvergence criteria are met, such as reaching a confidence in the\nestimated jammer position or a time constraint.\nIn our approach, we discretize the continuous domain X ⊂R2\ninto a finite set of uniformly spaced grid points denoted as G =\n{x(1), . . . , x(M)} ⊂X, where each x(i) corresponds to a grid\ncell. For notational simplicity, we will write x(i) simply as x. This\ngrid-based representation serves two purposes: first, it simplifies the\nfitting of the surrogate model and acquisition function evaluation,\nFig. 1. Overview of the proposed active localization framework.\nTop: GP posterior mean µn(x) and uncertainty σn(x) over F.\nBottom-right: UCB acquisition αUCB(x) used to select the next\nsensing target xc\nn (⋆). The black curve is the past trajectory and the\ngreen dashed line is the path planning toward xc\nn. Gray polygons are\nbuildings (non-traversable) and black dots are initial crowdsourced\nsamples D0.\nand second, it enables efficient path planning via graph-based algo-\nrithms. Some cells in G are obstructed by static obstacles. Let each\nindividual obstacle be defined as a subset Ok ⊂G for k = 1, . . . , K,\nand define the complete set of obstacle cells as O = SK\nk=1 Ok ⊂G.\nTherefore, the feasible set of grid locations where the agent can\nsafely navigate and take measurements is then given by F = G \\ O.\nAll agent trajectories and sensing decisions are constrained to lie\nwithin this set. We now describe each component of the framework\nin more detail.\nSensing. At each iteration n = 1, . . . , N of the BO loop, the agent\nselects a batch of bn feasible locations {xn,1, . . . xn,bn} ⊂F\n(see Sec.3) and records a corresponding set of noisy RSS mea-\nsurements according to (2). In this article, noise is assumed to\nbe Gaussian ξn ∼N(0, σ2). These new measurements are ap-\npended to the existing dataset, which evolves over time as Dn =\nDn−1 ∪{(xn,j , yn,j)}bn\nj=1. Here, D0 contains b0 initial crowd-\nsourced measurements available at time step zero, each also obtained\naccording to (2) by static agents.\nPredictive Model. We adopt a Gaussian Process (GP) [16] regres-\nsor as surrogate model of the interference-power field. GPs provide\na nonparametric Bayesian prior over functions that yields, after con-\nditioning on data, closed-form posterior predictive means and vari-\nances. In short, we assume the prior fsurr(˜x) ∼GP\n\u00000, kθ(˜x, ˜x′)\n\u0001\n,\nwhere at each location x = (px, py) we build the feature vector\n˜x = [px, py, zx], with px, py the normalized 2D coordinates and zx\nthe normalized building height at location x. This lightweight aug-\nmentation helps the GP capture systematic power variations induced\nby urban morphology without sampling inside obstacles.\nGiven the dataset Dn, the surrogate model follows a posterior\ndistribution fsurr,n|Dn, ˜x ∼N(µn(˜x), σ2\nn(˜x)) at any query loca-\ntion ˜x with mean µ(˜x) and uncertainty σ(˜x):\n\nµn(˜x) = kθ,n(˜x)⊤\u0000Kn + σ2\nηI\n\u0001−1yn\n(3)\nσ2\nn(˜x) = kθ(˜x, ˜x) −kθ,n(˜x)⊤\u0000Kn + σ2\nηI\n\u0001−1kθ,n(˜x)\n(4)\nwhere yn\n=\n[y1, . . . , y|Dn|]⊤denotes the observation vector\nand Kn\n∈\nR|Dn|×|Dn| defines the Gram matrix with entries\n[Kn]pq = kθ(˜xp, ˜x′\nq). For a new query ˜x, the associated kernel\nvector is kθ,n(˜x) = [kθ(˜x, ˜x1), . . . , kθ(˜x, ˜x|Dn|)]⊤. These two\nquantities are jointly exploited by the acquisition function (see\nSec. 3). To flexibly capture both short-range fluctuations (e.g., lo-\ncal multipath) and broader trends of the field, we use an additive,\nmulti-scale kernel with a learnable noise term:\nkθ(˜x, ˜x′) = kℓ\nθ(˜x, ˜x′) + ks\nθ(˜x, ˜x′) + σ2\nηδ˜x,˜x′\n(5)\nwith kι\nθ(˜x, ˜x′) = σ2\nι exp(−1\n2\nP\nd ˜x −˜x′)2/ℓ2\nι,d), and ι ∈{ℓ, s}\ncorresponding to the long- and short-length scales, explaining the\nsmooth and high-frequency components of the field. The white-noise\nterm σ2\nηδ˜x,˜x′ accounts for measurement noise and residual model\nerror, with δ˜x,˜x′ = 1 if ˜x = ˜x′ and 0 otherwise. We found this\ntwo-scale structure a good compromise between expressiveness and\nrobustness for urban fields. Our formulation is agnostic to this choice\nand can swap kernels without altering the rest of the pipeline.\nHyperparameters θ = (σ2\ns, σ2\nℓ, ℓs,d, ℓℓ,d, σ2\nη) are learned by\nmaximizing the GP log marginal likelihood with multiple random\nrestarts. Moreover, targets are normalized (zero-mean, unit-variance)\nand de-normalized at prediction time. Given Dn, the posterior pre-\ndictive is computed in closed form and supplied to the acquisition\n(see 3). Finally, the active sensing framework does not rely on\nGP-specific structure: any probabilistic surrogate able to produce\ncalibrated uncertainty (e.g., Bayesian neural networks) could re-\nplace the GP. We choose GPs here for their data efficiency in the\nlow-to-moderate sample regime and their well-calibrated uncertain-\nties, which are pivotal for acquisition-aware planning.\nAcquisition Function. To select the next candidate location, we\nadopt the Upper Confidence Bound (UCB) [17] acquisition strat-\negy. UCB selects points that maximize a weighted combination of\nthe surrogate model’s predictive mean and standard deviation:\nαUCB(x) = µ(x) + κ · σ(x)\n(6)\nwhere µ(x) and σ(x) are the predictive mean and standard devia-\ntion at location x, and κ > 0 is a tunable parameter that controls\nthe exploration–exploitation trade-off. Higher values of κ encourage\nmore exploration by favoring regions with high uncertainty, while\nlower values bias the search toward locations with high predicted\nsignal strength. At each iteration, the next candidate point is chosen\nas xc\nn = arg maxx∈F αUCB(x).\nPath Planning Strategy. While Bayesian optimization identifies\nthe next most valuable location xc\nn via an acquisition function, nav-\nigating directly to this target is often infeasible or suboptimal in\nreal environments. Urban landscapes impose hard constraints due\nto buildings, and moving in straight lines can lead to redundant\npaths that fail to extract new information. What is needed is a path\nplanning strategy that remains goal-oriented but also exploits the\nacquisition landscape along the way. In other words, at each BO it-\neration we set xc\nn as the intended destination, and instead of moving\ndirectly there, we compute a bounded-length path from the agent’s\ncurrent position that also prioritizes acquisition gain.\nTo this end, we modify the A* algorithm [18, 19], a graph-based\nsearch method for computing minimum-cost paths. For each node x,\nthe algorithm evaluates\nf(x) = g(x) + h(x),\n(7)\nwhere g(x) is the accumulated path cost from the start node (sum\nof edge costs cx,x′) and h(x) is an admissible heuristic that esti-\nmates the remaining cost to the goal. By always expanding the node\nwith minimum f(x), A* guarantees an optimal collision-free path\nin grid environments with static obstacles. While classical A* min-\nimizes only travel cost, yielding efficient but uninformed paths, we\nmodify the arc cost definition to incorporate values from the acqui-\nsition function. Specifically, for each edge connecting nodes x and\nx′, we define\ncx,x′ =\n\u0000λlen −λinfo · ¯α\n\u0001\n∥x −x′∥,\n(8)\nwhere ¯α = 1\n2\n\u0000αUCB(x) + αUCB(x′)\n\u0001\nis the mean acquisition value\nacross the edge. Here, λlen weighs travel cost, while λinfo biases the\nplanner toward promising regions. This modification transforms A*\ninto an acquisition-aware path planner A-UCB* that not only reaches\nthe selected target xc\nn but also collects valuable samples along the\nway. To respect mobility constraints, each path is limited to a maxi-\nmum length budget δ (implicitly accounting for a maximum allow-\nable velocity) and uniformly subsampled into bn waypoints per iter-\nation.\n4. EXPERIMENTS\n4.1. Data Generation\nWe synthesize RSS fields following the crowdsourced-jamming se-\ntups in prior work [6, 12], using MATLAB’s deterministic 3D ray-\ntracing engine [20, 21]. Unlike prior work, we adopt a denser sam-\npling strategy, placing receivers on a uniform grid with 2 m spacing\nbetween adjacent sampling points across the walkable workspace.\nBuilding maps are also preprocessed: interior courtyards without\nstreet access are removed so that each building becomes a compact\nobstacle polygon. We considered two representative urban layouts:\n• Chicago Downtown (dense urban core): a downtown sce-\nnario characterized by narrow streets and tall buildings, cre-\nating strong multipath and shadowing effects.\n• Boston Common (urban park with open-sky): a mixed en-\nvironment where open park areas provide long line-of-sight\ncorridors, while surrounding fac¸ades still generate significant\nreflections and occlusions.\n4.2. Quantitative Results\nWe evaluate the performance of our jammer localization framework\nunder various sampling andplanning strategies. The goal of this anal-\nysis is dual: first, to assess how accurately each method can localize\nthe jammer using a limited number of iterations, and second, to in-\nvestigate the role of acquisition-aware path planning in improving\nsample efficiency.\nBesides our method with a finite path budget, A-UCB* (δ =\n50), we consider: (i) A-UCB* with unlimited path length (δ = ∞)\nas an upper bound; (ii) Random Motion (RM), which moves uni-\nformly at random along the four cardinal directions with δ = 50\nsteps per iteration—this respects motion feasibility but ignores ac-\nquisition information, serving as a conservative lower bound; and\n(iii) Random i.i.d. Sampling (RIS), which draws queries uniformly\nfrom F without motion continuity or acquisition awareness.\n\nAll methods are evaluated over 100 independent trials. At each\ntrial, the agent starts from a randomly selected initial position in the\ngrid; this position is fixed across all methods within the same trial\nto ensure a fair comparison. In every BO iteration, all methods col-\nlect bn = 2 RSS measurements and all experiments use b0 = 35\ninitial crowdsourced samples uniformly drawn from F. The UCB\nexploration parameter is set to κ = 2, and the measurement noise\nvariance to σ2 = 2.5. For A-UCB*, the maximum path length bud-\nget is set to δ = 50. We fixed these values for all methods and trials.\nTo evaluate localization accuracy, we report two complementary\nerror metrics. The Surrogate Model Error (SME) measures the dis-\ntance between the true jammer location and the maximizer of the GP\nposterior mean field, reflecting the accuracy of the surrogate predic-\ntion. The Bayesian Optimization Error (BOE) instead measures the\ndistance to the grid point with the highest RSS value actually sam-\npled, which corresponds to the output of the proposed algorithm. The\nresults are summarized in Table 1, where we report the median and\ninterquartile range (25%-75%) of each error metric across the 100\ntrials. Furthermore, we illustrate the evolution of the BOE across the\n80 iterations in Fig.2. Specifically, these plots depict the distribution\nof BOE values at each iteration for both scenarios.\nTable 1. Localization error (median [25%–75%]) across 100 inde-\npendent trials for both Chicago Downtown and Boston Common\ndatasets. Errors are reported in meters for SME and BOE. Lower\n(↓) values indicate better localization performance.\nChicago Downtown\nBoston Common\nMethod\nSME (↓)\nBOE (↓)\nSME (↓)\nBOE (↓)\nRIS\n57.4 [27.6–120.1]\n65.6 [28.1–103.8]\n50.7 [25.4–93.6]\n35.9 [20.6–57.2]\nRM\n100.3 [36.9–226.5]\n102.5 [62.3–186.1]\n82.4 [46.0–132.4]\n83.9 [44.6–103.6]\nA-UCB* (δ = ∞)\n14.1 [6.5–24.5]\n14.1 [7.7–25.0]\n8.3 [5.3–12.4]\n8.3 [5.6–14.1]\nA-UCB* (δ = 50)\n13.4 [7.8–28.0]\n12.9 [7.8–24.0]\n12.8 [8.3–32.2]\n11.4 [6.9–23.2]\nFig. 2. Evolution of the BOE in the Chicago Downtown (top panel)\nand Boston Common (bottom panel) scenarios across 80 Bayesian\noptimization iterations. Boxplots at each iteration show the distribu-\ntion of localization errors across 100 independent trials. The inset\nzooms into the last 30 iterations highlight last convergence behavior.\nThe proposed acquisition-aware planning strategy consistently\nachieves faster convergence and lower localization errors compared\nto the uninformed baselines. Random motion is particularly inef-\nfective, as its tendency to revisit redundant regions leads to per-\nsistent errors even after many iterations. Uniform random sampling\nperforms slightly better due to broader spatial coverage, but it still\nlacks the adaptivity needed to consistently reduce error. In contrast,\nA-UCB* rapidly drives the optimization process toward the jam-\nmer location, with its error distribution narrowing significantly after\nabout 30 iterations—corresponding to fewer than 100 total measure-\nments when accounting for the initial 35 crowdsourced samples and\ntwo agent-collected samples per iteration. This outcome underscores\nthe effectiveness of acquisition-aware path planning in prioritizing\nhigh-utility measurements: rather than relying on exhaustive explo-\nration, the method focuses on high-value measurements, producing\nmore reliable field estimates minimizing wasted effort. Moreover,\neven when constrained by a finite path-length budget, the method\nretains most of its efficiency. Finally, the comparison between the\ntwo datasets emphasizes the role of the environment: Boston Com-\nmon, with its open-sky areas, yields lower error levels overall, while\nthe dense and obstructed Chicago Downtown setting poses greater\nchallenges. Despite this, A-UCB* maintains robust performance in\nboth scenarios, highlighting its adaptability in complex urban envi-\nronments.\n4.3. Sensitivity of κ\nVarying κ and measuring BOE at iteration 30 shows a shallow U-\nshaped trend in both datasets, as illustrated in Fig. 3. Very small\nκ (e.g., 0.1) is too exploitative, yielding large median errors and\nwide IQRs, where the agent’s initialization strongly influences per-\nformance. Very large κ (≥5) over-explores, increasing error and\nvariability at the fixed number of iterations. The lowest, most stable\nerrors occur for κ ∈[1, 3]. We therefore find κ = 2 a robust choice\nwithin the low-error plateau.\nFig. 3. Sensitivity analysis of the exploration–exploitation parameter\nκ in the UCB acquisition function. The plot shows the BOE after 30\niterations for both datasets. Each point corresponds to the median\nBOE across 100 trials, with error bars denoting the IQR.\n5. CONCLUSION\nWe introduced an acquisition-aware Bayesian optimization frame-\nwork for jammer localization that integrates probabilistic modeling\nwith adaptive path planning. Results in realistic urban environments\nconfirm that our method localizes interference sources with high\naccuracy using limited measurements, outperforming random base-\nlines in both convergence speed and robustness. Even under mo-\nbility constraints, the framework retains efficiency, highlighting its\nsuitability for real-world deployments. Future work will extend this\napproach to multi-jammer scenarios and explore alternative proba-\nbilistic surrogates beyond GP’s.\n\n6. 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Iskander, “Ray Tracing for Ra-\ndio Propagation Modeling: Principles and Applications,” IEEE\nAccess, vol. 3, pp. 1089–1100, 2015.\n[21] K.R. Schaubach, N.J. Davis, and T.S. Rappaport, “A ray trac-\ning method for predicting path loss and delay spread in mi-\ncrocellular environments,” in [1992 Proceedings] Vehicular\nTechnology Society 42nd VTS Conference - Frontiers of Tech-\nnology, 1992, pp. 932–935 vol.2.\n"},"cleaned":{"kind":"string","value":"ACTIVE JAMMER LOCALIZATION VIA ACQUISITION-AWARE PATH PLANNING Luis Gonz ́alez-Gudi ̃no1, Mariona Jaramillo-Civill2, Pau Closas2, Tales Imbiriba1 1Dept. of Computer Science, 2Dept. of Electrical & Computer Engineering, Northeastern University, Boston, MA, USA ABSTRACT We propose an active jammer localization framework that combines Bayesian optimization with acquisition-aware path planning. Unlike passive crowdsourced methods, our approach adaptively guides a mobile agent to collect high-utility Received Signal Strength measurements while accounting for urban obstacles and mobility constraints. For this, we modified the A* algorithm, A-UCB*, by incorporating acquisition values into trajectory costs, leading to high-acquisition planned paths. Simulations on realistic urban scenarios show that the proposed method achieves accurate localization with fewer measurements compared to uninformed baselines, demonstrating consistent performance under different environments. Index Terms- Jammer localization, GNSS interference, Bayesian optimization, Gaussian processes, Path planning 1. INTRODUCTION Global Navigation Satellite Systems (GNSS) such as GPS, Galileo, GLONASS and BeiDou provide critical position, navigation, and timing (PNT) services for a wide array of applications, from intelligent transportation and precision agriculture to timing-dependent infrastructures like banking systems and cellular networks [1]. However, the strong dependence on GNSS makes these systems vulnerable to both unintentional and intentional interference [2, 3]. Unintentional interference may originate from out-of-band sources such as terrestrial digital video broadcasting (DVB-T) or amateur radios, as well as from in-band sources like distance measuring equipment (DME) or civilian radars. Additionally, GNSS signals are susceptible to jamming by personal privacy devices (PPDs). These devices, which are inexpensive and readily available online despite being illegal, emit high-power signals in the L-band (the frequency band used by GNSS) and can disrupt reception over distances ranging from tens of meters to several kilometers. These interferences can overwhelm receiver front-ends and result in service denial, with incidents reported in ports and air traffic control zones [4]. Detecting and localizing such interferers is essential for resilient PNT operations. A cost-effective and scalable solution is crowdsourced data [5, 6, 7], especially in densely populated or high-traffic areas with many GNSS users. Crowdsourcing leverages existing GNSS-enabled devices (e.g., smartphones) to collect environmental data in a distributed way. A key measurement here is Received Signal Strength (RSS), which quantifies signal power at a given location and frequency. Under jamming, RSS can deviate markedly from expected GNSS levels, indicating abnormal activity. By analyzing spatially distributed RSS readings, it becomes possible to infer the presence and even approximate location of interference This work was partially supported by the National Science Foundation under Awards 1845833, 2326559 and 2530870. sources without requiring a dense deployment of dedicated monitoring stations. This principle has given rise to a variety of localization techniques. Many approaches in crowdsourced jammer localization typically rely on fitting power measurements, such as Carrier-toNoise-density ratio C/N0 or Automatic Gain Control (AGC) values, to a simple physical model [7, 8, 9, 10, 11]. These methods often assume a known path-loss propagation model that, while effective in open-sky scenarios, struggles in complex urban environments where multipath, shadowing and occlusions introduce significant deviations from the ideal path-loss function. To address these limitations, more recent work has explored data-driven approaches. Instead of assuming a fixed physical model, these methods use tools like neural networks to learn the complex, non-linear relationship between location and RSS directly from the data [6, 12]. A common thread in the aforementioned research is its passive nature. These methods rely on data collected by users pursuing their own objectives, meaning samples are gathered incidentally. This leads to inefficient localization, as measurements may be sparse, clustered in redundant areas, or fail to cover regions of highest uncertainty. As a result, many samples may be needed for a confident jammer estimate. Several recent works have explored alternative approaches to overcome these limitations. For instance, [13] propose a UAV-based system that scans the environment by hovering at preplanned waypoints and rotating a directional antenna to measure signal strength. However, the UAV follows a static plan and does not adapt its path to the data, limiting efficiency. In contrast, [14] deploy a UAV tailored for jamming scenarios that performs greedy bearing-based triangulation. This adds partial adaptivity by iteratively steering toward estimated jammer directions, but remains limited to short-horizon heuristics without global reasoning about promising regions. These limitations reveal a gap in the literature: the absence of adaptive strategies that guide data collection in a sample-efficient and environment-aware manner. To address this, we propose an active localization framework that combines Bayesian optimization with acquisition-aware path planning, enabling an autonomous agent to navigate complex environments while wisely selecting high-value measurements. To isolate the performance of our proposed framework, we focus on the localization of a single static jammer. However, with an appropriately designed multimodal surrogate model, the framework could be extended to handle multiple jammers. In summary, our main contributions are: • A novel Bayesian optimization framework for active jammer localization. • An acquisition-aware path planning strategy that balances movement cost and acquisition gain. • A sample-efficient strategy that accurately localizes the jammer with minimal measurements. 16 Oct 2025 2. PROBLEM FORMULATION In this work, we address the problem of localizing a single stationary jamming source in urban environments, using crowdsourced RSS measurements from static agents together with adaptive active sensing by an autonomous mobile agent. Our key assumption is that the jammer's true position xJ lies in a bounded two-dimensional area X ⊂R2 and induces an (unknown) interference-power field ftrue(x; xJ), whose global maximum corresponds to the location where the RSS from the jammer is strongest. Thus, our goal is to estimate it by finding the global maximizer of this field: ˆxJ = arg max x∈X ftrue(x; xJ) (1) Obtaining a closed-form analytical model for ftrue in urban scenarios is extremely challenging due to multipath, non-line-of-sight conditions, and shadowing effects that make the field highly nonconvex and dependent on city-specific characteristics like building layout and materials. As a consequence, ftrue is treated as a blackbox function. In this context, an autonomous agent is deployed to sequentially visit probing locations can collect data with the ultimate goal of solving (1). At each location x ∈X, the agent records a noisy RSS measurement: yn = ftrue(x ; xJ) + ξn, n = 1, . . . , N (2) with ξn representing additive measurement noise. Importantly, the agent operates under navigation constraints: some regions of X are inaccessible due to static obstacles such as buildings, walls, or restricted zones. We assume that these obstacles (and hence the feasible subset of X) are known a priori. Therefore, the challenge is not only to search for the global maximizer, but to decide where to measure next so that the maximizer can be identified with as few samples as possible. This motivates an approach that smartly balances exploration and exploitation. 3. FRAMEWORK FOR JAMMER LOCALIZATION To tackle this problem effectively, we require a strategy capable of navigating complex, noisy, and partially observable environments in a data-efficient manner. Bayesian optimization (BO) provides a natural fit for this setting: it is specifically designed for optimizing expensive, black-box functions with limited samples, while explicitly modeling uncertainty. In our context, BO enables the agent to reason about both the expected interference power and the confidence of that estimate across the environment, allowing it to actively trade off, through an acquisition function, between exploring uncertain regions and exploiting promising ones. For a detailed overview of the BO framework, we refer the reader to [15]. Inspired by BO, we adopt an iterative framework (see Fig.1) where at each iteration, the agent performs four main steps: (i) acquires a new measurement at a selected location, (ii) updates a probabilistic model of the interference field, (iii) computes an acquisition function that determines the next sensing location, and (iv) plans an acquisition-guided path to this location. This loop continues until convergence criteria are met, such as reaching a confidence in the estimated jammer position or a time constraint. In our approach, we discretize the continuous domain X ⊂R2 into a finite set of uniformly spaced grid points denoted as G = {x(1), . . . , x(M)} ⊂X, where each x(i) corresponds to a grid cell. For notational simplicity, we will write x(i) simply as x. This grid-based representation serves two purposes: first, it simplifies the fitting of the surrogate model and acquisition function evaluation, Fig. 1. Overview of the proposed active localization framework. Top: GP posterior mean μn(x) and uncertainty σn(x) over F. Bottom-right: UCB acquisition αUCB(x) used to select the next sensing target xc n (⋆). The black curve is the past trajectory and the green dashed line is the path planning toward xc n. Gray polygons are buildings (non-traversable) and black dots are initial crowdsourced samples D0. and second, it enables efficient path planning via graph-based algorithms. Some cells in G are obstructed by static obstacles. Let each individual obstacle be defined as a subset Ok ⊂G for k = 1, . . . , K, and define the complete set of obstacle cells as O = SK k=1 Ok ⊂G. Therefore, the feasible set of grid locations where the agent can safely navigate and take measurements is then given by F = G \\ O. All agent trajectories and sensing decisions are constrained to lie within this set. We now describe each component of the framework in more detail. Sensing. At each iteration n = 1, . . . , N of the BO loop, the agent selects a batch of bn feasible locations {xn,1, . . . xn,bn} ⊂F (see Sec.3) and records a corresponding set of noisy RSS measurements according to (2). In this article, noise is assumed to be Gaussian ξn ∼N(0, σ2). These new measurements are appended to the existing dataset, which evolves over time as Dn = Dn-1 ∪{(xn,j , yn,j)}bn j=1. Here, D0 contains b0 initial crowdsourced measurements available at time step zero, each also obtained according to (2) by static agents. Predictive Model. We adopt a Gaussian Process (GP) [16] regressor as surrogate model of the interference-power field. GPs provide a nonparametric Bayesian prior over functions that yields, after conditioning on data, closed-form posterior predictive means and variances. In short, we assume the prior fsurr( ̃x) ∼GP 0, kθ( ̃x, ̃x′) , where at each location x = (px, py) we build the feature vector ̃x = [px, py, zx], with px, py the normalized 2D coordinates and zx the normalized building height at location x. This lightweight augmentation helps the GP capture systematic power variations induced by urban morphology without sampling inside obstacles. Given the dataset Dn, the surrogate model follows a posterior distribution fsurr,n|Dn, ̃x ∼N(μn( ̃x), σ2 n( ̃x)) at any query location ̃x with mean μ( ̃x) and uncertainty σ( ̃x): μn( ̃x) = kθ,n( ̃x)⊤ Kn + σ2 ηI -1yn (3) σ2 n( ̃x) = kθ( ̃x, ̃x) -kθ,n( ̃x)⊤ Kn + σ2 ηI -1kθ,n( ̃x) (4) where yn = [y1, . . . , y|Dn|]⊤denotes the observation vector and Kn ∈ R|Dn|×|Dn| defines the Gram matrix with entries [Kn]pq = kθ( ̃xp, ̃x′ q). For a new query ̃x, the associated kernel vector is kθ,n( ̃x) = [kθ( ̃x, ̃x1), . . . , kθ( ̃x, ̃x|Dn|)]⊤. These two quantities are jointly exploited by the acquisition function (see Sec. 3). To flexibly capture both short-range fluctuations (e.g., local multipath) and broader trends of the field, we use an additive, multi-scale kernel with a learnable noise term: kθ( ̃x, ̃x′) = kl θ( ̃x, ̃x′) + ks θ( ̃x, ̃x′) + σ2 ηδ ̃x, ̃x′ (5) with kι θ( ̃x, ̃x′) = σ2 ι exp(-1 2 P d ̃x - ̃x′)2/l2 ι,d), and ι ∈{l, s} corresponding to the long- and short-length scales, explaining the smooth and high-frequency components of the field. The white-noise term σ2 ηδ ̃x, ̃x′ accounts for measurement noise and residual model error, with δ ̃x, ̃x′ = 1 if ̃x = ̃x′ and 0 otherwise. We found this two-scale structure a good compromise between expressiveness and robustness for urban fields. Our formulation is agnostic to this choice and can swap kernels without altering the rest of the pipeline. Hyperparameters θ = (σ2 s, σ2 l, ls,d, ll,d, σ2 η) are learned by maximizing the GP log marginal likelihood with multiple random restarts. Moreover, targets are normalized (zero-mean, unit-variance) and de-normalized at prediction time. Given Dn, the posterior predictive is computed in closed form and supplied to the acquisition (see 3). Finally, the active sensing framework does not rely on GP-specific structure: any probabilistic surrogate able to produce calibrated uncertainty (e.g., Bayesian neural networks) could replace the GP. We choose GPs here for their data efficiency in the low-to-moderate sample regime and their well-calibrated uncertainties, which are pivotal for acquisition-aware planning. Acquisition Function. To select the next candidate location, we adopt the Upper Confidence Bound (UCB) [17] acquisition strategy. UCB selects points that maximize a weighted combination of the surrogate model's predictive mean and standard deviation: αUCB(x) = μ(x) + κ · σ(x) (6) where μ(x) and σ(x) are the predictive mean and standard deviation at location x, and κ > 0 is a tunable parameter that controls the exploration-exploitation trade-off. Higher values of κ encourage more exploration by favoring regions with high uncertainty, while lower values bias the search toward locations with high predicted signal strength. At each iteration, the next candidate point is chosen as xc n = arg maxx∈F αUCB(x). Path Planning Strategy. While Bayesian optimization identifies the next most valuable location xc n via an acquisition function, navigating directly to this target is often infeasible or suboptimal in real environments. Urban landscapes impose hard constraints due to buildings, and moving in straight lines can lead to redundant paths that fail to extract new information. What is needed is a path planning strategy that remains goal-oriented but also exploits the acquisition landscape along the way. In other words, at each BO iteration we set xc n as the intended destination, and instead of moving directly there, we compute a bounded-length path from the agent's current position that also prioritizes acquisition gain. To this end, we modify the A* algorithm [18, 19], a graph-based search method for computing minimum-cost paths. For each node x, the algorithm evaluates f(x) = g(x) + h(x), (7) where g(x) is the accumulated path cost from the start node (sum of edge costs cx,x′) and h(x) is an admissible heuristic that estimates the remaining cost to the goal. By always expanding the node with minimum f(x), A* guarantees an optimal collision-free path in grid environments with static obstacles. While classical A* minimizes only travel cost, yielding efficient but uninformed paths, we modify the arc cost definition to incorporate values from the acquisition function. Specifically, for each edge connecting nodes x and x′, we define cx,x′ = λlen -λinfo · ̄α ∥x -x′∥, (8) where ̄α = 1 2 αUCB(x) + αUCB(x′) is the mean acquisition value across the edge. Here, λlen weighs travel cost, while λinfo biases the planner toward promising regions. This modification transforms A* into an acquisition-aware path planner A-UCB* that not only reaches the selected target xc n but also collects valuable samples along the way. To respect mobility constraints, each path is limited to a maximum length budget δ (implicitly accounting for a maximum allowable velocity) and uniformly subsampled into bn waypoints per iteration. 4. EXPERIMENTS 4.1. Data Generation We synthesize RSS fields following the crowdsourced-jamming setups in prior work [6, 12], using MATLAB's deterministic 3D raytracing engine [20, 21]. Unlike prior work, we adopt a denser sampling strategy, placing receivers on a uniform grid with 2 m spacing between adjacent sampling points across the walkable workspace. Building maps are also preprocessed: interior courtyards without street access are removed so that each building becomes a compact obstacle polygon. We considered two representative urban layouts: • Chicago Downtown (dense urban core): a downtown scenario characterized by narrow streets and tall buildings, creating strong multipath and shadowing effects. • Boston Common (urban park with open-sky): a mixed environment where open park areas provide long line-of-sight corridors, while surrounding fac ̧ades still generate significant reflections and occlusions. 4.2. Quantitative Results We evaluate the performance of our jammer localization framework under various sampling andplanning strategies. The goal of this analysis is dual: first, to assess how accurately each method can localize the jammer using a limited number of iterations, and second, to investigate the role of acquisition-aware path planning in improving sample efficiency. Besides our method with a finite path budget, A-UCB* (δ = 50), we consider: (i) A-UCB* with unlimited path length (δ = ∞) as an upper bound; (ii) Random Motion (RM), which moves uniformly at random along the four cardinal directions with δ = 50 steps per iteration-this respects motion feasibility but ignores acquisition information, serving as a conservative lower bound; and (iii) Random i.i.d. Sampling (RIS), which draws queries uniformly from F without motion continuity or acquisition awareness. All methods are evaluated over 100 independent trials. At each trial, the agent starts from a randomly selected initial position in the grid; this position is fixed across all methods within the same trial to ensure a fair comparison. In every BO iteration, all methods collect bn = 2 RSS measurements and all experiments use b0 = 35 initial crowdsourced samples uniformly drawn from F. The UCB exploration parameter is set to κ = 2, and the measurement noise variance to σ2 = 2.5. For A-UCB*, the maximum path length budget is set to δ = 50. We fixed these values for all methods and trials. To evaluate localization accuracy, we report two complementary error metrics. The Surrogate Model Error (SME) measures the distance between the true jammer location and the maximizer of the GP posterior mean field, reflecting the accuracy of the surrogate prediction. The Bayesian Optimization Error (BOE) instead measures the distance to the grid point with the highest RSS value actually sampled, which corresponds to the output of the proposed algorithm. The results are summarized in Table 1, where we report the median and interquartile range (25%-75%) of each error metric across the 100 trials. Furthermore, we illustrate the evolution of the BOE across the 80 iterations in Fig.2. Specifically, these plots depict the distribution of BOE values at each iteration for both scenarios. Table 1. Localization error (median [25%-75%]) across 100 independent trials for both Chicago Downtown and Boston Common datasets. Errors are reported in meters for SME and BOE. Lower (↓) values indicate better localization performance. Chicago Downtown Boston Common Method SME (↓) BOE (↓) SME (↓) BOE (↓) RIS 57.4 [27.6-120.1] 65.6 [28.1-103.8] 50.7 [25.4-93.6] 35.9 [20.6-57.2] RM 100.3 [36.9-226.5] 102.5 [62.3-186.1] 82.4 [46.0-132.4] 83.9 [44.6-103.6] A-UCB* (δ = ∞) 14.1 [6.5-24.5] 14.1 [7.7-25.0] 8.3 [5.3-12.4] 8.3 [5.6-14.1] A-UCB* (δ = 50) 13.4 [7.8-28.0] 12.9 [7.8-24.0] 12.8 [8.3-32.2] 11.4 [6.9-23.2] Fig. 2. Evolution of the BOE in the Chicago Downtown (top panel) and Boston Common (bottom panel) scenarios across 80 Bayesian optimization iterations. Boxplots at each iteration show the distribution of localization errors across 100 independent trials. The inset zooms into the last 30 iterations highlight last convergence behavior. The proposed acquisition-aware planning strategy consistently achieves faster convergence and lower localization errors compared to the uninformed baselines. Random motion is particularly ineffective, as its tendency to revisit redundant regions leads to persistent errors even after many iterations. Uniform random sampling performs slightly better due to broader spatial coverage, but it still lacks the adaptivity needed to consistently reduce error. In contrast, A-UCB* rapidly drives the optimization process toward the jammer location, with its error distribution narrowing significantly after about 30 iterations-corresponding to fewer than 100 total measurements when accounting for the initial 35 crowdsourced samples and two agent-collected samples per iteration. This outcome underscores the effectiveness of acquisition-aware path planning in prioritizing high-utility measurements: rather than relying on exhaustive exploration, the method focuses on high-value measurements, producing more reliable field estimates minimizing wasted effort. Moreover, even when constrained by a finite path-length budget, the method retains most of its efficiency. Finally, the comparison between the two datasets emphasizes the role of the environment: Boston Common, with its open-sky areas, yields lower error levels overall, while the dense and obstructed Chicago Downtown setting poses greater challenges. Despite this, A-UCB* maintains robust performance in both scenarios, highlighting its adaptability in complex urban environments. 4.3. Sensitivity of κ Varying κ and measuring BOE at iteration 30 shows a shallow Ushaped trend in both datasets, as illustrated in Fig. 3. Very small κ (e.g., 0.1) is too exploitative, yielding large median errors and wide IQRs, where the agent's initialization strongly influences performance. Very large κ (≥5) over-explores, increasing error and variability at the fixed number of iterations. The lowest, most stable errors occur for κ ∈[1, 3]. We therefore find κ = 2 a robust choice within the low-error plateau. Fig. 3. Sensitivity analysis of the exploration-exploitation parameter κ in the UCB acquisition function. The plot shows the BOE after 30 iterations for both datasets. Each point corresponds to the median BOE across 100 trials, with error bars denoting the IQR. 5. CONCLUSION We introduced an acquisition-aware Bayesian optimization framework for jammer localization that integrates probabilistic modeling with adaptive path planning. Results in realistic urban environments confirm that our method localizes interference sources with high accuracy using limited measurements, outperforming random baselines in both convergence speed and robustness. Even under mobility constraints, the framework retains efficiency, highlighting its suitability for real-world deployments. Future work will extend this approach to multi-jammer scenarios and explore alternative probabilistic surrogates beyond GP's. 6. REFERENCES [1] Y Jade Morton, Frank van Diggelen, James J Spilker Jr, Bradford W Parkinson, Sherman Lo, and Grace Gao, Position, navigation, and timing technologies in the 21st century: Integrated satellite navigation, sensor systems, and civil applications, volume 1, John Wiley & Sons, 2021. [2] Rigas Themistoklis Ioannides, Thomas Pany, and Glen Gibbons, \"Known Vulnerabilities of Global Navigation Satellite Systems, Status, and Potential Mitigation Techniques,\" Proceedings of the IEEE, vol. 104, no. 6, pp. 1174-1194, 2016. [3] Moeness G. Amin, Pau Closas, Ali Broumandan, and John L. Volakis, \"Vulnerabilities, threats, and authentication in satellite-based navigation systems [scanning the issue],\" Proceedings of the IEEE, vol. 104, no. 6, pp. 1169-1173, 2016. [4] J. Coffed, \"The Threat of GPS Jamming: The Risk to an Information Utility,\" Tech. Rep., Exelis, Inc., Feb. 2015, Online. https://rntfnd.org/wp-content/uploads/ExelisGPS-Vulnerability-Assessment-February2014.pdf. [5] Luka Strizic, Dennis M. Akos, and Sherman Lo, \"Crowdsourcing GNSS Jammer Detection and Localization,\" in Proceedings of the 2018 International Technical Meeting of The Institute of Navigation, Reston, Virginia, 01 2018, pp. 626-641. [6] Andrea Nardin, Tales Imbiriba, and Pau Closas, \"Jamming Source Localization Using Augmented Physics-Based Model,\" in ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1-5. [7] Borio D, Gioia C, Stern A, Dimc F, and Baldini G, Jammer Localization: from Crowdsourcing to Synthetic Detection, Institute of Navigation, 2016. [8] Gl ̈adje Karl Olsson, Sara Nilsson, Erik Axell, Erik G. Larsson, and Panos Papadimitratos, \"Using Mobile Phones for Participatory Detection and Localization of a GNSS Jammer,\" 2023. [9] Hangyu Bai, Tao Zhang, and Wenxian Yu, \"A Map Reconstruction Based Method for Localizing Multiple GNSS Jammers with Unknown Number,\" in 2024 43rd Chinese Control Conference (CCC), 2024, pp. 3839-3844. [10] Sriramya Bhamidipati and Grace Xingxin Gao, \"Simultaneous localization of multiple jammers and receivers using probability hypothesis density,\" in 2018 IEEE/ION Position, Location and Navigation Symposium (PLANS), 2018, pp. 940-944. [11] Benon Gattis, Ethan Pyke, Dennis Akos, Mark Crews, and Stephen Robertson, \"Stress-Testing Flagship Smartphone Models With Real-World GNSS RFI to Determine Real-Time Emitter Localization Capabilities,\" in 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), 2025, pp. 1566-1574. [12] Mariona Jaramillo-Civill, Peng Wu, Andrea Nardin, Tales Imbiriba, and Pau Closas, \"Jammer Source Localization with Federated Learning,\" in 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), 2025, pp. 362-371. [13] Marco Spanghero, Filip Geib, Ronny Panier, and Panos Papadimitratos, \"GNSS Jammer Localization and Identification With Airborne Commercial GNSS Receivers,\" IEEE Transactions on Information Forensics and Security, vol. 20, pp. 35503565, 2025. [14] Adrien Perkins, Louis Dressel, Sherman Lo, Tyler Reid, Kazuma Gunning, and Per Enge, \"Demonstration of UAV Based GPS Jammer Localization During a Live Interference Exercise,\" in Proceedings of the 29th International Technical Meeting of the Satellite Division of The (ION GNSS+), Portland, Oregon, Sept. 2016, The . 3094-3106. [15] Eric Brochu, Vlad M. Cora, and Nando de Freitas, \"A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning,\" 2010. [16] Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, The MIT Press, 11 2005. [17] Peter Auer, \"Using confidence bounds for exploitationexploration trade-offs,\" J. Mach. Learn. Res., vol. 3, no. null, pp. 397-422, Mar. 2003. [18] Peter E. Hart, Nils J. Nilsson, and Bertram Raphael, \"A Formal Basis for the Heuristic Determination of Minimum Cost Paths,\" IEEE Transactions on Systems Science and Cybernetics, vol. 4, no. 2, pp. 100-107, 1968. [19] Rina Dechter and Judea Pearl, \"Generalized best-first search strategies and the optimality of A*,\" J. ACM, vol. 32, no. 3, pp. 505-536, July 1985. [20] Zhengqing Yun and Magdy F. Iskander, \"Ray Tracing for Radio Propagation Modeling: Principles and Applications,\" IEEE Access, vol. 3, pp. 1089-1100, 2015. [21] K.R. Schaubach, N.J. Davis, and T.S. Rappaport, \"A ray tracing method for predicting path loss and delay spread in microcellular environments,\" in [1992 Proceedings] Vehicular Technology Society 42nd VTS Conference - Frontiers of Technology, 1992, pp. 932-935 vol.2."}}},{"rowIdx":192,"cells":{"id":{"kind":"string","value":"2510.14791"},"raw":{"kind":"string","value":"MNRAS 000, 1–15 (2025)\nPreprint 17 October 2025\nCompiled using MNRAS LATEX style file v3.3\nStellar population astrophysics (SPA) with the TNG. The Phosphorus\nabundance on the young side of Milky Way★\nMingjie Jian (简明杰) ,1† Xiaoting Fu (符晓婷) ,2,3 Valentina D’Orazi,4,5 Angela Bragaglia,3\nS. Bijavara Seshashayana,6,7 He Zhao (赵赫) ,8 Ziyi Guo (郭子怡) ,9,10 Karin Lind,1\nNoriyuki Matsunaga (松永典之) ,11 Antonino Nunnari,12,4 Giuseppe Bono,4 Nicoletta Sanna,13\nDonatella Romano,3 and Marina Dal Ponte5\n1Department of Astronomy, Stockholm University, AlbaNova University Center, Roslagstullsbacken 21, 114 21 Stockholm, Sweden\n2Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China\n3INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via P. Gobetti 93/3, 40129 Bologna, Italy\n4Department of Physics, University of Rome Tor Vergata, via della Ricerca Scientifica 1, 00133, Rome, Italy\n5INAF - Osservatorio Astronomico di Padova, Vicolo dell’ Osservatorio 5, 35122 Padova, Italy\n6Materials Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden\n7Nordic Optical Telescope, Rambla José Ana Fernández Pérez 7, ES-38711 Breña Baja, Spain\n8Departamento de Ciencias Fisicas, Universidad Andres Bello, Republica 220, 8320000 Santiago, Chile\n9School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China\n10Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China\n11 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\n12 INAF – Astronomic Observatory of Rome, Via Frascati 33, 00078 Monte Porzio Catone, Italy\n13 INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe present phosphorus abundance measurements for a total of 102 giant stars, including 82 stars in 24 open clusters and 20\nCepheids, based on high-resolution near-infrared spectra obtained with GIANO-B. Evolution of phosphorus abundance, despite\nits astrophysical and biological significance, remains poorly understood due to a scarcity of observational data. By combining\nprecise stellar parameters from the optical, a robust line selection and measurement method, we measure phosphorus abundances\nusing available P i lines. Our analysis confirms a declining trend in [P/Fe] with increasing [Fe/H] around solar metallicity for\nclusters and Cepheids, consistent with previous studies. We also report a [P/Fe]–age relation among open clusters older than\n1 Gyr, indicating a time-dependent enrichment pattern. Such pattern can be explained by the different stellar formation history\nof their parental gas, with more efficient stellar formation in the gas of older clusters (thus with higher phosphorus abundances).\n[P/Fe] shows a flat trend among cepheids and clusters younger than 1 Gyr (along with three Cepheids inside open clusters),\npossibly hinting at the phosphorus contribution from the previous-generation low-mass stars. Such trend suggests that the young\nclusters share a nearly common chemical history, with a mild increase in phosphorus production by low-mass stars.\nKey words: stars: abundances – stars: late-type – open clusters and associations: general – stars: variables: Cepheids\n1 INTRODUCTION\nSpectroscopic analysis of stars allows us to determine the abundances\nof various elements in their atmospheres, effectively revealing their\nchemical compositions. These elemental signatures reflect the com-\nposition of the molecular cloud from which the stars were born,\ncapturing the initial conditions of star formation. Consequently, mea-\nsuring the chemical compositions of stars across different elements\n★Based on observations made with the Italian Telescopio Nazionale Galileo\n(TNG) operated on the island of La Palma by the Fundación Galileo Galilei\nof the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio\ndel Roque de los Muchachos of the Instituto de Astrofisica de Canarias\n† E-mail: jian-mingjie@outlook.com\nprovides crucial constraints on the evolutionary history of the Milky\nWay and constitutes one of the central goals of Galactic Archaeology\n(or Galactic Paleontology; see e.g. Tolstoy 2011). For instance, ob-\nservations of 𝛼-elements (e.g., O, Mg, Si, S, Ca and Ti) in the Milky\nWay help distinguish different components of the Galactic disk (e.g.,\nthe thin and thick disk; Hayden et al. 2015; Anders et al. 2017). When\ncombined with our understanding of the nucleosynthetic origins of\nthese elements (e.g., Kobayashi et al. 2020), such measurements of-\nfer key insights into the chemical evolution of our host galaxy, solar\nsystem or even life (Fernández-García et al. 2017).\nHowever, the chemical evolution of some elements remains elusive\ndue to a lack of observational constraints. Phosphorus is one such\nelement. The observable spectral lines of phosphorus are mostly lo-\ncated in the ultraviolet and infrared wavelength regions, making it\n© 2025 The Authors\narXiv:2510.14791v1 [astro-ph.SR] 16 Oct 2025\n\n2\nM. Jian et al.\nimpossible to measure its abundance using optical spectra. Given\nthat ultraviolet observations are heavily affected by interstellar ex-\ntinction, the infrared becomes the most suitable wavelength range for\nstudying phosphorus abundance and thus provides the most useful\ninformation for Galactic archaeology. In contrast, many spectral lines\nof other elements are found in the optical, which makes it easier to\ndetermine stellar parameters using optical spectra. These parameters\ncan then be used to infer the abundances of elements whose lines are\nonly available in the infrared. Simultaneously obtained optical and\ninfrared spectra are particularly well-suited for this approach (see,\ne.g., Jian et al. 2024).\nObservational studies of phosphorus began relatively recently,\nwith the pioneering work of Caffau et al. (2011), who measured the P\nabundances of 20 F-type dwarf stars. They found that the [P/Fe] ratio\ndecreases from approximately +0.4 to 0 as metallicity increases from\n[Fe/H] = −1 to solar. This trend contrasts with that of other light\nodd-𝑍elements, such as Na and Al, and also deviates from the theo-\nretical predictions of Kobayashi et al. (2006). Subsequently, Roederer\net al. (2014) extended the observational data towards the metal-poor\nregime. Using the Hubble Space Telescope Imaging Spectrograph,\nthey measured phosphorus abundances in 14 dwarf stars with metal-\nlicities ranging from [Fe/H] = −4 to −0.1 dex. [P/Fe] remains\napproximately solar in the metallicity range [Fe/H] ∼−4 to −2 dex,\nthen increases to [P/Fe] ∼0.5 at [Fe/H] ≈−1.1, before decreas-\ning back to near-solar levels at higher metallicities. Most subsequent\nstudies have confirmed this general trend. Hawkins et al. (2016) per-\nformed the first large-scale measurement of phosphorus abundances\nusing data from the APOGEE survey, significantly expanding the\nsample around solar metallicity. Later Hayes et al. (2022) measured\nphosphorus abundances for more than 120 000 stars using APOGEE\nDR17, though only the upper-limit can be estimated for the majority\nof the stars (∼87 000) thus the completeness in metallicity is largely\naffected. Over 40 additional measurements were later contributed\nby Caffau et al. (2016) and Caffau et al. (2019), using the high-\nresolution spectrograph GIANO. Nandakumar et al. (2022) used\nthe IGRINS spectrograph to reproduce the [P/Fe] trend within the\nmetallicity range of [Fe/H] = −1.2 to +0.3 dex. In addition, Maas\net al. (2022) found that the thick disc stars with the metallicity ranges\nfrom [Fe/H] = −1 to −0.4 dex tend to exhibit higher phosphorus\nabundances. Interestingly, Weinberg et al. (2019) reported that the\n[P/Mg] ratio varies among stars with different magnesium abun-\ndances. Since Mg is almost a “pure” 𝛼element that is mainly pro-\nduced by core-collapse supernovae (CCSN), the results of Weinberg\net al. (2019) contradict the theoretical prediction that phosphorus is\npredominantly produced by CCSN.\nA number of giant stars with peculiar phosphorus abundances\nhave also been identified. Masseron et al. (2020a) discovered 15 stars\nexhibiting extremely high phosphorus enhancements ([P/Fe] ≳1.5)\nwith unusual overabundances O, Mg, Si, Al, and Ce, and further\nexamined the detailed abundance patterns of three of these stars\nin a follow-up study (Masseron et al. 2020b). This sample was later\nexpanded to 78 stars by Brauner et al. (2023). The unusual abundance\npatterns observed in these objects continue to pose a challenge to\ncurrent Galactic chemical evolution (GCE) models.\nGCE models suggest that phosphorus is primarily produced in\nmassive stars. Figure 1 presents the stellar total yield of phosphorus\n(the amount of element newly-produced plus that present in the star\nat birth, in contrast with the net yield, those newly-produced by\nstellar nucleosynthesis) after being weighted by the Kroupa stellar\ninitial mass function (IMF, Kroupa 2001). We consider three types\nof yields here: low-mass stars (from Karakas 2010), massive stars\n(from Nomoto et al. 2013) and Type Ia supernovae (SNIa, from\n0\n20\n40\n60\n80\n100\nStellar mass (M\n)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nIMF weighted total yield (M\n)\n1e\n7\niniZ = 0.0001\niniZ = 0.004\niniZ = 0.01\nFigure 1. Stellar total yield of phosphorus in three different initial metallicities\n(ini𝑍). The yield has been weighted by Kroupa IMF, and the stellar population\nhas been normalized to 1 M⊙.\nIwamoto et al. 1999). The massive stars contribute over 85% of\nphosphorus in a single stellar population in all three metallicities\nwe consider for a representative initial metallicities for Population I\nand II progenitor stars in galactic chemical evolution models. Also,\nthe yield for metal-rich stars is higher than that for metal poor stars\nin most of the mass range, since in general stellar wind increases\nas metallicity increases (Vink et al. 2001). Cescutti et al. (2012)\ncompares their phosphorus measurement on stars with [Fe/H] > −1\nwith the GCE models, and concluded that the major producer of P\nshould be CCSN, though their adopted yields need to be multiplied\nby 3 to match the [P/Fe] values, and their trend does not match the\nmetal-poor observation from Roederer et al. (2014). Recently Bekki\n& Tsujimoto (2024) suggested that the P production of oxygen-neon\nnovae (ONe novae) needs to be included to explain the solar [P/Fe]\nvalues in the range of [Fe/H] < −2.5 dex as well as the increase\nbetween −2.5 < [Fe/H] < −1.\nTo unfold the evolutionary history of phosphorus in the Milky\nWay, stellar age is a key parameter. Despite recent observational\nprogress, the age–phosphorus relation remains poorly understood.\nDetermining the ages of individual field stars in the Milky Way with\nhigh precision is challenging. Maas et al. (2019) attempted to explore\nthis relation by constructing a P–age diagram, but the stellar ages\nin their sample suffer from large uncertainties. They emphasised\nthat more accurate age determinations are essential to unveil the\nevolutionary behaviour of phosphorus in the thin disc of the Milky\nWay.\nCluster member stars, on the other hand, provide more reliable age\nestimates by fitting isochrones to the cluster colour–magnitude dia-\ngram (CMD). This offers a promising avenue for constructing a more\nprecise P–age relation. In this work, we present phosphorus abun-\ndance measurements for members of 24 open clusters as well as 20\nCepheids in the field, located in different directions across the Galaxy\nand the Galactocentric distance (𝑅gc) between 7 to 10 kpc, aiming\nto place new constraints on the chemical evolution of phosphorus in\nthe Milky Way.\nThis paper is organised as follows. Section 2 describes the data and\nthe reduction process. Section 3 outlines the method used to deter-\nmine phosphorus abundances. The results are presented in Section 4,\nfollowed by a discussion in Section 5 and conclusions in Section 6.\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n3\n2 DATA AND REDUCTION\nThe data used in this study consist of near-infrared (NIR) spectra\ncovering the wavelength range 0.9–2.45 𝜇m, obtained with a spectral\nresolution of 𝑅= 50,000 using the GIANO-B spectrograph (Oliva\net al. 2012a,b; Origlia et al. 2014). These NIR spectra were acquired\nsimultaneously with optical spectra spanning 0.383–0.690 𝜇m at a\nresolution of 𝑅= 115,000, obtained with the HARPS-N spectro-\ngraph (Cosentino et al. 2012) in GIARPS mode (Tozzi et al. 2016;\nClaudi et al. 2017) at the Telescopio Nazionale Galileo (TNG), a 3.58-\nm optical/infrared telescope located in the Roque de los Muchachos\nObservatory in La Palma, Canary Islands. This configuration effec-\ntively eliminates temporal variations between the NIR and optical\nspectral lines. The dataset is part of the Large Programme Stel-\nlar Population Astrophysics (SPA), which aims to derive detailed,\nage-resolved chemical abundances across the Milky Way disc (pro-\ngramme ID A37TAC_31, PI: L. Origlia). The programme began in\n2018 and was awarded observing time with both the HARPS-N and\nGIANO-B high-resolution echelle spectrographs at the TNG.\nOur dataset consists of 82 giant stars located in open clusters,\nsupplemented by 20 field Cepheids and two standard stars: the Sun\nand Arcturus. Table 1 provides the observational log of the target\nstars, and Figure 2 displays the CMDs of our sample. We adopted the\nages of the clusters from Cantat-Gaudin et al. (2020) in this study.\nFor the Cepheids, we derived the ages from their period using the\n𝑍= 0.02 period-age relation presented in Bono et al. (2005), with\ntheir pulsation mode (fundamental or first overtone) distinguished.\nThree of the Cepheids, DL Cas, SV Vul and X Vul, are classified as\nopen cluster members (e.g., in Hao et al. 2022). SV Vul and X Vul\nhave similar ages from their pulsation period and host clusters (with\ndifferences < 12 Myr). DL Cas, however, has a age from period-age\nrelation of 51 ± 9 Myr, while its host cluster, NGC 129 is ∼130 Myr\nold. Considering that only three of our Cepheids have ages from their\nhost clusters, we adopt their ages from period-age relation to increase\nthe number of stars with age measurement.\nAlthough the SPA project also includes many dwarf stars in these\nclusters, measuring phosphorus abundances for them is particularly\nchallenging. This is because, in dwarfs, phosphorus lines are signif-\nicantly weaker than in other types of stars (see Figure 2), while their\ntypically higher 𝑣sin𝑖values broaden the lines. This combination\nhighly reduce the P line detectability in dwarfs. In addition, dwarfs\nare intrinsically fainter, and the P lines are often smeared out by noise\nin their spectra. Thus we limit our target stars to giants in this study.\n2.1 Telluric correction and normalization\nThe GIANO-B spectra were reduced using the GOFIO data reduc-\ntion pipeline (Rainer et al. 2018). For each spectral order, the pipeline\nperforms bad pixel and cosmic ray removal, sky and dark subtrac-\ntion, and corrections for flat-field and blaze effects, followed by the\nextraction of one-dimensional spectra. We further processed the ex-\ntracted spectra using the pre-analysis pipeline giano_ct1, which\ncarries out continuum normalisation and telluric absorption correc-\ntion. Continuum normalisation is performed using the “alpha-roll”\nmethod described by Xu et al. (2019) and Cretignier et al. (2020).\nIn this method, a circle with radius 𝛼is rolled along the top of the\nspectrum, and the contact points between the circle and the flux curve\nare selected as continuum points. The value of 𝛼is adaptively deter-\nmined based on the difference between the smoothed and observed\n1 https://github.com/MingjieJian/giano_ct\nspectra, increasing in regions where absorption lines are present. Tel-\nluric correction is then applied using the TelFit package (Gullikson\net al. 2014), by fitting the model telluric spectrum to the observed\nspectrum, and the individual spectral orders are merged into a sin-\ngle, continuum-normalised spectrum. The best-fit telluric model is\nalso saved for later analysis. We note that certain wavelength re-\ngions (13530–14350 Å and 18020–19400 Å) are heavily affected by\ntelluric absorption and are therefore excluded during the correction\nprocess.\n3 MEASUREMENT OF P ABUNDANCE\n3.1 Stellar parameters\nStellar parameters (i.e., effective temperature 𝑇eff, surface gravity\nlog 𝑔, iron abundance [Fe/H] as a proxy of metallicity, and microtur-\nbulance velocity 𝑉mic) for our target stars were independently deter-\nmined by Dal Ponte et al. (2025) based on HARPS-N spectra, and\nby Bijavara-Seshashayana et al. (in prep.) using GIANO-B spectra.\nThe two sets of parameters are generally consistent, although some\ndiscrepancies in log 𝑔are present for individual stars. In this work,\nwe adopt the stellar parameters from Dal Ponte et al. (2025).\nThe spectral synthesis in the following analysis is performed using\nthe PySME code (Wehrhahn et al. 2023). Spectral synthesis is carried\nout by the SME library (Piskunov & Valenti 2017; Valenti & Piskunov\n1996), based on a one-dimensional stellar atmosphere model, under\nthe assumption of local thermodynamic equilibrium (LTE). We note\nthat although non-LTE corrections are available for 16 elements\n(Amarsi et al. 2020; Mallinson et al. 2024), there is currently no\ngrid of departure coefficients for phosphorus. PySME is capable of\nsynthesising model spectra and determining stellar parameters and\nabundances by 𝜒2 minimisation between the observed and synthetic\nspectra within user-defined masks. The current version of PySME\ntakes all input line lists into account during synthesis, which can be\ntime- and memory-consuming when dealing with spectra that span a\nwide wavelength range. Our recent developments in PySME include\na more efficient synthesis via the dynamic removal of weak spectral\nlines, measurement of stellar parameters using pre-defined masks,\nand abundance measurements for multiple elements, all performed\nin an automated manner. These improvements will serve as part of the\npipeline for future large-scale spectroscopic surveys such as 4MOST\n(de Jong et al. 2019). The first two developments will be discussed\nand tested in detail in Jian et al. (in prep.), while we focus on the\nabundance measurement aspect in this article.\nThe line list, covering the wavelength range from 9360 to\n24, 240 Å, is extracted from the VALD3 database version 3750M\n(Piskunov et al. 1995; Ryabchikova et al. 2015). We used the “Ex-\ntract All” mode with hyperfine structure included. All molecular lines\nacross the full GIANO wavelength range are included, except for TiO\nlines. The enormous number of TiO lines makes it nearly impossible\nboth to download and to include them in the synthesis. However, for\ntarget stars with 𝑇eff ⪅4500 K (14 stars in our sample), the blending\nof TiO lines may not be negligible. Thus, we included the TiO lines\nwithin ±20 Å of the P lines used for abundance measurements for\nour target giants (i.e., those in Table 2 with 𝑁giant not marked as\n‘-’). We adopt MARCS model atmospheres (Gustafsson et al. 2008)\nas input, and solar abundances are set according to Grevesse et al.\n(2007, except for phosphorus; see Section 4.1).\nThe broadening velocity, including macroturbulence and projected\nrotational velocity, 𝑉broad, is not provided in Dal Ponte et al. (2025).\nWe therefore measure 𝑉broad using the GIANO-B spectra. For each\nMNRAS 000, 1–15 (2025)\n\n4\nM. Jian et al.\nTable 1. Observation log of our target stars, with the columns of (left to right) star name, Gaia DR3 ID, observation date in reduced heliocentric Julian dates\n(RHJD = HJD −2400000), exposure time, and the mean signal-to-noise ratio per pixel. Only the first six stars from cluster giant and Cepheid sample and the\nstandard stars are listed, and the full table is available in CDS.\nStar name\nGaia DR3 ID\nRHJD\n𝑡exp (s)\n𝑆/𝑁mean\nAlessi1-2\n402506369136008832\n58471.820\n300 × 12\n212\nAlessi1-3\n402505991180022528\n58471.872\n300 × 12\n225\nAlessi1-5\n402867593065772288\n58470.864\n300 × 8\n233\nAlessi1-6\n402880684126058880\n58471.926\n300 × 10\n199\nAlessi Teutsch11-1\n2184332753719499904\n58710.863\n300 × 2\n377\nBasel11b-1\n3424056131485038592\n58513.903\n300 × 10\n196\n...\n...\n...\n...\n...\nDL Cas (NGC 129)\n428620663657823232\n58354.228\n300 × 4\n253\nSV Vul (UBC 130)\n2027951173435143680\n58425.869\n300 × 2\n545\nX Vul (UBC 129)\n2027263738133623168\n58349.853\n300 × 4\n375\nCO Aur\n3451987987438434944\n58426.248\n300 × 4\n398\nRX Aur\n200708636406382720\n58429.227\n300 × 4\n425\nRW Cam\n473043922712140928\n58354.187\n300 × 4\n388\n...\n...\n...\n...\n...\nSun (Ganymede)\n–\n58325.880\n30 × 2\n130\nArcturus\n–\n58301.894\n60 × 2\n612\nTable 2. List of P i lines used in this study, including their air wavelengths\n(𝜆air), lower excitation potentials (EP), and oscillator strengths (log 𝑔𝑓).\nColumns 𝑁giant and 𝑁Cepheid indicates the number of stars using the cor-\nresponding line for P measurement (excluding cases where the line is blended\nwith telluric features or only provides an upper limit), and the number in the\nparentheses (if presents) indicates the number of stars being removed (see the\ndescription in section 3.3). The final column (‘Sun’) marks whether the line\nwas also used in the solar abundance determination.\n𝜆air\nEP\nlog 𝑔𝑓\n𝑁giant\n𝑁Cepheid\nSun\n(Å)\n(eV)\n9609.036\n6.9356\n−1.05\n-\n0 (1)\n9676.222\n8.0785\n0.00\n-\n4\n9734.755\n6.9543\n−0.36\n-\n3\n9750.748\n6.9543\n−0.18\n12\n1\nY\n9790.194\n7.1758\n−0.69\n-\n2\n9796.828\n6.9852\n0.27\n-\n1\nY\n9903.671\n7.1758\n−0.30\n-\n7\nY\n9976.681\n6.9852\n−0.29\n-\n1 (5)\nY\n10084.277\n7.2127\n0.14\n-\n4\n10204.716\n7.2127\n−0.52\n-\n4\n10511.588\n6.9356\n−0.13\n11\n1\n10529.524\n6.9543\n0.24\n22\n1\nY\n10581.577\n6.9852\n0.45\n34\n1\nY\n10596.903\n6.9356\n−0.21\n29 (1)\n4\nY\n10681.406\n6.9543\n−0.19\n6\n5\nY\n10769.511\n6.9543\n−1.07\n-\n2\n10813.141\n6.9852\n−0.41\n0 (12)\n5\n10932.724\n8.0785\n0.31\n-\n1\n10967.373\n8.0783\n0.12\n-\n7\n11183.240\n7.2127\n0.40\n8\n-\n16254.749\n8.2255\n0.00\n-\n7\n16482.932\n7.2127\n−0.29\n42\n5\nY\n16590.045\n8.2276\n0.50\n-\n0 (6)\n16738.681\n8.2864\n0.00\n-\n0 (4)\n17112.447\n8.2504\n0.50\n0 (51)\n0 (7)\n17286.920\n8.2864\n0.00\n-\n3\n17423.670\n8.2864\n0.00\n0 (2)\n4\nstar, we first fix the other stellar parameters and select isolated Fe I\nlines based on the synthetic spectrum (the details of the line selection\nmethod are described in Section 3.2). The𝑉broad value is then fitted for\neach selected Fe I line, and the weighted mean and standard deviation\nacross all lines are adopted as the final broadening velocity and its\nassociated uncertainty.\nThe adopted stellar parameters are listed in Table 3 and plotted\nin Figure 3. The theoretical equivalent width (EW) of the P i line\nat 10529.524 Å – one of the strongest phosphorus lines in the near-\ninfrared – is also shown as a contour in the diagram. Most infrared P\nlines exhibit EW contour shapes similar to this line, with maximum\nstrengths occurring at approximately 𝑇eff ≈6000 K and low log 𝑔,\nand decreasing as𝑇eff moves away from this peak or as log 𝑔increases.\nAs a result, most of our Cepheids exhibit relatively strong P lines\n(see also Elgueta et al. 2024). In contrast, our cluster giants lie far\nfrom the EW peak region, and the EWs of the 10529.524 Å line are\ntypically around 15 mÅ. Therefore, high signal-to-noise ratio spectra\nare required to derive reliable phosphorus abundances for these stars.\n3.2 Line selection\nGiven the relatively large sample size of stars in this study, as well as\ntheir wide range of stellar parameters, it is essential to select suitable\nP i lines using the synthetic spectra. The VALD database contains\n229 P i lines within the relevant wavelength range. However, whether\na given line can be used for phosphorus abundance determination\ndepends on several factors: whether the line appears in the observed\nspectrum, the degree of blending with nearby lines, and whether\nthe line can be reliably measured given the quality of the observed\nspectra. Therefore, a careful line selection process is required prior\nto performing the abundance analysis.\nThe goal of the line selection process is to identify all P lines that\nare present in the spectra and exhibit minimal blending, based on syn-\nthetic spectra. All P i lines are first extracted from the line list. For\neach star, we define a total broadening velocity,𝑉broad, which accounts\nfor the broadening of microturbulence velocity (𝑉mic), macroturbu-\nlence velocity (𝑉mac), projected rotation (𝑣sin𝑖), and instrumental\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n5\nFigure 2. CMD of the target clusters. The grey dots represent stars with membership probabilities greater than 0.5 from Cantat-Gaudin et al. (2020), while the\nblue circles highlight our selected cluster giant stars.\nresolution (𝑅), as:\n𝑉broad =\n√︂\n𝑉2\nmic + 𝑉2mac + (𝑣sin𝑖)2 +\n\u0010 𝑐\n𝑅\n\u00112\n,\n(1)\nwith 𝑐as the speed of light. A wavelength region with a width of\n4𝑉broad/𝑐is assigned to each target line, defining a feature. If two\nfeatures overlap in wavelength, they are merged and treated as a\nsingle feature.\nSynthetic spectra based on the stellar parameters mentioned in\nSection 3.1 are then generated for each feature, and three variables\nare used to evaluate the suitability of each P line for abundance\nanalysis:\n• Feature-dominance: the ratio of the EW of the phosphorus\nline(s) to the total EW of all lines within the feature;\n• Feature-depth: the maximum depth of the feature in the nor-\nmalised flux;\n• Telluric-depth: the maximum depth of telluric absorption in\nthe corresponding wavelength region.\nWe require that the selected features satisfy the following thresh-\nolds: feature-dominance greater than 0.4, feature-depth greater\nthan 0.006, and telluric-depth less than 0.1. The adopted feature-\ndominance threshold does not require the feature to be completely\ndominated by phosphorus lines. We found that the primary contam-\nMNRAS 000, 1–15 (2025)\n\n6\nM. Jian et al.\nTable 3. Stellar parameters of the target stars. Only the first six stars from cluster giant and Cepheid sample and the standard stars are listed, and the full table is\navailable in CDS.\nStar name\n𝑇eff\nlog 𝑔\n[Fe/H]\n𝑉mic\n𝑉broad\nAlessi1-2\n4986 ± 30\n2.83 ± 0.07\n−0.001 ± 0.010\n1.297 ± 0.013\n6.7 ± 0.9\nAlessi1-3\n4996 ± 30\n2.84 ± 0.07\n−0.007 ± 0.010\n1.276 ± 0.014\n6.6 ± 0.9\nAlessi1-5\n4939 ± 31\n2.73 ± 0.08\n−0.028 ± 0.010\n1.353 ± 0.013\n6.7 ± 0.9\nAlessi1-6\n4985 ± 32\n2.87 ± 0.08\n−0.013 ± 0.010\n1.278 ± 0.014\n6.4 ± 0.8\nAlessi Teutsch 11-1\n4517 ± 35\n2.15 ± 0.11\n−0.040 ± 0.013\n1.756 ± 0.014\n7.2 ± 1.1\nBasel 11b-1\n4950 ± 35\n2.51 ± 0.09\n0.007 ± 0.010\n1.808 ± 0.010\n8.4 ± 1.1\n...\n...\n...\n...\n...\n...\nDL Cas\n5622 ± 18\n1.73 ± 0.04\n0.050 ± 0.010\n3.920 ± 0.040\n25.92 ± 0.17\nSV Vul\n5676 ± 13\n1.00 ± 0.03\n0.190 ± 0.010\n4.240 ± 0.020\n15.19 ± 0.06\nX Vul\n6294 ± 12\n1.95 ± 0.02\n0.170 ± 0.010\n4.410 ± 0.020\n18.04 ± 0.06\nCO Aur\n7076 ± 9\n2.77 ± 0.01\n0.180 ± 0.010\n2.670 ± 0.020\n13.52 ± 0.05\nRX Aur\n6466 ± 10\n1.52 ± 0.01\n−0.040 ± 0.010\n4.040 ± 0.020\n26.62 ± 0.09\nRW Cam\n5672 ± 16\n1.63 ± 0.04\n0.170 ± 0.010\n3.180 ± 0.020\n17.51 ± 0.09\n...\n...\n...\n...\n...\n...\nSun\n5772\n4.44\n0.00\n1.0\n2.8\nArcturus\n4286\n1.6\n−0.52\n1.74\n4.6\nFigure 3. The Kiel diagram of our sample stars. The underlying contours\nshow the EW (mÅ) of P i line at 10529.524 Å at solar metallicity.\ninants in P features are CN (at a wavelength range of ⪅12000 Å)\nand CO (⪆12000 Å) molecular lines. To minimize the blending of\nthese molecular lines, we altered the C abundance to fit the spectra\nof ±2 Å around the P line and fix it during the fit for the P abun-\ndance. These C abundances are presented in appendix A and the\ntables therein. The feature-depth threshold is determined based on\nthe default synthesis precision of PySME. The telluric-depth thresh-\nold of 0.1 is relatively strict. Given that P lines are intrinsically weak,\nthe profiles of P lines blended with deep telluric absorption are often\nsignificantly distorted.\nA total of 11 and 26 P i lines passed the automatic selection criteria\nfor cluster giants and Cepheids, respectively, but no P ii line passed\nour selection. Table 2 lists the line parameters and indicates which\ngroup of stars each line is used for. The phosphorus lines that meet\nthe threshold requirements for each target are subsequently included\nto derive its phosphorus abundance. Minor refinements to the line\nselection are made following individual abundance measurements for\neach line. The oscillator strength, or log 𝑔𝑓value, is a key parameter\nthat determines the strength of a spectral line in synthetic spectra.\nAccurate abundance measurements therefore rely critically on the\nprecision of the adopted log 𝑔𝑓values. While Elgueta et al. (2024)\nrecalibrated the log 𝑔𝑓values for seven of the P lines included in our\nselection, all of the selected lines already have laboratory-measured\nlog 𝑔𝑓values. To maintain consistency across all line parameters,\nwe adopt the log 𝑔𝑓values provided by the VALD database for all P\nlines used in this work.\n3.3 Abundance measurement\nFor each phosphorus line, the pixels within the full feature re-\ngion—defined as a width of 4𝑉broad/𝑐centred on the line—are used\nto fit the phosphorus abundance with PySME. Figures 4 and 5 show\nan example of the line fitting procedure, for our standard star the Sun\nand Alessi 1-2, respectively. Synthetic spectra spanning a broader\nwidth of 4𝑉broad/𝑐+ 4 Å are generated, as illustrated in the upper\npanel. Within the fitting region (vertical orange shaded band), if\nthe blue curve (representing synthetic spectra generated using the P\nline only) lies between the shaded blue area (synthetic spectra with\nall lines, with varied [P/Fe]) and the blending by other species (the\nspectra with thin solid lines) is small, the feature is considered to be\ndominated by the P line.\nThe lower panel displays the configuration and result of the actual\nfitting. To obtain a more accurate continuum level, we extend the\ninput observed spectra to a total width of 4𝑉broad/𝑐+ 10 Å, centred\non the P line. Pixels are classified into three types of masks:\n• Continuum mask: pixels that satisfy all of the following con-\nditions: (1) their depth in the synthetic spectrum is less than 0.025;\n(2) their depth in the telluric model is less than 0.1; and (3) the abso-\nlute difference between the observed and synthetic fluxes is less than\ntwice the standard deviation of the residuals;\n• Line mask: pixels within 4𝑉broad/𝑐of the P line centre;\n• Bad mask: all remaining pixels.\nThis masking strategy ensures that pixels affected by other spectral\nlines or strong telluric absorption are excluded from continuum esti-\nmation. As the continuum normalisation is generally well performed\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n7\nin Section 2, PySME applies only a constant scaling to the synthetic\nspectrum, determined by the mean flux level of the continuum pix-\nels. The phosphorus abundance is then derived by minimising the 𝜒2\nover the line mask region only.\nWe adopt the best-fit phosphorus abundance and the correspond-\ning fit_uncertainties (the statistical uncertainties of the fitting)\noutput from PySME as our final measurement and its associated un-\ncertainty. Upper limits are identified following the method similar to\nthe one described in Wang et al. (2024): if the EW of a fitted line\nat its best-fit abundance 𝐴(P)𝜆is smaller than three times the EW\nobtained at 𝐴(P)𝜆+𝜎𝐴(P)𝜆, the measurement is classified as an upper\nlimit and excluded from further analysis. When measurement uncer-\ntainties properly reflect the noise level in the spectrum, this criterion\nhelps mitigate selection bias by avoiding the preferential inclusion of\nstronger lines influenced by noise.\nWe expect the phosphorus abundances derived from all selected\nlines for a given star to be consistent within their respective uncer-\ntainties. However, we find that for a few lines, the derived abundances\nsystematically deviate from the mean value obtained from the other\nlines. This discrepancy may arise from several factors, such as inac-\ncurate log 𝑔𝑓values, improper continuum placement, or undetected\nblending features that are not captured by the synthetic spectra. These\nproblematic lines are marked with 𝑁giant or 𝑁Cepheid as 0 following\nby parentheses in Table 2 and are excluded from subsequent anal-\nysis. Most of these lines are used in only a few stars, and we defer\ndiscussion of specific cases where the affected lines are present in\nthe majority of stars to Appendix B.\nFinally, the weighted average and standard deviation of the 𝐴(P)\nfrom all available P i lines are taken as the final 𝐴(P) and its corre-\nsponding error.\n4 RESULTS\nWe present the results of phosphorus abundance measurements for\nindividual stars and clusters in this section.\n4.1 Phosphorus Abundance for the standard stars\nOur measurement of the solar spectrum from GIANO shows good\nconsistency with previous determinations, as shown in Figure 6. The\nsame line selection procedure as for other stars was applied to the\nstandard stars. A total of 13 phosphorus lines were initially selected.\nAmong them, one line is too weak to provide reliable measurements\nand is therefore treated as upper limits, while another three are sig-\nnificantly blended with telluric absorption features or not showing\na clear absorption profile. It is evident that the lines affected by\ntelluric contamination tend to deviate from the remaining measure-\nments (i.e., the 9525.741 Å and 11183.24 Å line in Figure 6), which\njustifies their exclusion from the final abundance calculation. The\nfinal phosphorus abundance, 5.46, is obtained as the weighted mean\nof the remaining reliable P lines. This value is in good agreement\nwith the solar phosphorus abundance derived using 1D LTE spectral\nsynthesis in Scott et al. (2015), 5.38. The estimated uncertainty of\n0.2 dex likely arises from a combination of factors, including lower\nresolution and signal-to-noise ratio compared to the solar spectrum\nused in Scott et al. (2015), uncertainties in the adopted log 𝑔𝑓values,\nand imperfect continuum placement.\nFor Arcturus, no phosphorus lines pass our line selection criteria.\nCompared to the Sun, Arcturus is cooler and has significantly lower\nsurface gravity (log 𝑔), which leads to weaker P lines (see Figure 3).\nIn addition, its relatively low metallicity ([Fe/H] = −0.51) further\nreduces the line strengths, rendering all available phosphorus lines\ntoo weak for reliable abundance measurement.\nOur results from the Sun and Arcturus help to define an approxi-\nmate detection boundary for phosphorus abundance measurements at\nthe resolution and signal-to-noise ratio (S/N) of our GIANO-B spec-\ntra. Specifically, a metallicity of [Fe/H] ∼−0.5 appears to be the\nlower limit for detecting P lines in cool giant stars, as demonstrated\nby the non-detection in Arcturus. For warmer giants and dwarfs, our\ntest using the solar stellar parameters suggests that only one sin-\ngle P line at 10529.524 Å remains detectable at [Fe/H] = −1.0,\nindicating a rough detection limit under such conditions. This con-\nclusion is consistent with previous high-resolution studies, particu-\nlarly those by Caffau et al. (2011), whose sample consists of dwarf\nstars with the most metal-poor star having a metallicity just above\n[Fe/H] = −1.0. To extend phosphorus measurements to lower metal-\nlicities, stars within the optimum 𝑇eff and log 𝑔range (see Fig 3)\nshould be observed at very high SNR, or observations should be\ndone in ultraviolet (see Roederer et al. 2014).\n4.2 Phosphorus Abundance for our sample stars\nFigure 7 shows an example of the final phosphorus abundance de-\ntermination for the star Alessi 1-2. Eight lines were selected for the\nmeasurement, two of which exhibit weak absorption features and are\ntherefore treated as upper limits. One additional line was excluded\nbased on the criteria described in Section 3.3. The final phosphorus\nabundance is calculated as the weighted average of the abundances\nderived from the remaining lines.\nTables 4 and 5 summarise the phosphorus abundance measure-\nments for cluster giants and Cepheids, respectively. The uncertain-\nties in the averaged P abundances for the giants are typically around\n0.1 dex, while those for the Cepheids are mostly lower, indicating\nthat P lines are generally stronger in Cepheid spectra. The NLTE\neffect on the phosphorus lines may also contribute, at least in part,\nto the uncertainties in our target stars. However, there is currently no\navailable NLTE departure coefficient for phosphorus, and thus it is\nnot yet possible to quantify or correct this effect. We then compute\nthe average phosphorus abundance for each cluster using its member\nstars, as listed in Table 6.\nThe [P/Fe]–[Fe/H] distribution of our cluster and Cepheid sample\nis broadly consistent with previous studies, as shown in Figure 8. Our\ntargets span a relatively narrow metallicity range, from [Fe/H] =\n−0.1 to +0.15. Given this limited coverage, we adopt the results\nfrom Nandakumar et al. (2022) as the primary comparison, and refer\nthe reader to their Figure 4 for a comprehensive compilation of earlier\nstudies. Both our field Cepheids and cluster stars exhibit a modest\ndecreasing trend in [P/Fe] from ∼0.3 to ∼0 over the range [Fe/H] =\n−0.1 to 0. At the metal-rich end, [P/Fe] appears to flatten near zero,\nalthough this trend is not well constrained due to the small number\nof stars in this regime. Overall, our observations fall within the same\n[P/Fe] range as reported by Nandakumar et al. (2022), supporting\nthe consistency of our results with previous measurements.\nThe [P/Fe]–age relation observed in our cluster sample shows\ngeneral consistency with previous studies, while also providing new\ninsights at the younger end. As shown in Figure 9, the phosphorus\nabundance (both in the form of absolute P abundance 𝐴(P) and the\nrelative [P/Fe]) of clusters with ages greater than 1 Gyr increases\nwith age. This trend is consistent with the findings of Maas et al.\n(2019) and Feuillet et al. (2018), although we note that the stellar ages\nfrom Maas et al. (2019) (with mean error of ∼1.8 Gyr) were derived\nfrom isochrone fitting of field stars, a method notoriously subject to\nlarger uncertainties than when it is applied to stellar clusters (with\nMNRAS 000, 1–15 (2025)\n\n8\nM. Jian et al.\nTable 4. P measurement for cluster giants, with the star name, mean P abundance from all the normal-flag lines, number of normal-flag line and the abundance\nof the first six lines. The abundances with superscript of u or t are those with upper limit or blended by telluric lines. Only the first 10 stars and first 6 lines are\nlisted, and the full table is available in CDS.\nstar name\n𝐴(P)mean\n𝑁\n𝐴(P)9750.748\n𝐴(P)10511.588\n𝐴(P)10529.524\n𝐴(P)10581.577\n𝐴(P)10596.903\n𝐴(P)11183.24\nAlessi 1-2\n5.49 ± 0.12\n5\n5.68 ± 0.24u\n5.01 ± 0.18u\n5.56 ± 0.14\n5.43 ± 0.07\n5.51 ± 0.11\n5.77 ± 0.13\nAlessi 1-3\n5.50 ± 0.14\n7\n5.80 ± 0.12\n5.56 ± 0.07\n5.20 ± 0.09\n5.52 ± 0.05\n5.55 ± 0.12\n5.56 ± 0.22\nAlessi 1-5\n5.46 ± 0.21\n5\n5.62 ± 0.16\n4.98 ± 0.18u\n5.37 ± 0.09\n-\n5.39 ± 0.08\n5.72 ± 0.09\nAlessi 1-6\n5.51 ± 0.23\n4\n5.62 ± 0.25u\n5.16 ± 0.19u\n5.28 ± 0.18\n-\n5.37 ± 0.17\n5.84 ± 0.15\nAlessi Teutsch 11-1\n5.49 ± 0.05\n2\n1.28 ± 5.77t\n5.04 ± 0.22u\n5.45 ± 0.14\n-\n5.49 ± 0.06\n-\nBasel 11b-1\n5.41 ± 0.13\n4\n4.76 ± 0.82u\n4.95 ± 0.31u\n5.45 ± 0.17\n-\n5.44 ± 0.13\n5.53 ± 0.16\nBasel 11b-2\n5.35 ± 0.19\n6\n5.92 ± 0.14\n5.36 ± 0.14\n5.27 ± 0.22u\n5.30 ± 0.04\n5.31 ± 0.10\n5.73 ± 0.16\nBasel 11b-3\n5.46 ± 0.14\n2\n-\n-\n-\n5.51 ± 0.09\n-\n-\nCOIN-Gaia 30-1\n-\n0\n−1.36 ± 37.53t\n−2.10 ± 8.58\n5.42 ± 0.28u\n-\n4.70 ± 0.41r\n5.82 ± 0.11t\nCollinder 350-1\n-\n0\n-\n-\n-\n-\n-\n-\nCollinder 350-2\n5.30 ± 0.04\n2\n-\n-\n-\n5.30 ± 0.03\n-\n-\n...\n...\n...\n...\n...\n...\n...\n...\n...\nTable 5. Similar to Table 4, but for the sun and Cepheid sample, with ages coming from the period-age relation. Only the first 11 Cepheids are listed, and the\nfull table is available in CDS.\nstar name\nAge (Gyr)\n𝐴(P)mean\n𝑁line\n𝐴(P)9734.755\n𝐴(P)9750.748\n𝐴(P)9790.194\n𝐴(P)9796.828\n𝐴(P)9903.671\nA(P)9976.681\nSV Vul\n0.016 ± 0.003\n5.88 ± 0.09\n1\n6.24 ± 0.04r\n-\n-\n-\n-\n-\nDL Cas\n0.051 ± 0.009\n5.52 ± 0.23\n4\n-\n-\n5.78 ± 0.07\n-\n5.61 ± 0.11t\n-\nCO Aur\n0.096 ± 0.009\n5.73 ± 0.17\n4\n-\n4.63 ± 0.93\n-\n-\n-\n-\nRX Aur\n0.039 ± 0.007\n5.61 ± 0.07\n4\n-\n-\n5.69 ± 0.03\n-\n-\n-\nV351 Cep\n0.080 ± 0.008\n5.97 ± 0.08\n2\n-\n−1.76 ± 6.86\n-\n-\n-\n-\nCD Cyg\n0.031 ± 0.006\n5.65 ± 0.08\n2\n-\n-\n5.70 ± 0.21t\n-\n5.75 ± 0.11\n-\nV1334 Cyg\n0.075 ± 0.008\n5.61 ± 0.13\n5\n-\n-\n5.74 ± 0.10t\n-\n-\n-\nVZ Cyg\n0.071 ± 0.013\n5.65 ± 0.25\n4\n6.19 ± 0.22t\n-\n5.44 ± 0.45t\n-\n-\n-\nX Cyg\n0.031 ± 0.006\n5.43 ± 0.07\n2\n-\n-\n5.63 ± 0.13t\n-\n5.98 ± 0.08t\n-\nW Gem\n0.051 ± 0.009\n5.39 ± 0.18\n7\n-\n-\n-\n-\n-\n-\nRR Lac\n0.059 ± 0.011\n5.54 ± 0.09\n5\n-\n-\n-\n3.11 ± 29.35t\n-\n-\n...\n...\n...\n...\n...\n...\n...\n...\n...\n...\nSun\n-\n5.46 ± 0.20\n9\n-\n-\n-\n5.50 ± 0.12\n-\n5.74 ± 0.09\nTable 6. The mean Phosphorus abundance of our target clusters, with the number of stars for the cluster listed.\nCluster\nAge (Gyr)\n[Fe/H]mean\n𝐴(P)mean\n[P/Fe]mean\n𝑁\nAlt Name\nGulliver 18\n0.04\n−0.01 ± 0.02\n5.59 ± 0.11\n0.25 ± 0.11\n1\nCollinder 416\nCollinder 463\n0.11\n−0.06 ± 0.01\n5.46 ± 0.19\n0.16 ± 0.19\n2\nUPK 219\n0.15\n0.07 ± 0.01\n5.33 ± 0.06\n−0.10 ± 0.06\n1\nTombaugh 5\n0.19\n0.02 ± 0.06\n5.31 ± 0.14\n−0.08 ± 0.15\n3\nNGC 7086\n0.19\n−0.08 ± 0.03\n5.61 ± 0.17\n0.33 ± 0.17\n2\nCollinder 437\nUBC 194\n0.23\n0.06 ± 0.01\n5.44 ± 0.12\n0.02 ± 0.13\n1\nBasel 11b\n0.23\n0.03 ± 0.03\n5.41 ± 0.05\n0.02 ± 0.06\n3\nFSR 877\nNGC 2437\n0.30\n0.01 ± 0.06\n5.45 ± 0.28\n0.08 ± 0.29\n4\nM 46, Melotte75\nUBC 169\n0.30\n0.05 ± 0.03\n5.40 ± 0.15\n−0.01 ± 0.16\n2\nNGC 2548\n0.40\n0.05 ± 0.05\n5.30 ± 0.07\n−0.11 ± 0.09\n3\nM 48, Melotte 85\nStock 2\n0.40\n−0.00 ± 0.04\n5.30 ± 0.12\n−0.06 ± 0.12\n7\nNGC 7209\n0.43\n0.04 ± 0.01\n5.63 ± 0.05\n0.22 ± 0.05\n1\nMelotte 238, Collinder 444\nCollinder 350\n0.59\n0.08 ± 0.01\n5.30 ± 0.04\n−0.15 ± 0.04\n1\nNGC 2632\n0.68\n0.13 ± 0.01\n5.62 ± 0.18\n0.13 ± 0.19\n1\nM 44, Praesepe\nNGC 752\n1.17\n0.08 ± 0.02\n5.30 ± 0.08\n−0.13 ± 0.09\n3\nMelotte 12, Theia 1214\nIC 4756\n1.29\n−0.00 ± 0.04\n5.39 ± 0.05\n0.03 ± 0.06\n6\nCollinder386, Melotte 210\nAlessi 1\n1.45\n−0.01 ± 0.01\n5.49 ± 0.01\n0.14 ± 0.02\n4\nCasado-Alessi 1\nNGC 6991\n1.55\n0.03 ± 0.06\n5.42 ± 0.15\n0.03 ± 0.16\n5\nUBC 141\n2.09\n−0.02 ± 0.01\n5.38 ± 0.23\n0.05 ± 0.23\n1\nUBC 577\n2.75\n−0.04 ± 0.04\n5.41 ± 0.12\n0.09 ± 0.13\n3\nAlessi 191\nRuprecht 171\n2.75\n−0.04 ± 0.05\n5.71 ± 0.08\n0.39 ± 0.09\n4\nNGC 2682\n4.27\n0.02 ± 0.01\n5.66 ± 0.53\n0.28 ± 0.53\n1\nM67\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n9\n10527\n10528\n10529\n10530\n10531\n10532\n0.95\n0.96\n0.97\n0.98\n0.99\n1.00\nNormalized flux\nSun, P 1 (10529.524 Å)\nSynthetic spectra with [P/Fe]±0.1\nSynthetic spectra with P 1 line only\nSynthetic spectra of other species\n10524\n10526\n10528\n10530\n10532\n10534\nWavelength (Å)\n0.80\n0.85\n0.90\n0.95\n1.00\nNormalized flux\nFitted A(P)=5.26±0.11, EWsynth, all=10.94±2.25 mÅ, normal\nbad\nline\ncont\nmask (dashed)\nSynthesized spectrum\nTelluric spectrum\nObserved spectrum\nFigure 4. Phosphorus abundance determination from the P i 10529.524 Å line in the solar spectrum. Top panel: Synthetic spectra after varying the abundances\nof P by ±0.1,dex (blue shaded profiles) together with spectra that contains only the P i transition (solid blue), and other species (gray). The orange dotted\nvertical line marks the line centre, while the orange shaded strip marks the wavelength range used in the fit. Bottom panel: Observed spectrum (blue points)\ncompared with the best–fitting synthetic spectrum (solid orange) and models corresponding to ±1𝜎uncertainty in 𝐴(P) (orange dashed). The light-grey dashed\n“stair-step” curve (right-hand 𝑦-axis) shows the pixel mask: cont (continuum windows), line (pixels included in the fit), and bad (pixels excluded). The orange\nshaded region is the same fitting window as in the top panel. Similar plots for the other analysed P lines are available in the Supporting Information.\nrelative error around 25%; Cantat-Gaudin et al. 2020). In contrast,\nthe trend at younger ages seems to be flat. We performed linear fit\nto the clusters with ages younger and older than 1 Gyr separately in\nFigure 9. The linear fit for the older clusters show a positive slope\nand significant Pearson 𝑝-values less than 0.05, in both 𝐴(P) and\n[P/Fe]. Those for the young clusters present a negative slope with\n𝑝-values larger than 0.05.\nMost clusters with their ages less than 1 Gyr have near-solar\nphosphorus abundances, while the clusters and Cepheids with even\nyounger ages show either solar or enhanced [P/Fe] values. They re-\nduce the slope of linear fitting and further decrease the 𝑝-values to\nless than 0.05. The measurement of the Cepheids fill the blank in the\nyoungest side of the trend. They extends the negative trend to the age\nof 0.015 Gyrs, and raises the fitted Phosphorus abundance to ∼5.7.\n4.3 Spatial distribution of Phosphorus\nOne of the goal of Galactic archaeology is to map the spatial distribu-\ntion of elements in our Galaxy. Since OCs represents a well-defined\nsingle stellar population, their spatial distribution allows us to inves-\ntigate the phosphorus distribution among different groups of stars in\nvarious directions across the Galactic plane. Classical Cepheids in\nour sample can also provide similar information, as they are relatively\nyoung and their period–luminosity relation enables precise distance\nmeasurements.\nFigure 10 displays the spatial distribution of our target clusters and\nCepheids in heliocentric Cartesian coordinates, overlaid with a dust\nextinction map (Dharmawardena et al. 2024) to represent the current\nstructure of the interstellar medium (ISM). We adopted Solar 𝑋and\n𝑍values of 8122 and 20.8 pc, respectively (GRAVITY Collaboration\net al. 2018; Bennett & Bovy 2019).\nDespite the clear inhomogeneous in the present-day ISM, we find\nno statistically significant correlation between the absolute phospho-\nrus abundance, A(P), and the clusters’ current locations, be it their\nGalactocentric radius or their position relative to local dust struc-\ntures. This apparent lack of correlation is not a deficiency in our data\nbut is, in fact, a key piece of evidence for the dynamic nature of\nthe Milky Way’s disk. Over timescales of tens of millions to several\nbillion years, processes such as radial migration and vertical heating\nact to redistribute stars and clusters from their birthplaces (see e.g.\nViscasillas Vázquez et al. 2023). Consequently, a cluster’s current\nlocation is often decoupled from the chemical composition of the gas\nfrom which it formed.\n5 DISCUSSION\nOur analysis of OCs ages reveals a distinct “V-shape” in the phospho-\nrus enrichment history of the Milky Way’s thin disk (see Figure 9).\nWhile previous studies have established a general trend of [P/Fe]\nMNRAS 000, 1–15 (2025)\n\n10\nM. Jian et al.\n10579\n10580\n10581\n10582\n10583\n10584\n0.92\n0.94\n0.96\n0.98\n1.00\nNormalized flux\nAlessi1-2, P 1 (10581.577 Å)\nSynthetic spectra with [P/Fe]±0.1\nSynthetic spectra with P 1 line only\nSynthetic spectra of CN 1\nSynthetic spectra of other species\n10576\n10578\n10580\n10582\n10584\n10586\nWavelength (Å)\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nNormalized flux\nFitted A(P)=5.43±0.07, EWsynth, all=37.19±2.15 mÅ, normal\nbad\nline\ncont\nmask (dashed)\nSynthesized spectrum\nTelluric spectrum\nObserved spectrum\nFigure 5. Similar as Figure 4, but for the line at 10581.577 Å of Alessi 1-2. Here several CN lines appear around the target P line, and their synthetic spectrum\nis also plotted in the upper panel. Similar plots for other lines and other stars are available in the Supporting Information.\nFigure 6. The Phosphorus abundance measurement result from all the lines for the Sun. Blue filled circles give the abundance 𝐴(P) derived for each line\n(wavelength range shown on the 𝑥-axis); their vertical blue error bars mark the 1𝜎fitting uncertainty. Deep-red plus symbols plot the line-by-line abundances\nreported by Scott et al. (2015) for those that are used in both studies, enabling a direct comparison. Red crosses indicate lines discarded from our final mean\nbecause of severe telluric blending or being manually excluded, while blue downward arrows denote lines for which only an upper limit on 𝐴(P) could be set.\nThe horizontal blue dashed line is the weighted mean abundance from all accepted (“normal-flag”) lines, with its 1𝜎dispersion shown by the light-blue band.\nFor reference, the red dashed line marks the solar value 𝐴(P) = 5.38 adopted by Scott et al. (2015). The panel title lists the mean abundance and its standard\ndeviation obtained in this work.\nwith [Fe/H] (Caffau et al. 2011, 2016; Maas et al. 2019, 2022),\nthe high-precision and wide range of ages with a similar [Fe/H]\nof our cluster sample allow us to dissect the phosphorus evolution\nin time, uncovering two distinct evolutionary trends. The “V-shape”\nin the 𝐴(P)-age plane is not a single evolutionary track, but rather\nthe superposition of two different regimes of GCE. The clusters\nyounger than ∼1 Gyr appear to follow a path consistent with local,\nrecent enrichment, while the older clusters unveil a fossil record of\nthe diverse and dynamic star formation histories from the Galaxy’s\npast. We note that similar but milder [Mg/Fe]–age and [Mg/H]–age\ntrend can also be found using the Mg abundances from Dal Ponte\net al. (2025) where only open clusters are analysed. A systematic\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n11\n9750.156-\n9751.340Å\n10510.949-\n10512.227Å\n10528.884-\n10530.164Å\n10580.934-\n10582.220Å\n10596.259-\n10597.547Å\n11182.560-\n11183.920Å\n16481.930-\n16483.934Å\n17111.407-\n17113.487Å\n5.0\n5.2\n5.4\n5.6\n5.8\n6.0\n6.2\nA(P)\nAlessi1-2, A(P)=5.49±0.12\nP 1 line\nScott+2015 adopted value: 5.38\nFitted value\nFitted std\nFigure 7. Similar as Figure 6, but for Alessi 2-1. Similar figures for other stars are available in the Supporting Information.\nFigure 8. [P/Fe] versus [Fe/H] for OC giants in our sample (left panel, the average [P/Fe] for each cluster) and Cepheid stars (right panel). For comparison, the\n[P/Fe] measurement of the giants from Nandakumar et al. (2022) is also plotted.\ncomparison of age–abundance trends for other elements with similar\nnucleosynthetic origins would be highly informative; however, such\nan analysis is beyond the scope of the present study and is deferred\nto future work.\n5.1 Old OCs as a fossil record of diverse star formation\nhistories and P enrichment\nFor OCs older than ∼1 Gyr, we observe that the absolute phospho-\nrus abundance 𝐴(P) decreases as stellar age decreases from ∼4 Gyr\nto ∼1 Gyr. This trend is contrary to the predictions of a simple,\nsingle-zone GCE model where metal abundance should monoton-\nically increase over the evolution time. The explanation lies in the\nfact that the Milky Way is not a single, uniform entity. The solar\nneighbourhood today contains a mix of stellar populations born in\ndifferent locations and at different times, each with a unique star\nformation history. This is a key feature of multi-zone GCE.\nThe clusters in our sample, despite having a similar near-Solar\nmetallicity (probed by [Fe/H]), reached this metallicity at vastly dif-\nferent cosmic epochs (see Table 6). For a cluster like M67 which\nhas achieved the solar metallicity by about 4 Gyr ago, its parent gas\ncloud must have undergone a very rapid and intense period of en-\nrichment. In contrast, for another cluster in the old OC group, NGC\n752, its parent gas cloud reached a similar metallicity much more\ngradually, about 1 Gyr ago. This difference in enrichment speed im-\nplies a fundamentally different Star Formation History. The rapid\nenrichment required for the older clusters necessitates either a sig-\nnificantly higher Star Formation Rate or a more “top-heavy” IMF\nthat produces more massive stars per generation. In either scenario,\nthe integrated contribution from massive stars—the primary produc-\ners of phosphorus—is far greater (see Figure 1). Therefore, it is an\ninevitable consequence that the 4 Gyr old M67 formed from gas that\nwas more abundant in absolute phosphorus 𝐴(P) than the gas that\nformed the 1 Gyr-old NGC 752. The declining 𝐴(P) trend for clus-\nters older than ∼1 Gyr is thus a direct manifestation of observing a\nsequence of clusters born from environments with progressively less\nintense star formation histories.\n5.2 Young OCs and Cepheids as evidence for local, quiescent P\nEnrichments\nOpen clusters younger than ∼1 Gyr and Cepheids in Figure 9 exhibit a\nflat or slightly increasing 𝐴(P) (or [P/Fe]) with decreasing age. This\nbehaviour aligns well with the expectations for a single-zone GCE\nMNRAS 000, 1–15 (2025)\n\n12\nM. Jian et al.\nFigure 9. Cluster or stellar age versus phosphorus abundance for our targets.\nThe A(P) and [P/Fe] measurement from Maas et al. (2019) are plotted in gray\nas a reference. The blue lines and transparent strip represent separate linear\nfits and its 95% confidential interval to the cluster data, applied independently\nto those younger and older than 1 Gyr, while the orange ones are the linear fit\nto the cluster and Cepheid sample in the young side. The [P/Fe]–age trend\nfrom Feuillet et al. (2018) is plotted as a reference.\nmodel. These clusters and stars are young and located in the Solar\nneighbourhood, meaning they were likely born locally and share a\nnearly common, recent chemical history. They represent the current,\nmore quiescent phase of evolution in our part of the Galaxy. In this\nlocal environment, we see a slow enrichment of phosphorus over the\nlast ∼1 Gyr.\nSuch gentle phosphorus enrichment may be explained by the con-\ntribution from low-mass stars. Figure 11 shows the evolution of phos-\nphorus in single stellar population at two different initial metallicity\n(ini𝑍) values, following the same set-up as the yields in Figure 1.\nOverall, the main phosphorus production at both lower and higher\nmetallicities originates from massive stars, which is consistent with\nthe conclusions of previous studies (e.g. Cescutti et al. 2012). The\ncontribution from lower mass stars, i.e. AGB stars, reached a level\nof 10−8 𝑀⊙in the later evolution time at ini𝑍= 0.0001. How-\never, the contribution of AGB stars to the phosphorus production at\nini𝑍= 0.01 is an order of magnitude higher than in the metal-poor\ncase, making it more significant at later times in the evolution of the\nsingle stellar population. This increased phosphorus production by\nAGB stars is reflected in the peak of the IMF-weighted total yield\nat the lowest stellar masses, as seen for ini𝑍= 0.004 and 0.01 in\nFigure 1. These slight sub-Solar metallicities are also proximally\ncorresponding to the previous generation of the OCs and Cepheids\nin our study. In the relative absence of vigorous massive star for-\nmation in the immediate Solar vicinity, the enrichment from these\nlower-mass stars can become a noticeable contributor to the ongo-\ning chemical evolution, explaining the slight upward trend in 𝐴(P)\ntowards the present day.\nWe note that this interpretation is sensitive to the uncertainties in\nthe nucleosynthetic yields adopted here, highlighting the need for\nmore precise yield determinations. Also the contribution of ONe\nnova, recently discussed by Bekki & Tsujimoto (2024), is not con-\nsidered here, but it may contribute to some extent to phosphorus\nproduction.\n6 SUMMARY\nIn this work, we present a detailed abundance analysis of phosphorus\nfor 82 giant stars in 24 open clusters, together with 20 Cepheids (17\nin the Galactic field and 3 in open clusters). The analysis is based\non high-resolution near-infrared spectra obtained with the GIANO-\nB spectrograph. To achieve this, we developed and implemented a\nrobust line selection and fitting procedure, which carefully accounts\nfor blending, telluric contamination, and line-strength variations, en-\nsuring reliable abundance measurements.\nOur key findings include:\n• Our results confirm the previously observed modest decreas-\ning trend of [P/Fe] with increasing [Fe/H] for stars around solar\nmetallicity.\n• Using the high precision age of open clusters, we uncover a\ndistinct “V-shape” in the phosphorus-age relation, revealing two\nseparate chemical evolution pathways and mixtures for the Milky\nWay’s thin disk:\n– For clusters older than ∼1 Gyr in our sample, we find that\nphosphorus abundance increases with stellar age. We interpret\nthis trend as a fossil record of diverse star formation histories and\nphosphorus enrichments. With a similar metallicity, older clusters\nformed from a gas that was more rapidly enriched, a consequence\nof more intense star formation in the earlier epochs of the Galactic\ndisk.\n– For clusters younger than ∼1 Gyr and Cepheids in our sample,\nthe [P/Fe] and 𝐴(P) trend with age is nearly flat with a small\nnegative slope. This points to a more quiescent, local enrichment\nhistory, where the gentle increase in phosphorus of the younger\nclusters may be attributable to the contribution from low-mass\nstars at solar metallicities.\n• We find no correlation between the 𝐴(P) abundance and the\ncurrent spatial location of open clusters and cepheids.\nFuture observations of more young open cluster giant stars or\nCepheids may further clarify the [P/Fe]–age trend in the very young\npopulation of our Milky Way.\nACKNOWLEDGEMENTS\nA.B., M.J., and V.D. acknowledge funding from INAF Mini-Grant\n2022 (High resolution spectroscopy of open clusters). X.F. acknowl-\nedges the support of the National Natural Science Foundation of\nChina (NSFC) No. 12203100. G.B., G.F. and A.N. thank the sup-\nport from Project PRIN MUR 2022 (code 2022ARWP9C) ‘Early\nFormation and Evolution of Bulge and HalO (EFEBHO)’ (PI: M.\nMarconi), funded by the European Union—Next Generation EU,\nand from the Large grant INAF 2023 MOVIE (PI: M. Marconi).\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n13\n−0.5\n0.0\n0.5\nZ [kpc]\n−2.0 −1.5 −1.0 −0.5\n0.0\n0.5\n1.0\n1.5\n2.0\nX [kpc]\n−1.0\n−0.5\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nY [kpc]\n⊙\n−0.5\n0.0\n0.5\nZ [kpc]\n5.2\n5.3\n5.4\n5.5\n5.6\n5.7\n5.8\nA(P)\nFigure 10. Space distribution of the target clusters (dots) and Cepheids (stars), colour-coded by their phosphorus abundance. The overlaid extinction maps for\neach plane are derived from Dharmawardena et al. (2024). A maximum 𝐴0 of 0.8 mag was set (represented by darkest black), with a contour (black dashed line)\nat 0.2 mag for the XY plane, while a maximum 𝐴0 of 3 mag was applied for the XZ and YZ planes. The coloured curves represent the Galactic log-periodic\nspiral arms, defined by the parameters from (Reid et al. 2019): the Carina–Sagittarius arm in purple, the Local arm in black, and the Perseus arm in green.\nAdditionally, the spur between the Local and Sagittarius–Carina arms is indicated by the blue curve.\nPart of the research activities described in this paper were carried\nout with the contribution of NextGenerationEU funds within the Na-\ntional Recovery and Resilience Plan (PNRR), Mission 4–Education\nand Research, Component 2–From Research to Business (M4C2),\nInvestment Line 3.1–Strengthening and creation of Research Infras-\ntructures, Project IR0000034— “STILES–Strengthening the Italian\nLeadership in ELT and SKA”, CUP C33C22000640006. This work\nhas made use of the VALD database, operated at Uppsala University,\nthe Institute of Astronomy RAS in Moscow, and the University of\nVienna.\nDATA AVAILABILITY\nFull version of Figures 4, 5 and 7 are available in the Supporting\nInformation, and Tables 1, 3–5, A1 and A2 are available at the CDS\nvia anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5),\nor via https://cdsarc.cds.unistra.fr. The data underlying\nthis article will be shared on reasonable request to the corresponding\nauthor.\nREFERENCES\nAmarsi A. M., et al., 2020, A&A, 642, A62\nAnders F., et al., 2017, A&A, 600, A70\nBekki K., Tsujimoto T., 2024, ApJ, 967, L1\nBennett M., Bovy J., 2019, MNRAS, 482, 1417\nBono G., Marconi M., Cassisi S., Caputo F., Gieren W., Pietrzynski G., 2005,\nApJ, 621, 966\nBrauner M., Masseron T., García-Hernández D. A., Pignatari M., Womack\nK. A., Lugaro M., Hayes C. 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S., et al., 2019, The Messenger, 175, 3\nMNRAS 000, 1–15 (2025)\n\nSPA P abundance\n15\nAPPENDIX A: CARBON ABUNDANCES FROM\nBLENDING MOLECULAR FEATURES\nThis section presents Table A1 and A2 of C abundance used to fit\nthe molecular features around the phosphorus lines, for cluster giants\nand Cepheids respectively.\nAPPENDIX B: THE P LINES WITH BIASED ABUNDANCE\nAs described in Section 3.3 and summarised in Table 2, four lines\nselected for cluster giants and four lines for Cepheids were excluded\nfrom the analysis due to systematic biases in their derived phosphorus\nabundances. Among these, two lines for the giants and one for the\nCepheids were used in more than five stars. Here, we discuss these\nlines individually.\nThe P i line at 10813.141 Å is intrinsically very weak, with a typ-\nical line depth of only ∼0.01 in most stars. Its abundance sensitivity\nis highly dependent on accurate continuum placement. However, the\ncontinuum in the fitting region is often depressed below unity, and\nsince PySME applies only a constant scaling over a wide region dur-\ning abundance fitting, local misplacement of the continuum across a\nnarrow wavelength range cannot be corrected. As a result, the phos-\nphorus abundances derived from this line tend to be overestimated.\nAlso, the situation for P i line at 17423.670 Å is similar but with\nunderestimated abundance, thus it is also being removed.\nThe line at 17112.447 Å consistently yields phosphorus abun-\ndances that are approximately 0.3 dex higher than the average derived\nfrom other lines. Inspection of the line fitting reveals the presence\nof an unidentified absorption feature on the red side of the P line.\nThis feature is absent from the synthetic spectra (otherwise it would\nhave been accounted for), but it appears consistently in all observed\nspectra with a similar relative position, suggesting it originates from\nthe stellar spectrum. Due to the severity of this blending, the fitting\nroutine attempts to reproduce the composite feature by increasing the\nphosphorus abundance, leading to a systematic overestimation.\nSix P i lines were excluded from the Cepheid sample. Their be-\nhavior is similar to that of the 10813.141 Å line, for which accurate\ncontinuum placement is challenging in this wavelength region, com-\npromising the reliability of the abundance fitting. As a result, these\nlines were excluded from the analysis.\nThis paper has been typeset from a TEX/LATEX file prepared by the author.\nMNRAS 000, 1–15 (2025)\n\n16\nM. Jian et al.\nTable A1. Carbon abundance used to fit molecular features around the phosphorus lines for cluster giants. Only the first 10 stars and first 6 lines are listed, and\nthe full table is available in CDS.\nstar name\n𝐴(C)9750.748\n𝐴(C)10511.588\n𝐴(C)10529.524\n𝐴(C)10581.577\n𝐴(C)10596.903\n𝐴(C)11183.24\nAlessi 1-2\n8.30 ± 0.05\n8.14 ± 0.38\n8.56 ± 0.10\n8.33 ± 0.24\n8.06 ± 0.22\n8.31 ± 0.08\nAlessi 1-3\n8.27 ± 0.06\n8.25 ± 0.28\n8.64 ± 0.09\n8.40 ± 0.22\n8.42 ± 0.08\n8.25 ± 0.08\nAlessi 1-5\n8.38 ± 0.05\n8.04 ± 0.37\n8.61 ± 0.08\n-\n8.40 ± 0.06\n8.30 ± 0.07\nAlessi 1-6\n8.26 ± 0.07\n7.68 ± 0.78\n8.71 ± 0.08\n-\n7.63 ± 0.16\n8.20 ± 0.09\nAlessi Teutsch 11-1\n8.25 ± 0.05\n8.35 ± 0.11\n8.52 ± 0.05\n-\n8.32 ± 0.03\n-\nBasel 11b-1\n8.35 ± 0.07\n8.47 ± 0.14\n8.79 ± 0.07\n-\n8.15 ± 0.25\n8.33 ± 0.09\nBasel 11b-2\n8.42 ± 0.06\n8.37 ± 0.19\n8.79 ± 0.07\n8.67 ± 0.12\n8.51 ± 0.08\n8.38 ± 0.08\nBasel 11b-3\n-\n-\n-\n8.54 ± 0.09\n-\n-\nCOIN-Gaia 30-1\n7.87 ± 0.17\n8.12 ± 0.29\n8.52 ± 0.09\n-\n8.38 ± 0.07\n8.45 ± 0.05\nCollinder 350-1\n-\n-\n-\n-\n-\n-\nCollinder 350-2\n-\n-\n-\n8.55 ± 0.14\n-\n-\n...\n...\n...\n...\n...\n...\n...\nTable A2. Carbon abundance used by fitting the molecular features around the phosphorus lines for Cepheids. Only the first 10 stars and first 6 lines are listed,\nand the full table is available in CDS.\nstar name\n𝐴(C)9493.571\n𝐴(C)9609.036\n𝐴(C)10932.724\n𝐴(C)10967.373\n𝐴(C)11186.753\n𝐴(C)15711.522\nSV Vul\n8.08 ± 0.18\n8.64 ± 0.04\n8.52 ± 0.11\n8.63 ± 0.02\n8.41 ± 0.10\n10.27 ± 0.09\nDL Cas\n-\n-\n-\n-\n-\n-\nCO Aur\n6.35 ± 2.36\n-\n-\n8.35 ± 3.05\n-\n-\nRX Aur\n-\n-\n-\n-\n-\n-\nV 351 Cep\n-\n-\n-\n8.33 ± 2.73\n10.33 ± 0.31\n-\nCD Cyg\n-\n-\n8.25 ± 0.02\n-\n-\n-\nV 1334 Cyg\n6.09 ± 5.05\n-\n-\n-\n-\n-\nVZ Cyg\n8.07 ± 20.49\n10.06 ± 0.61\n-\n8.07 ± 4.22\n-\n-\nX Cyg\n-\n-\n-\n-\n-\n-\nW Gem\n-\n-\n-\n-\n-\n-\nRR Lac\n-\n-\n-\n-\n-\n-\n...\n...\n...\n...\n...\n...\n...\nMNRAS 000, 1–15 (2025)\n"},"cleaned":{"kind":"string","value":"MNRAS 000, 1-15 (2025) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.3 Stellar population astrophysics (SPA) with the TNG. The Phosphorus abundance on the young side of Milky Way★ Mingjie Jian (简明杰) ,1† Xiaoting Fu (符晓婷) ,2,3 Valentina D'Orazi,4,5 Angela Bragaglia,3 S. Bijavara Seshashayana,6,7 He Zhao (赵赫) ,8 Ziyi Guo (郭子怡) ,9,10 Karin Lind,1 Noriyuki Matsunaga (松永典之) ,11 Antonino Nunnari,12,4 Giuseppe Bono,4 Nicoletta Sanna,13 Donatella Romano,3 and Marina Dal Ponte5 1 21, 114 21 Stockholm, Sweden 2Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China 3INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via P. Gobetti 93/3, 40129 Bologna, Italy 4 1, 00133, Rome, Italy 5INAF - Osservatorio Astronomico di Padova, Vicolo dell' Osservatorio 5, 35122 Padova, Italy 6Materials Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden 7Nordic Optical Telescope, Rambla José Ana Fernández Pérez 7, ES-38711 Breña Baja, Spain 8Departamento de Ciencias Fisicas, Universidad Andres Bello, Republica 220, 8320000 Santiago, Chile 9 210093, China 10Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China 11 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 12 INAF - Astronomic Observatory of Rome, Via Frascati 33, 00078 Monte Porzio Catone, Italy 13 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy Accepted XXX. Received YYY; in original form ZZZ ABSTRACT We present phosphorus abundance measurements for a total of 102 giant stars, including 82 stars in 24 open clusters and 20 Cepheids, based on high-resolution near-infrared spectra obtained with GIANO-B. Evolution of phosphorus abundance, despite its astrophysical and biological significance, remains poorly understood due to a scarcity of observational data. By combining precise stellar parameters from the optical, a robust line selection and measurement method, we measure phosphorus abundances using available P i lines. Our analysis confirms a declining trend in [P/Fe] with increasing [Fe/H] around solar metallicity for clusters and Cepheids, consistent with previous studies. We also report a [P/Fe]-age relation among open clusters older than 1 Gyr, indicating a time-dependent enrichment pattern. Such pattern can be explained by the different stellar formation history of their parental gas, with more efficient stellar formation in the gas of older clusters (thus with higher phosphorus abundances). [P/Fe] shows a flat trend among cepheids and clusters younger than 1 Gyr (along with three Cepheids inside open clusters), possibly hinting at the phosphorus contribution from the previous-generation low-mass stars. Such trend suggests that the young clusters share a nearly common chemical history, with a mild increase in phosphorus production by low-mass stars. Key words: stars: abundances - stars: late-type - open clusters and associations: general - stars: variables: Cepheids 1 INTRODUCTION Spectroscopic analysis of stars allows us to determine the abundances of various elements in their atmospheres, effectively revealing their chemical compositions. These elemental signatures reflect the composition of the molecular cloud from which the stars were born, capturing the initial conditions of star formation. Consequently, measuring the chemical compositions of stars across different elements ★Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias † E-mail: provides crucial constraints on the evolutionary history of the Milky Way and constitutes one of the central goals of Galactic Archaeology (or Galactic Paleontology; see e.g. Tolstoy 2011). For instance, observations of α-elements (e.g., O, Mg, Si, S, Ca and Ti) in the Milky Way help distinguish different components of the Galactic disk (e.g., the thin and thick disk; Hayden et al. 2015; Anders et al. 2017). When combined with our understanding of the nucleosynthetic origins of these elements (e.g., Kobayashi et al. 2020), such measurements offer key insights into the chemical evolution of our host galaxy, solar system or even life (Fernández-García et al. 2017). However, the chemical evolution of some elements remains elusive due to a lack of observational constraints. Phosphorus is one such element. The observable spectral lines of phosphorus are mostly located in the ultraviolet and infrared wavelength regions, making it The Authors 16 Oct 2025 2 M. Jian et al. impossible to measure its abundance using optical spectra. Given that ultraviolet observations are heavily affected by interstellar extinction, the infrared becomes the most suitable wavelength range for studying phosphorus abundance and thus provides the most useful information for Galactic archaeology. In contrast, many spectral lines of other elements are found in the optical, which makes it easier to determine stellar parameters using optical spectra. These parameters can then be used to infer the abundances of elements whose lines are only available in the infrared. Simultaneously obtained optical and infrared spectra are particularly well-suited for this approach (see, e.g., Jian et al. 2024). Observational studies of phosphorus began relatively recently, with the pioneering work of Caffau et al. (2011), who measured the P abundances of 20 F-type dwarf stars. They found that the [P/Fe] ratio decreases from approximately +0.4 to 0 as metallicity increases from [Fe/H] = -1 to solar. This trend contrasts with that of other light odd-Zelements, such as Na and Al, and also deviates from the theoretical predictions of Kobayashi et al. (2006). Subsequently, Roederer et al. (2014) extended the observational data towards the metal-poor regime. Using the Hubble Space Telescope Imaging Spectrograph, they measured phosphorus abundances in 14 dwarf stars with metallicities ranging from [Fe/H] = -4 to -0.1 dex. [P/Fe] remains approximately solar in the metallicity range [Fe/H] ∼-4 to -2 dex, then increases to [P/Fe] ∼0.5 at [Fe/H] ≈-1.1, before decreasing back to near-solar levels at higher metallicities. Most subsequent studies have confirmed this general trend. Hawkins et al. (2016) performed the first large-scale measurement of phosphorus abundances using data from the APOGEE survey, significantly expanding the sample around solar metallicity. Later Hayes et al. (2022) measured phosphorus abundances for more than 120 000 stars using APOGEE DR17, though only the upper-limit can be estimated for the majority of the stars (∼87 000) thus the completeness in metallicity is largely affected. Over 40 additional measurements were later contributed by Caffau et al. (2016) and Caffau et al. (2019), using the highresolution spectrograph GIANO. Nandakumar et al. (2022) used the IGRINS spectrograph to reproduce the [P/Fe] trend within the metallicity range of [Fe/H] = -1.2 to +0.3 dex. In addition, Maas et al. (2022) found that the thick disc stars with the metallicity ranges from [Fe/H] = -1 to -0.4 dex tend to exhibit higher phosphorus abundances. Interestingly, Weinberg et al. (2019) reported that the [P/Mg] ratio varies among stars with different magnesium abundances. Since Mg is almost a \"pure\" αelement that is mainly produced by core-collapse supernovae (CCSN), the results of Weinberg et al. (2019) contradict the theoretical prediction that phosphorus is predominantly produced by CCSN. A number of giant stars with peculiar phosphorus abundances have also been identified. Masseron et al. (2020a) discovered 15 stars exhibiting extremely high phosphorus enhancements ([P/Fe] ≳1.5) with unusual overabundances O, Mg, Si, Al, and Ce, and further examined the detailed abundance patterns of three of these stars in a follow-up study (Masseron et al. 2020b). This sample was later expanded to 78 stars by Brauner et al. (2023). The unusual abundance patterns observed in these objects continue to pose a challenge to current Galactic chemical evolution (GCE) models. GCE models suggest that phosphorus is primarily produced in massive stars. Figure 1 presents the stellar total yield of phosphorus (the amount of element newly-produced plus that present in the star at birth, in contrast with the net yield, those newly-produced by stellar nucleosynthesis) after being weighted by the Kroupa stellar initial mass function (IMF, Kroupa 2001). We consider three types of yields here: low-mass stars (from Karakas 2010), massive stars (from Nomoto et al. 2013) and Type Ia supernovae (SNIa, from 0 20 40 60 80 100 Stellar mass (M ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 IMF weighted total yield (M ) 1e 7 iniZ = 0.0001 iniZ = 0.004 iniZ = 0.01 Figure 1. Stellar total yield of phosphorus in three different initial metallicities (iniZ). The yield has been weighted by Kroupa IMF, and the stellar population has been normalized to 1 M⊙. Iwamoto et al. 1999). The massive stars contribute over 85% of phosphorus in a single stellar population in all three metallicities we consider for a representative initial metallicities for Population I and II progenitor stars in galactic chemical evolution models. Also, the yield for metal-rich stars is higher than that for metal poor stars in most of the mass range, since in general stellar wind increases as metallicity increases (Vink et al. 2001). Cescutti et al. (2012) compares their phosphorus measurement on stars with [Fe/H] > -1 with the GCE models, and concluded that the major producer of P should be CCSN, though their adopted yields need to be multiplied by 3 to match the [P/Fe] values, and their trend does not match the metal-poor observation from Roederer et al. (2014). Recently Bekki & Tsujimoto (2024) suggested that the P production of oxygen-neon novae (ONe novae) needs to be included to explain the solar [P/Fe] values in the range of [Fe/H] < -2.5 dex as well as the increase between -2.5 < [Fe/H] < -1. To unfold the evolutionary history of phosphorus in the Milky Way, stellar age is a key parameter. Despite recent observational progress, the age-phosphorus relation remains poorly understood. Determining the ages of individual field stars in the Milky Way with high precision is challenging. Maas et al. (2019) attempted to explore this relation by constructing a P-age diagram, but the stellar ages in their sample suffer from large uncertainties. They emphasised that more accurate age determinations are essential to unveil the evolutionary behaviour of phosphorus in the thin disc of the Milky Way. Cluster member stars, on the other hand, provide more reliable age estimates by fitting isochrones to the cluster colour-magnitude diagram (CMD). This offers a promising avenue for constructing a more precise P-age relation. In this work, we present phosphorus abundance measurements for members of 24 open clusters as well as 20 Cepheids in the field, located in different directions across the Galaxy and the Galactocentric distance (Rgc) between 7 to 10 kpc, aiming to place new constraints on the chemical evolution of phosphorus in the Milky Way. This paper is organised as follows. Section 2 describes the data and the reduction process. Section 3 outlines the method used to determine phosphorus abundances. The results are presented in Section 4, followed by a discussion in Section 5 and conclusions in Section 6. MNRAS 000, 1-15 (2025) SPA P abundance 3 2 DATA AND REDUCTION The data used in this study consist of near-infrared (NIR) spectra covering the wavelength range 0.9-2.45 μm, obtained with a spectral resolution of R= 50,000 using the GIANO-B spectrograph (Oliva et al. 2012a,b; Origlia et al. 2014). These NIR spectra were acquired simultaneously with optical spectra spanning 0.383-0.690 μm at a resolution of R= 115,000, obtained with the HARPS-N spectrograph (Cosentino et al. 2012) in GIARPS mode (Tozzi et al. 2016; Claudi et al. 2017) at the Telescopio Nazionale Galileo (TNG), a 3.58m optical/infrared telescope located in the Roque de los Muchachos Observatory in La Palma, Canary Islands. This configuration effectively eliminates temporal variations between the NIR and optical spectral lines. The dataset is part of the Large Programme Stellar Population Astrophysics (SPA), which aims to derive detailed, age-resolved chemical abundances across the Milky Way disc (programme ID A37TAC_31, PI: L. Origlia). The programme began in 2018 and was awarded observing time with both the HARPS-N and GIANO-B high-resolution echelle spectrographs at the TNG. Our dataset consists of 82 giant stars located in open clusters, supplemented by 20 field Cepheids and two standard stars: the Sun and Arcturus. Table 1 provides the observational log of the target stars, and Figure 2 displays the CMDs of our sample. We adopted the ages of the clusters from Cantat-Gaudin et al. (2020) in this study. For the Cepheids, we derived the ages from their period using the Z= 0.02 period-age relation presented in Bono et al. (2005), with their pulsation mode (fundamental or first overtone) distinguished. Three of the Cepheids, DL Cas, SV Vul and X Vul, are classified as open cluster members (e.g., in Hao et al. 2022). SV Vul and X Vul have similar ages from their pulsation period and host clusters (with differences < 12 Myr). DL Cas, however, has a age from period-age relation of 51 ± 9 Myr, while its host cluster, NGC 129 is ∼130 Myr old. Considering that only three of our Cepheids have ages from their host clusters, we adopt their ages from period-age relation to increase the number of stars with age measurement. Although the SPA project also includes many dwarf stars in these clusters, measuring phosphorus abundances for them is particularly challenging. This is because, in dwarfs, phosphorus lines are significantly weaker than in other types of stars (see Figure 2), while their typically higher vsinivalues broaden the lines. This combination highly reduce the P line detectability in dwarfs. In addition, dwarfs are intrinsically fainter, and the P lines are often smeared out by noise in their spectra. Thus we limit our target stars to giants in this study. 2.1 Telluric correction and normalization The GIANO-B spectra were reduced using the GOFIO data reduction pipeline (Rainer et al. 2018). For each spectral order, the pipeline performs bad pixel and cosmic ray removal, sky and dark subtraction, and corrections for flat-field and blaze effects, followed by the extraction of one-dimensional spectra. We further processed the extracted spectra using the pre-analysis pipeline giano_ct1, which carries out continuum normalisation and telluric absorption correction. Continuum normalisation is performed using the \"alpha-roll\" method described by Xu et al. (2019) and Cretignier et al. (2020). In this method, a circle with radius αis rolled along the top of the spectrum, and the contact points between the circle and the flux curve are selected as continuum points. The value of αis adaptively determined based on the difference between the smoothed and observed 1 https://github.com/MingjieJian/giano_ct spectra, increasing in regions where absorption lines are present. Telluric correction is then applied using the TelFit package (Gullikson et al. 2014), by fitting the model telluric spectrum to the observed spectrum, and the individual spectral orders are merged into a single, continuum-normalised spectrum. The best-fit telluric model is also saved for later analysis. We note that certain wavelength regions (13530-14350 Å and 18020-19400 Å) are heavily affected by telluric absorption and are therefore excluded during the correction process. 3 MEASUREMENT OF P ABUNDANCE 3.1 Stellar parameters Stellar parameters (i.e., effective temperature Teff, surface gravity log g, iron abundance [Fe/H] as a proxy of metallicity, and microturbulance velocity Vmic) for our target stars were independently determined by Dal Ponte et al. (2025) based on HARPS-N spectra, and by Bijavara-Seshashayana et al. (in prep.) using GIANO-B spectra. The two sets of parameters are generally consistent, although some discrepancies in log gare present for individual stars. In this work, we adopt the stellar parameters from Dal Ponte et al. (2025). The spectral synthesis in the following analysis is performed using the PySME code (Wehrhahn et al. 2023). Spectral synthesis is carried out by the SME library (Piskunov & Valenti 2017; Valenti & Piskunov 1996), based on a one-dimensional stellar atmosphere model, under the assumption of local thermodynamic equilibrium (LTE). We note that although non-LTE corrections are available for 16 elements (Amarsi et al. 2020; Mallinson et al. 2024), there is currently no grid of departure coefficients for phosphorus. PySME is capable of synthesising model spectra and determining stellar parameters and abundances by χ2 minimisation between the observed and synthetic spectra within user-defined masks. The current version of PySME takes all input line lists into account during synthesis, which can be time- and memory-consuming when dealing with spectra that span a wide wavelength range. Our recent developments in PySME include a more efficient synthesis via the dynamic removal of weak spectral lines, measurement of stellar parameters using pre-defined masks, and abundance measurements for multiple elements, all performed in an automated manner. These improvements will serve as part of the pipeline for future large-scale spectroscopic surveys such as 4MOST (de Jong et al. 2019). The first two developments will be discussed and tested in detail in Jian et al. (in prep.), while we focus on the abundance measurement aspect in this article. The line list, covering the wavelength range from 9360 to 24, 240 Å, is extracted from the VALD3 database version 3750M (Piskunov et al. 1995; Ryabchikova et al. 2015). We used the \"Extract All\" mode with hyperfine structure included. All molecular lines across the full GIANO wavelength range are included, except for TiO lines. The enormous number of TiO lines makes it nearly impossible both to download and to include them in the synthesis. However, for target stars with Teff ⪅4500 K (14 stars in our sample), the blending of TiO lines may not be negligible. Thus, we included the TiO lines within ±20 Å of the P lines used for abundance measurements for our target giants (i.e., those in Table 2 with Ngiant not marked as '-'). We adopt MARCS model atmospheres (Gustafsson et al. 2008) as input, and solar abundances are set according to Grevesse et al. (2007, except for phosphorus; see Section 4.1). The broadening velocity, including macroturbulence and projected rotational velocity, Vbroad, is not provided in Dal Ponte et al. (2025). We therefore measure Vbroad using the GIANO-B spectra. For each MNRAS 000, 1-15 (2025) 4 M. Jian et al. Table 1. Observation log of our target stars, with the columns of (left to right) star name, Gaia DR3 ID, observation date in reduced heliocentric Julian dates (RHJD = HJD -2400000), exposure time, and the mean signal-to-noise ratio per pixel. Only the first six stars from cluster giant and Cepheid sample and the standard stars are listed, and the full table is available in CDS. Star name Gaia DR3 ID RHJD texp (s) S/Nmean Alessi1-2 402506369136008832 58471.820 300 × 12 212 Alessi1-3 402505991180022528 58471.872 300 × 12 225 Alessi1-5 402867593065772288 58470.864 300 × 8 233 Alessi1-6 402880684126058880 58471.926 300 × 10 199 Alessi Teutsch11-1 2184332753719499904 58710.863 300 × 2 377 Basel11b-1 3424056131485038592 58513.903 300 × 10 196 ... ... ... ... ... DL Cas (NGC 129) 428620663657823232 58354.228 300 × 4 253 SV Vul (UBC 130) 2027951173435143680 58425.869 300 × 2 545 X Vul (UBC 129) 2027263738133623168 58349.853 300 × 4 375 CO Aur 3451987987438434944 58426.248 300 × 4 398 RX Aur 200708636406382720 58429.227 300 × 4 425 RW Cam 473043922712140928 58354.187 300 × 4 388 ... ... ... ... ... Sun (Ganymede) - 58325.880 30 × 2 130 Arcturus - 58301.894 60 × 2 612 Table 2. List of P i lines used in this study, including their air wavelengths (λair), lower excitation potentials (EP), and oscillator strengths (log gf). Columns Ngiant and NCepheid indicates the number of stars using the corresponding line for P measurement (excluding cases where the line is blended with telluric features or only provides an upper limit), and the number in the parentheses (if presents) indicates the number of stars being removed (see the description in section 3.3). The final column ('Sun') marks whether the line was also used in the solar abundance determination. λair EP log gf Ngiant NCepheid Sun (Å) (eV) 9609.036 6.9356 -1.05 - 0 (1) 9676.222 8.0785 0.00 - 4 9734.755 6.9543 -0.36 - 3 9750.748 6.9543 -0.18 12 1 Y 9790.194 7.1758 -0.69 - 2 9796.828 6.9852 0.27 - 1 Y 9903.671 7.1758 -0.30 - 7 Y 9976.681 6.9852 -0.29 - 1 (5) Y 10084.277 7.2127 0.14 - 4 10204.716 7.2127 -0.52 - 4 10511.588 6.9356 -0.13 11 1 10529.524 6.9543 0.24 22 1 Y 10581.577 6.9852 0.45 34 1 Y 10596.903 6.9356 -0.21 29 (1) 4 Y 10681.406 6.9543 -0.19 6 5 Y 10769.511 6.9543 -1.07 - 2 10813.141 6.9852 -0.41 0 (12) 5 10932.724 8.0785 0.31 - 1 10967.373 8.0783 0.12 - 7 11183.240 7.2127 0.40 8 - 16254.749 8.2255 0.00 - 7 16482.932 7.2127 -0.29 42 5 Y 16590.045 8.2276 0.50 - 0 (6) 16738.681 8.2864 0.00 - 0 (4) 17112.447 8.2504 0.50 0 (51) 0 (7) 17286.920 8.2864 0.00 - 3 17423.670 8.2864 0.00 0 (2) 4 star, we first fix the other stellar parameters and select isolated Fe I lines based on the synthetic spectrum (the details of the line selection method are described in Section 3.2). TheVbroad value is then fitted for each selected Fe I line, and the weighted mean and standard deviation across all lines are adopted as the final broadening velocity and its associated uncertainty. The adopted stellar parameters are listed in Table 3 and plotted in Figure 3. The theoretical equivalent width (EW) of the P i line at 10529.524 Å - one of the strongest phosphorus lines in the nearinfrared - is also shown as a contour in the diagram. Most infrared P lines exhibit EW contour shapes similar to this line, with maximum strengths occurring at approximately Teff ≈6000 K and low log g, and decreasing asTeff moves away from this peak or as log gincreases. As a result, most of our Cepheids exhibit relatively strong P lines (see also Elgueta et al. 2024). In contrast, our cluster giants lie far from the EW peak region, and the EWs of the 10529.524 Å line are typically around 15 mÅ. Therefore, high signal-to-noise ratio spectra are required to derive reliable phosphorus abundances for these stars. 3.2 Line selection Given the relatively large sample size of stars in this study, as well as their wide range of stellar parameters, it is essential to select suitable P i lines using the synthetic spectra. The VALD database contains 229 P i lines within the relevant wavelength range. However, whether a given line can be used for phosphorus abundance determination depends on several factors: whether the line appears in the observed spectrum, the degree of blending with nearby lines, and whether the line can be reliably measured given the quality of the observed spectra. Therefore, a careful line selection process is required prior to performing the abundance analysis. The goal of the line selection process is to identify all P lines that are present in the spectra and exhibit minimal blending, based on synthetic spectra. All P i lines are first extracted from the line list. For each star, we define a total broadening velocity,Vbroad, which accounts for the broadening of microturbulence velocity (Vmic), macroturbulence velocity (Vmac), projected rotation (vsini), and instrumental MNRAS 000, 1-15 (2025) SPA P abundance 5 Figure 2. CMD of the target clusters. The grey dots represent stars with membership probabilities greater than 0.5 from Cantat-Gaudin et al. (2020), while the blue circles highlight our selected cluster giant stars. resolution (R), as: Vbroad = √︂ V2 mic + V2mac + (vsini)2 + c R 2 , (1) with cas the speed of light. A wavelength region with a width of 4Vbroad/cis assigned to each target line, defining a feature. If two features overlap in wavelength, they are merged and treated as a single feature. Synthetic spectra based on the stellar parameters mentioned in Section 3.1 are then generated for each feature, and three variables are used to evaluate the suitability of each P line for abundance analysis: • Feature-dominance: the ratio of the EW of the phosphorus line(s) to the total EW of all lines within the feature; • Feature-depth: the maximum depth of the feature in the normalised flux; • Telluric-depth: the maximum depth of telluric absorption in the corresponding wavelength region. We require that the selected features satisfy the following thresholds: feature-dominance greater than 0.4, feature-depth greater than 0.006, and telluric-depth less than 0.1. The adopted featuredominance threshold does not require the feature to be completely dominated by phosphorus lines. We found that the primary contamMNRAS 000, 1-15 (2025) 6 M. Jian et al. Table 3. Stellar parameters of the target stars. Only the first six stars from cluster giant and Cepheid sample and the standard stars are listed, and the full table is available in CDS. Star name Teff log g [Fe/H] Vmic Vbroad Alessi1-2 4986 ± 30 2.83 ± 0.07 -0.001 ± 0.010 1.297 ± 0.013 6.7 ± 0.9 Alessi1-3 4996 ± 30 2.84 ± 0.07 -0.007 ± 0.010 1.276 ± 0.014 6.6 ± 0.9 Alessi1-5 4939 ± 31 2.73 ± 0.08 -0.028 ± 0.010 1.353 ± 0.013 6.7 ± 0.9 Alessi1-6 4985 ± 32 2.87 ± 0.08 -0.013 ± 0.010 1.278 ± 0.014 6.4 ± 0.8 Alessi Teutsch 11-1 4517 ± 35 2.15 ± 0.11 -0.040 ± 0.013 1.756 ± 0.014 7.2 ± 1.1 Basel 11b-1 4950 ± 35 2.51 ± 0.09 0.007 ± 0.010 1.808 ± 0.010 8.4 ± 1.1 ... ... ... ... ... ... DL Cas 5622 ± 18 1.73 ± 0.04 0.050 ± 0.010 3.920 ± 0.040 25.92 ± 0.17 SV Vul 5676 ± 13 1.00 ± 0.03 0.190 ± 0.010 4.240 ± 0.020 15.19 ± 0.06 X Vul 6294 ± 12 1.95 ± 0.02 0.170 ± 0.010 4.410 ± 0.020 18.04 ± 0.06 CO Aur 7076 ± 9 2.77 ± 0.01 0.180 ± 0.010 2.670 ± 0.020 13.52 ± 0.05 RX Aur 6466 ± 10 1.52 ± 0.01 -0.040 ± 0.010 4.040 ± 0.020 26.62 ± 0.09 RW Cam 5672 ± 16 1.63 ± 0.04 0.170 ± 0.010 3.180 ± 0.020 17.51 ± 0.09 ... ... ... ... ... ... Sun 5772 4.44 0.00 1.0 2.8 Arcturus 4286 1.6 -0.52 1.74 4.6 Figure 3. The Kiel diagram of our sample stars. The underlying contours show the EW (mÅ) of P i line at 10529.524 Å at solar metallicity. inants in P features are CN (at a wavelength range of ⪅12000 Å) and CO (⪆12000 Å) molecular lines. To minimize the blending of these molecular lines, we altered the C abundance to fit the spectra of ±2 Å around the P line and fix it during the fit for the P abundance. These C abundances are presented in appendix A and the tables therein. The feature-depth threshold is determined based on the default synthesis precision of PySME. The telluric-depth threshold of 0.1 is relatively strict. Given that P lines are intrinsically weak, the profiles of P lines blended with deep telluric absorption are often significantly distorted. A total of 11 and 26 P i lines passed the automatic selection criteria for cluster giants and Cepheids, respectively, but no P ii line passed our selection. Table 2 lists the line parameters and indicates which group of stars each line is used for. The phosphorus lines that meet the threshold requirements for each target are subsequently included to derive its phosphorus abundance. Minor refinements to the line selection are made following individual abundance measurements for each line. The oscillator strength, or log gfvalue, is a key parameter that determines the strength of a spectral line in synthetic spectra. Accurate abundance measurements therefore rely critically on the precision of the adopted log gfvalues. While Elgueta et al. (2024) recalibrated the log gfvalues for seven of the P lines included in our selection, all of the selected lines already have laboratory-measured log gfvalues. To maintain consistency across all line parameters, we adopt the log gfvalues provided by the VALD database for all P lines used in this work. 3.3 Abundance measurement For each phosphorus line, the pixels within the full feature region-defined as a width of 4Vbroad/ccentred on the line-are used to fit the phosphorus abundance with PySME. Figures 4 and 5 show an example of the line fitting procedure, for our standard star the Sun and Alessi 1-2, respectively. Synthetic spectra spanning a broader width of 4Vbroad/c+ 4 Å are generated, as illustrated in the upper panel. Within the fitting region (vertical orange shaded band), if the blue curve (representing synthetic spectra generated using the P line only) lies between the shaded blue area (synthetic spectra with all lines, with varied [P/Fe]) and the blending by other species (the spectra with thin solid lines) is small, the feature is considered to be dominated by the P line. The lower panel displays the configuration and result of the actual fitting. To obtain a more accurate continuum level, we extend the input observed spectra to a total width of 4Vbroad/c+ 10 Å, centred on the P line. Pixels are classified into three types of masks: • Continuum mask: pixels that satisfy all of the following conditions: (1) their depth in the synthetic spectrum is less than 0.025; (2) their depth in the telluric model is less than 0.1; and (3) the absolute difference between the observed and synthetic fluxes is less than twice the standard deviation of the residuals; • Line mask: pixels within 4Vbroad/cof the P line centre; • Bad mask: all remaining pixels. This masking strategy ensures that pixels affected by other spectral lines or strong telluric absorption are excluded from continuum estimation. As the continuum normalisation is generally well performed MNRAS 000, 1-15 (2025) SPA P abundance 7 in Section 2, PySME applies only a constant scaling to the synthetic spectrum, determined by the mean flux level of the continuum pixels. The phosphorus abundance is then derived by minimising the χ2 over the line mask region only. We adopt the best-fit phosphorus abundance and the corresponding fit_uncertainties (the statistical uncertainties of the fitting) output from PySME as our final measurement and its associated uncertainty. Upper limits are identified following the method similar to the one described in Wang et al. (2024): if the EW of a fitted line at its best-fit abundance A(P)λis smaller than three times the EW obtained at A(P)λ+σA(P)λ, the measurement is classified as an upper limit and excluded from further analysis. When measurement uncertainties properly reflect the noise level in the spectrum, this criterion helps mitigate selection bias by avoiding the preferential inclusion of stronger lines influenced by noise. We expect the phosphorus abundances derived from all selected lines for a given star to be consistent within their respective uncertainties. However, we find that for a few lines, the derived abundances systematically deviate from the mean value obtained from the other lines. This discrepancy may arise from several factors, such as inaccurate log gfvalues, improper continuum placement, or undetected blending features that are not captured by the synthetic spectra. These problematic lines are marked with Ngiant or NCepheid as 0 following by parentheses in Table 2 and are excluded from subsequent analysis. Most of these lines are used in only a few stars, and we defer discussion of specific cases where the affected lines are present in the majority of stars to Appendix B. Finally, the weighted average and standard deviation of the A(P) from all available P i lines are taken as the final A(P) and its corresponding error. 4 RESULTS We present the results of phosphorus abundance measurements for individual stars and clusters in this section. 4.1 Phosphorus Abundance for the standard stars Our measurement of the solar spectrum from GIANO shows good consistency with previous determinations, as shown in Figure 6. The same line selection procedure as for other stars was applied to the standard stars. A total of 13 phosphorus lines were initially selected. Among them, one line is too weak to provide reliable measurements and is therefore treated as upper limits, while another three are significantly blended with telluric absorption features or not showing a clear absorption profile. It is evident that the lines affected by telluric contamination tend to deviate from the remaining measurements (i.e., the 9525.741 Å and 11183.24 Å line in Figure 6), which justifies their exclusion from the final abundance calculation. The final phosphorus abundance, 5.46, is obtained as the weighted mean of the remaining reliable P lines. This value is in good agreement with the solar phosphorus abundance derived using 1D LTE spectral synthesis in Scott et al. (2015), 5.38. The estimated uncertainty of 0.2 dex likely arises from a combination of factors, including lower resolution and signal-to-noise ratio compared to the solar spectrum used in Scott et al. (2015), uncertainties in the adopted log gfvalues, and imperfect continuum placement. For Arcturus, no phosphorus lines pass our line selection criteria. Compared to the Sun, Arcturus is cooler and has significantly lower surface gravity (log g), which leads to weaker P lines (see Figure 3). In addition, its relatively low metallicity ([Fe/H] = -0.51) further reduces the line strengths, rendering all available phosphorus lines too weak for reliable abundance measurement. Our results from the Sun and Arcturus help to define an approximate detection boundary for phosphorus abundance measurements at the resolution and signal-to-noise ratio (S/N) of our GIANO-B spectra. Specifically, a metallicity of [Fe/H] ∼-0.5 appears to be the lower limit for detecting P lines in cool giant stars, as demonstrated by the non-detection in Arcturus. For warmer giants and dwarfs, our test using the solar stellar parameters suggests that only one single P line at 10529.524 Å remains detectable at [Fe/H] = -1.0, indicating a rough detection limit under such conditions. This conclusion is consistent with previous high-resolution studies, particularly those by Caffau et al. (2011), whose sample consists of dwarf stars with the most metal-poor star having a metallicity just above [Fe/H] = -1.0. To extend phosphorus measurements to lower metallicities, stars within the optimum Teff and log grange (see Fig 3) should be observed at very high SNR, or observations should be done in ultraviolet (see Roederer et al. 2014). 4.2 Phosphorus Abundance for our sample stars Figure 7 shows an example of the final phosphorus abundance determination for the star Alessi 1-2. Eight lines were selected for the measurement, two of which exhibit weak absorption features and are therefore treated as upper limits. One additional line was excluded based on the criteria described in Section 3.3. The final phosphorus abundance is calculated as the weighted average of the abundances derived from the remaining lines. Tables 4 and 5 summarise the phosphorus abundance measurements for cluster giants and Cepheids, respectively. The uncertainties in the averaged P abundances for the giants are typically around 0.1 dex, while those for the Cepheids are mostly lower, indicating that P lines are generally stronger in Cepheid spectra. The NLTE effect on the phosphorus lines may also contribute, at least in part, to the uncertainties in our target stars. However, there is currently no available NLTE departure coefficient for phosphorus, and thus it is not yet possible to quantify or correct this effect. We then compute the average phosphorus abundance for each cluster using its member stars, as listed in Table 6. The [P/Fe]-[Fe/H] distribution of our cluster and Cepheid sample is broadly consistent with previous studies, as shown in Figure 8. Our targets span a relatively narrow metallicity range, from [Fe/H] = -0.1 to +0.15. Given this limited coverage, we adopt the results from Nandakumar et al. (2022) as the primary comparison, and refer the reader to their Figure 4 for a comprehensive compilation of earlier studies. Both our field Cepheids and cluster stars exhibit a modest decreasing trend in [P/Fe] from ∼0.3 to ∼0 over the range [Fe/H] = -0.1 to 0. At the metal-rich end, [P/Fe] appears to flatten near zero, although this trend is not well constrained due to the small number of stars in this regime. Overall, our observations fall within the same [P/Fe] range as reported by Nandakumar et al. (2022), supporting the consistency of our results with previous measurements. The [P/Fe]-age relation observed in our cluster sample shows general consistency with previous studies, while also providing new insights at the younger end. As shown in Figure 9, the phosphorus abundance (both in the form of absolute P abundance A(P) and the relative [P/Fe]) of clusters with ages greater than 1 Gyr increases with age. This trend is consistent with the findings of Maas et al. (2019) and Feuillet et al. (2018), although we note that the stellar ages from Maas et al. (2019) (with mean error of ∼1.8 Gyr) were derived from isochrone fitting of field stars, a method notoriously subject to larger uncertainties than when it is applied to stellar clusters (with MNRAS 000, 1-15 (2025) 8 M. Jian et al. Table 4. P measurement for cluster giants, with the star name, mean P abundance from all the normal-flag lines, number of normal-flag line and the abundance of the first six lines. The abundances with superscript of u or t are those with upper limit or blended by telluric lines. Only the first 10 stars and first 6 lines are listed, and the full table is available in CDS. star name A(P)mean N A(P)9750.748 A(P)10511.588 A(P)10529.524 A(P)10581.577 A(P)10596.903 A(P)11183.24 Alessi 1-2 5.49 ± 0.12 5 5.68 ± 0.24u 5.01 ± 0.18u 5.56 ± 0.14 5.43 ± 0.07 5.51 ± 0.11 5.77 ± 0.13 Alessi 1-3 5.50 ± 0.14 7 5.80 ± 0.12 5.56 ± 0.07 5.20 ± 0.09 5.52 ± 0.05 5.55 ± 0.12 5.56 ± 0.22 Alessi 1-5 5.46 ± 0.21 5 5.62 ± 0.16 4.98 ± 0.18u 5.37 ± 0.09 - 5.39 ± 0.08 5.72 ± 0.09 Alessi 1-6 5.51 ± 0.23 4 5.62 ± 0.25u 5.16 ± 0.19u 5.28 ± 0.18 - 5.37 ± 0.17 5.84 ± 0.15 Alessi Teutsch 11-1 5.49 ± 0.05 2 1.28 ± 5.77t 5.04 ± 0.22u 5.45 ± 0.14 - 5.49 ± 0.06 - Basel 11b-1 5.41 ± 0.13 4 4.76 ± 0.82u 4.95 ± 0.31u 5.45 ± 0.17 - 5.44 ± 0.13 5.53 ± 0.16 Basel 11b-2 5.35 ± 0.19 6 5.92 ± 0.14 5.36 ± 0.14 5.27 ± 0.22u 5.30 ± 0.04 5.31 ± 0.10 5.73 ± 0.16 Basel 11b-3 5.46 ± 0.14 2 - - - 5.51 ± 0.09 - - COIN-Gaia 30-1 - 0 -1.36 ± 37.53t -2.10 ± 8.58 5.42 ± 0.28u - 4.70 ± 0.41r 5.82 ± 0.11t Collinder 350-1 - 0 - - - - - - Collinder 350-2 5.30 ± 0.04 2 - - - 5.30 ± 0.03 - - ... ... ... ... ... ... ... ... ... Table 5. Similar to Table 4, but for the sun and Cepheid sample, with ages coming from the period-age relation. Only the first 11 Cepheids are listed, and the full table is available in CDS. star name Age (Gyr) A(P)mean Nline A(P)9734.755 A(P)9750.748 A(P)9790.194 A(P)9796.828 A(P)9903.671 A(P)9976.681 SV Vul 0.016 ± 0.003 5.88 ± 0.09 1 6.24 ± 0.04r - - - - - DL Cas 0.051 ± 0.009 5.52 ± 0.23 4 - - 5.78 ± 0.07 - 5.61 ± 0.11t - CO Aur 0.096 ± 0.009 5.73 ± 0.17 4 - 4.63 ± 0.93 - - - - RX Aur 0.039 ± 0.007 5.61 ± 0.07 4 - - 5.69 ± 0.03 - - - V351 Cep 0.080 ± 0.008 5.97 ± 0.08 2 - -1.76 ± 6.86 - - - - CD Cyg 0.031 ± 0.006 5.65 ± 0.08 2 - - 5.70 ± 0.21t - 5.75 ± 0.11 - V1334 Cyg 0.075 ± 0.008 5.61 ± 0.13 5 - - 5.74 ± 0.10t - - - VZ Cyg 0.071 ± 0.013 5.65 ± 0.25 4 6.19 ± 0.22t - 5.44 ± 0.45t - - - X Cyg 0.031 ± 0.006 5.43 ± 0.07 2 - - 5.63 ± 0.13t - 5.98 ± 0.08t - W Gem 0.051 ± 0.009 5.39 ± 0.18 7 - - - - - - RR Lac 0.059 ± 0.011 5.54 ± 0.09 5 - - - 3.11 ± 29.35t - - ... ... ... ... ... ... ... ... ... ... Sun - 5.46 ± 0.20 9 - - - 5.50 ± 0.12 - 5.74 ± 0.09 Table 6. The mean Phosphorus abundance of our target clusters, with the number of stars for the cluster listed. Cluster Age (Gyr) [Fe/H]mean A(P)mean [P/Fe]mean N Alt Name Gulliver 18 0.04 -0.01 ± 0.02 5.59 ± 0.11 0.25 ± 0.11 1 Collinder 416 Collinder 463 0.11 -0.06 ± 0.01 5.46 ± 0.19 0.16 ± 0.19 2 UPK 219 0.15 0.07 ± 0.01 5.33 ± 0.06 -0.10 ± 0.06 1 Tombaugh 5 0.19 0.02 ± 0.06 5.31 ± 0.14 -0.08 ± 0.15 3 NGC 7086 0.19 -0.08 ± 0.03 5.61 ± 0.17 0.33 ± 0.17 2 Collinder 437 UBC 194 0.23 0.06 ± 0.01 5.44 ± 0.12 0.02 ± 0.13 1 Basel 11b 0.23 0.03 ± 0.03 5.41 ± 0.05 0.02 ± 0.06 3 FSR 877 NGC 2437 0.30 0.01 ± 0.06 5.45 ± 0.28 0.08 ± 0.29 4 M 46, Melotte75 UBC 169 0.30 0.05 ± 0.03 5.40 ± 0.15 -0.01 ± 0.16 2 NGC 2548 0.40 0.05 ± 0.05 5.30 ± 0.07 -0.11 ± 0.09 3 M 48, Melotte 85 Stock 2 0.40 -0.00 ± 0.04 5.30 ± 0.12 -0.06 ± 0.12 7 NGC 7209 0.43 0.04 ± 0.01 5.63 ± 0.05 0.22 ± 0.05 1 Melotte 238, Collinder 444 Collinder 350 0.59 0.08 ± 0.01 5.30 ± 0.04 -0.15 ± 0.04 1 NGC 2632 0.68 0.13 ± 0.01 5.62 ± 0.18 0.13 ± 0.19 1 M 44, Praesepe NGC 752 1.17 0.08 ± 0.02 5.30 ± 0.08 -0.13 ± 0.09 3 Melotte 12, Theia 1214 IC 4756 1.29 -0.00 ± 0.04 5.39 ± 0.05 0.03 ± 0.06 6 Collinder386, Melotte 210 Alessi 1 1.45 -0.01 ± 0.01 5.49 ± 0.01 0.14 ± 0.02 4 Casado-Alessi 1 NGC 6991 1.55 0.03 ± 0.06 5.42 ± 0.15 0.03 ± 0.16 5 UBC 141 2.09 -0.02 ± 0.01 5.38 ± 0.23 0.05 ± 0.23 1 UBC 577 2.75 -0.04 ± 0.04 5.41 ± 0.12 0.09 ± 0.13 3 Alessi 191 Ruprecht 171 2.75 -0.04 ± 0.05 5.71 ± 0.08 0.39 ± 0.09 4 NGC 2682 4.27 0.02 ± 0.01 5.66 ± 0.53 0.28 ± 0.53 1 M67 MNRAS 000, 1-15 (2025) SPA P abundance 9 10527 10528 10529 10530 10531 10532 0.95 0.96 0.97 0.98 0.99 1.00 Normalized flux Sun, P 1 (10529.524 Å) Synthetic spectra with [P/Fe]±0.1 Synthetic spectra with P 1 line only Synthetic spectra of other species 10524 10526 10528 10530 10532 10534 Wavelength (Å) 0.80 0.85 0.90 0.95 1.00 Normalized flux Fitted A(P)=5.26±0.11, EWsynth, all=10.94±2.25 mÅ, normal bad line cont mask (dashed) Synthesized spectrum Telluric spectrum Observed spectrum Figure 4. Phosphorus abundance determination from the P i 10529.524 Å line in the solar spectrum. Top panel: Synthetic spectra after varying the abundances of P by ±0.1,dex (blue shaded profiles) together with spectra that contains only the P i transition (solid blue), and other species (gray). The orange dotted vertical line marks the line centre, while the orange shaded strip marks the wavelength range used in the fit. Bottom panel: Observed spectrum (blue points) compared with the best-fitting synthetic spectrum (solid orange) and models corresponding to ±1σuncertainty in A(P) (orange dashed). The light-grey dashed \"stair-step\" curve (right-hand y-axis) shows the pixel mask: cont (continuum windows), line (pixels included in the fit), and bad (pixels excluded). The orange shaded region is the same fitting window as in the top panel. Similar plots for the other analysed P lines are available in the Supporting Information. relative error around 25%; Cantat-Gaudin et al. 2020). In contrast, the trend at younger ages seems to be flat. We performed linear fit to the clusters with ages younger and older than 1 Gyr separately in Figure 9. The linear fit for the older clusters show a positive slope and significant Pearson p-values less than 0.05, in both A(P) and [P/Fe]. Those for the young clusters present a negative slope with p-values larger than 0.05. Most clusters with their ages less than 1 Gyr have near-solar phosphorus abundances, while the clusters and Cepheids with even younger ages show either solar or enhanced [P/Fe] values. They reduce the slope of linear fitting and further decrease the p-values to less than 0.05. The measurement of the Cepheids fill the blank in the youngest side of the trend. They extends the negative trend to the age of 0.015 Gyrs, and raises the fitted Phosphorus abundance to ∼5.7. 4.3 Spatial distribution of Phosphorus One of the goal of Galactic archaeology is to map the spatial distribution of elements in our Galaxy. Since OCs represents a well-defined single stellar population, their spatial distribution allows us to investigate the phosphorus distribution among different groups of stars in various directions across the Galactic plane. Classical Cepheids in our sample can also provide similar information, as they are relatively young and their period-luminosity relation enables precise distance measurements. Figure 10 displays the spatial distribution of our target clusters and Cepheids in heliocentric Cartesian coordinates, overlaid with a dust extinction map (Dharmawardena et al. 2024) to represent the current structure of the interstellar medium (ISM). We adopted Solar Xand Zvalues of 8122 and 20.8 pc, respectively (GRAVITY Collaboration et al. 2018; Bennett & Bovy 2019). Despite the clear inhomogeneous in the present-day ISM, we find no statistically significant correlation between the absolute phosphorus abundance, A(P), and the clusters' current locations, be it their Galactocentric radius or their position relative to local dust structures. This apparent lack of correlation is not a deficiency in our data but is, in fact, a key piece of evidence for the dynamic nature of the Milky Way's disk. Over timescales of tens of millions to several billion years, processes such as radial migration and vertical heating act to redistribute stars and clusters from their birthplaces (see e.g. Viscasillas Vázquez et al. 2023). Consequently, a cluster's current location is often decoupled from the chemical composition of the gas from which it formed. 5 DISCUSSION Our analysis of OCs ages reveals a distinct \"V-shape\" in the phosphorus enrichment history of the Milky Way's thin disk (see Figure 9). While previous studies have established a general trend of [P/Fe] MNRAS 000, 1-15 (2025) 10 M. Jian et al. 10579 10580 10581 10582 10583 10584 0.92 0.94 0.96 0.98 1.00 Normalized flux Alessi1-2, P 1 (10581.577 Å) Synthetic spectra with [P/Fe]±0.1 Synthetic spectra with P 1 line only Synthetic spectra of CN 1 Synthetic spectra of other species 10576 10578 10580 10582 10584 10586 Wavelength (Å) 0.5 0.6 0.7 0.8 0.9 1.0 Normalized flux Fitted A(P)=5.43±0.07, EWsynth, all=37.19±2.15 mÅ, normal bad line cont mask (dashed) Synthesized spectrum Telluric spectrum Observed spectrum Figure 5. Similar as Figure 4, but for the line at 10581.577 Å of Alessi 1-2. Here several CN lines appear around the target P line, and their synthetic spectrum is also plotted in the upper panel. Similar plots for other lines and other stars are available in the Supporting Information. Figure 6. The Phosphorus abundance measurement result from all the lines for the Sun. Blue filled circles give the abundance A(P) derived for each line (wavelength range shown on the x-axis); their vertical blue error bars mark the 1σfitting uncertainty. Deep-red plus symbols plot the line-by-line abundances reported by Scott et al. (2015) for those that are used in both studies, enabling a direct comparison. Red crosses indicate lines discarded from our final mean because of severe telluric blending or being manually excluded, while blue downward arrows denote lines for which only an upper limit on A(P) could be set. The horizontal blue dashed line is the weighted mean abundance from all accepted (\"normal-flag\") lines, with its 1σdispersion shown by the light-blue band. For reference, the red dashed line marks the solar value A(P) = 5.38 adopted by Scott et al. (2015). The panel title lists the mean abundance and its standard deviation obtained in this work. with [Fe/H] (Caffau et al. 2011, 2016; Maas et al. 2019, 2022), the high-precision and wide range of ages with a similar [Fe/H] of our cluster sample allow us to dissect the phosphorus evolution in time, uncovering two distinct evolutionary trends. The \"V-shape\" in the A(P)-age plane is not a single evolutionary track, but rather the superposition of two different regimes of GCE. The clusters younger than ∼1 Gyr appear to follow a path consistent with local, recent enrichment, while the older clusters unveil a fossil record of the diverse and dynamic star formation histories from the Galaxy's past. We note that similar but milder [Mg/Fe]-age and [Mg/H]-age trend can also be found using the Mg abundances from Dal Ponte et al. (2025) where only open clusters are analysed. A systematic MNRAS 000, 1-15 (2025) SPA P abundance 11 9750.1569751.340Å 10510.94910512.227Å 10528.88410530.164Å 10580.93410582.220Å 10596.25910597.547Å 11182.56011183.920Å 16481.93016483.934Å 17111.40717113.487Å 5.0 5.2 5.4 5.6 5.8 6.0 6.2 A(P) Alessi1-2, A(P)=5.49±0.12 P 1 line Scott+2015 adopted value: 5.38 Fitted value Fitted std Figure 7. Similar as Figure 6, but for Alessi 2-1. Similar figures for other stars are available in the Supporting Information. Figure 8. [P/Fe] versus [Fe/H] for OC giants in our sample (left panel, the average [P/Fe] for each cluster) and Cepheid stars (right panel). For comparison, the [P/Fe] measurement of the giants from Nandakumar et al. (2022) is also plotted. comparison of age-abundance trends for other elements with similar nucleosynthetic origins would be highly informative; however, such an analysis is beyond the scope of the present study and is deferred to future work. 5.1 Old OCs as a fossil record of diverse star formation histories and P enrichment For OCs older than ∼1 Gyr, we observe that the absolute phosphorus abundance A(P) decreases as stellar age decreases from ∼4 Gyr to ∼1 Gyr. This trend is contrary to the predictions of a simple, single-zone GCE model where metal abundance should monotonically increase over the evolution time. The explanation lies in the fact that the Milky Way is not a single, uniform entity. The solar neighbourhood today contains a mix of stellar populations born in different locations and at different times, each with a unique star formation history. This is a key feature of multi-zone GCE. The clusters in our sample, despite having a similar near-Solar metallicity (probed by [Fe/H]), reached this metallicity at vastly different cosmic epochs (see Table 6). For a cluster like M67 which has achieved the solar metallicity by about 4 Gyr ago, its parent gas cloud must have undergone a very rapid and intense period of enrichment. In contrast, for another cluster in the old OC group, NGC 752, its parent gas cloud reached a similar metallicity much more gradually, about 1 Gyr ago. This difference in enrichment speed implies a fundamentally different Star Formation History. The rapid enrichment required for the older clusters necessitates either a significantly higher Star Formation Rate or a more \"top-heavy\" IMF that produces more massive stars per generation. In either scenario, the integrated contribution from massive stars-the primary producers of phosphorus-is far greater (see Figure 1). Therefore, it is an inevitable consequence that the 4 Gyr old M67 formed from gas that was more abundant in absolute phosphorus A(P) than the gas that formed the 1 Gyr-old NGC 752. The declining A(P) trend for clusters older than ∼1 Gyr is thus a direct manifestation of observing a sequence of clusters born from environments with progressively less intense star formation histories. 5.2 Young OCs and Cepheids as evidence for local, quiescent P Enrichments Open clusters younger than ∼1 Gyr and Cepheids in Figure 9 exhibit a flat or slightly increasing A(P) (or [P/Fe]) with decreasing age. This behaviour aligns well with the expectations for a single-zone GCE MNRAS 000, 1-15 (2025) 12 M. Jian et al. Figure 9. Cluster or stellar age versus phosphorus abundance for our targets. The A(P) and [P/Fe] measurement from Maas et al. (2019) are plotted in gray as a reference. The blue lines and transparent strip represent separate linear fits and its 95% confidential interval to the cluster data, applied independently to those younger and older than 1 Gyr, while the orange ones are the linear fit to the cluster and Cepheid sample in the young side. The [P/Fe]-age trend from Feuillet et al. (2018) is plotted as a reference. model. These clusters and stars are young and located in the Solar neighbourhood, meaning they were likely born locally and share a nearly common, recent chemical history. They represent the current, more quiescent phase of evolution in our part of the Galaxy. In this local environment, we see a slow enrichment of phosphorus over the last ∼1 Gyr. Such gentle phosphorus enrichment may be explained by the contribution from low-mass stars. Figure 11 shows the evolution of phosphorus in single stellar population at two different initial metallicity (iniZ) values, following the same set-up as the yields in Figure 1. Overall, the main phosphorus production at both lower and higher metallicities originates from massive stars, which is consistent with the conclusions of previous studies (e.g. Cescutti et al. 2012). The contribution from lower mass stars, i.e. AGB stars, reached a level of 10-8 M⊙in the later evolution time at iniZ= 0.0001. However, the contribution of AGB stars to the phosphorus production at iniZ= 0.01 is an order of magnitude higher than in the metal-poor case, making it more significant at later times in the evolution of the single stellar population. This increased phosphorus production by AGB stars is reflected in the peak of the IMF-weighted total yield at the lowest stellar masses, as seen for iniZ= 0.004 and 0.01 in Figure 1. These slight sub-Solar metallicities are also proximally corresponding to the previous generation of the OCs and Cepheids in our study. In the relative absence of vigorous massive star formation in the immediate Solar vicinity, the enrichment from these lower-mass stars can become a noticeable contributor to the ongoing chemical evolution, explaining the slight upward trend in A(P) towards the present day. We note that this interpretation is sensitive to the uncertainties in the nucleosynthetic yields adopted here, highlighting the need for more precise yield determinations. Also the contribution of ONe nova, recently discussed by Bekki & Tsujimoto (2024), is not considered here, but it may contribute to some extent to phosphorus production. 6 SUMMARY In this work, we present a detailed abundance analysis of phosphorus for 82 giant stars in 24 open clusters, together with 20 Cepheids (17 in the Galactic field and 3 in open clusters). The analysis is based on high-resolution near-infrared spectra obtained with the GIANOB spectrograph. To achieve this, we developed and implemented a robust line selection and fitting procedure, which carefully accounts for blending, telluric contamination, and line-strength variations, ensuring reliable abundance measurements. Our key findings include: • Our results confirm the previously observed modest decreasing trend of [P/Fe] with increasing [Fe/H] for stars around solar metallicity. • Using the high precision age of open clusters, we uncover a distinct \"V-shape\" in the phosphorus-age relation, revealing two separate chemical evolution pathways and mixtures for the Milky Way's thin disk: - For clusters older than ∼1 Gyr in our sample, we find that phosphorus abundance increases with stellar age. We interpret this trend as a fossil record of diverse star formation histories and phosphorus enrichments. With a similar metallicity, older clusters formed from a gas that was more rapidly enriched, a consequence of more intense star formation in the earlier epochs of the Galactic disk. - For clusters younger than ∼1 Gyr and Cepheids in our sample, the [P/Fe] and A(P) trend with age is nearly flat with a small negative slope. This points to a more quiescent, local enrichment history, where the gentle increase in phosphorus of the younger clusters may be attributable to the contribution from low-mass stars at solar metallicities. • We find no correlation between the A(P) abundance and the current spatial location of open clusters and cepheids. Future observations of more young open cluster giant stars or Cepheids may further clarify the [P/Fe]-age trend in the very young population of our Milky Way. ACKNOWLEDGEMENTS A.B., M.J., and V.D. acknowledge funding from INAF Mini-Grant 2022 (High resolution spectroscopy of open clusters). X.F. acknowledges the support of the National Natural Science Foundation of China (NSFC) No. 12203100. G.B., G.F. and A.N. thank the support from Project PRIN MUR 2022 (code 2022ARWP9C) 'Early Formation and Evolution of Bulge and HalO (EFEBHO)' (PI: M. Marconi), funded by the European Union-Next Generation EU, and from the Large grant INAF 2023 MOVIE (PI: M. Marconi). MNRAS 000, 1-15 (2025) SPA P abundance 13 -0.5 0.0 0.5 Z [kpc] -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 X [kpc] -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Y [kpc] ⊙ -0.5 0.0 0.5 Z [kpc] 5.2 5.3 5.4 5.5 5.6 5.7 5.8 A(P) Figure 10. Space distribution of the target clusters (dots) and Cepheids (stars), colour-coded by their phosphorus abundance. The overlaid extinction maps for each plane are derived from Dharmawardena et al. (2024). A maximum A0 of 0.8 mag was set (represented by darkest black), with a contour (black dashed line) at 0.2 mag for the XY plane, while a maximum A0 of 3 mag was applied for the XZ and YZ planes. The coloured curves represent the Galactic log-periodic spiral arms, defined by the parameters from (Reid et al. 2019): the Carina-Sagittarius arm in purple, the Local arm in black, and the Perseus arm in green. Additionally, the spur between the Local and Sagittarius-Carina arms is indicated by the blue curve. Part of the research activities described in this paper were carried out with the contribution of NextGenerationEU funds within the National Recovery and Resilience Plan (PNRR), Mission 4-Education and Research, Component 2-From Research to Business (M4C2), Investment Line 3.1-Strengthening and creation of Research Infrastructures, Project IR0000034- \"STILES-Strengthening the Italian Leadership in ELT and SKA\", CUP C33C22000640006. This work has made use of the VALD database, operated at Uppsala University, the . 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The P i line at 10813.141 Å is intrinsically very weak, with a typical line depth of only ∼0.01 in most stars. Its abundance sensitivity is highly dependent on accurate continuum placement. However, the continuum in the fitting region is often depressed below unity, and since PySME applies only a constant scaling over a wide region during abundance fitting, local misplacement of the continuum across a narrow wavelength range cannot be corrected. As a result, the phosphorus abundances derived from this line tend to be overestimated. Also, the situation for P i line at 17423.670 Å is similar but with underestimated abundance, thus it is also being removed. The line at 17112.447 Å consistently yields phosphorus abundances that are approximately 0.3 dex higher than the average derived from other lines. Inspection of the line fitting reveals the presence of an unidentified absorption feature on the red side of the P line. This feature is absent from the synthetic spectra (otherwise it would have been accounted for), but it appears consistently in all observed spectra with a similar relative position, suggesting it originates from the stellar spectrum. Due to the severity of this blending, the fitting routine attempts to reproduce the composite feature by increasing the phosphorus abundance, leading to a systematic overestimation. Six P i lines were excluded from the Cepheid sample. Their behavior is similar to that of the 10813.141 Å line, for which accurate continuum placement is challenging in this wavelength region, compromising the reliability of the abundance fitting. As a result, these lines were excluded from the analysis. This paper has been typeset from a TEX/LATEX file prepared by the author. MNRAS 000, 1-15 (2025) 16 M. Jian et al. Table A1. Carbon abundance used to fit molecular features around the phosphorus lines for cluster giants. Only the first 10 stars and first 6 lines are listed, and the full table is available in CDS. star name A(C)9750.748 A(C)10511.588 A(C)10529.524 A(C)10581.577 A(C)10596.903 A(C)11183.24 Alessi 1-2 8.30 ± 0.05 8.14 ± 0.38 8.56 ± 0.10 8.33 ± 0.24 8.06 ± 0.22 8.31 ± 0.08 Alessi 1-3 8.27 ± 0.06 8.25 ± 0.28 8.64 ± 0.09 8.40 ± 0.22 8.42 ± 0.08 8.25 ± 0.08 Alessi 1-5 8.38 ± 0.05 8.04 ± 0.37 8.61 ± 0.08 - 8.40 ± 0.06 8.30 ± 0.07 Alessi 1-6 8.26 ± 0.07 7.68 ± 0.78 8.71 ± 0.08 - 7.63 ± 0.16 8.20 ± 0.09 Alessi Teutsch 11-1 8.25 ± 0.05 8.35 ± 0.11 8.52 ± 0.05 - 8.32 ± 0.03 - Basel 11b-1 8.35 ± 0.07 8.47 ± 0.14 8.79 ± 0.07 - 8.15 ± 0.25 8.33 ± 0.09 Basel 11b-2 8.42 ± 0.06 8.37 ± 0.19 8.79 ± 0.07 8.67 ± 0.12 8.51 ± 0.08 8.38 ± 0.08 Basel 11b-3 - - - 8.54 ± 0.09 - - COIN-Gaia 30-1 7.87 ± 0.17 8.12 ± 0.29 8.52 ± 0.09 - 8.38 ± 0.07 8.45 ± 0.05 Collinder 350-1 - - - - - - Collinder 350-2 - - - 8.55 ± 0.14 - - ... ... ... ... ... ... ... Table A2. Carbon abundance used by fitting the molecular features around the phosphorus lines for Cepheids. Only the first 10 stars and first 6 lines are listed, and the full table is available in CDS. star name A(C)9493.571 A(C)9609.036 A(C)10932.724 A(C)10967.373 A(C)11186.753 A(C)15711.522 SV Vul 8.08 ± 0.18 8.64 ± 0.04 8.52 ± 0.11 8.63 ± 0.02 8.41 ± 0.10 10.27 ± 0.09 DL Cas - - - - - - CO Aur 6.35 ± 2.36 - - 8.35 ± 3.05 - - RX Aur - - - - - - V 351 Cep - - - 8.33 ± 2.73 10.33 ± 0.31 - CD Cyg - - 8.25 ± 0.02 - - - V 1334 Cyg 6.09 ± 5.05 - - - - - VZ Cyg 8.07 ± 20.49 10.06 ± 0.61 - 8.07 ± 4.22 - - X Cyg - - - - - - W Gem - - - - - - RR Lac - - - - - - ... ... ... ... ... ... ... MNRAS 000, 1-15 (2025)"}}},{"rowIdx":193,"cells":{"id":{"kind":"string","value":"2510.14787"},"raw":{"kind":"string","value":"A Human-Vector Susceptible–Infected–Susceptible Model for Analyzing\nand Controlling the Spread of Vector-Borne Diseases\nLorenzo Zino, Alessandro Casu, and Alessandro Rizzo\nAbstract— We propose an epidemic model for the spread\nof vector-borne diseases. The model, which is built extending\nthe classical susceptible–infected–susceptible model, accounts\nfor two populations —humans and vectors— and for cross-\ncontagion between the two species, whereby humans become in-\nfected upon interaction with carrier vectors, and vectors become\ncarriers after interaction with infected humans. We formulate\nthe model as a system of ordinary differential equations and\nleverage monotone systems theory to rigorously characterize\nthe epidemic dynamics. Specifically, we characterize the global\nasymptotic behavior of the disease, determining conditions for\nquick eradication of the disease (i.e., for which all trajectories\nconverge to a disease-free equilibrium), or convergence to\na (unique) endemic equilibrium. Then, we incorporate two\ncontrol actions: namely, vector control and incentives to adopt\nprotection measures. Using the derived mathematical tools, we\nassess the impact of these two control actions and determine\nthe optimal control policy.\nI. INTRODUCTION\nIn the last decade, mathematical models of epidemic\ndiseases have gained traction within the systems and control\ncommunity [1]–[5]. In fact, the development of increas-\ningly refined models has allowed to accurately predict the\ncourse of an epidemic outbreak and, ultimately, to design\nand assess intervention policies [1], [4], [6]. In particular,\nthe latest epidemiological threats, such as the outbreaks of\nEbola, COVID-19, and seasonal flu have provided further\nmotivation to pursue these studies, yielding tailored versions\nof these general epidemic models [7]–[11].\nTypical modeling setups deal with human-to-human conta-\ngion mechanisms [1]–[5]. However, according to the World\nHealth Organization, more than 17% of all infectious dis-\neases are vector-borne [12]. This means that they are not\ntransmitted through human-to-human interactions, but by\narthropod vectors (such as mosquitoes, fleas, or ticks) that\ncan carry pathogens and transmit them to humans [12].\nVector-borne diseases (including dengue, malaria, and West\nNile fever) pose a significant threat to our society, being\ncausing more than 700,000 deaths annually [12]. Moreover,\nthe ongoing climate change crisis exacerbates concerns on\nthe prevention of these diseases, as vectors adapt to new habi-\ntats [13]. This is the case, e.g., of Aedes aegypti (responsible\nfor the transmission of several diseases, including dengue,\nZika, and yellow fever), which is predicted to infest many\nregions of Europe if the temperature increases by 2°C [14].\nThe authors are with the Department of Electronics and Telecommunica-\ntions, Politecnico di Torino, Torino, Italy (lorenzo.zino@polito.it,\nalessandro.casu@studenti.polito.it,\nalessandro.rizzo@polito.it).\nNumerous mathematical models of vector-borne diseases\nhave been proposed and studied, particularly in response to\nthe increasing concern for dengue fever, leading to a rich\nbody of research [15], [16]. However, most of these models,\ndeveloped by computational epidemiologists as complex\nsimulation tools, offer limited analytical tractability [17].\nConversely, there is a scarcity of parsimonious models that\nefficiently balance accuracy and interpretability.\nHere, we fill in this gap by proposing a novel mathematical\nmodel for vector-borne diseases. Our model, grounded in\ndynamical systems theory, considers two interacting popula-\ntions of humans and vectors. Through such interactions, the\npathogen is transmitted from carrier vectors to susceptible\nhumans and from infectious humans to vectors, establishing\na positive feedback loop of contagion. Formally, we cast our\nmodel as a system of nonlinear ordinary differential equa-\ntions (ODEs), in which we couple i) an epidemic model for\nhumans, inspired by the Susceptible–Infected–Susceptible\n(SIS) model [2], ii) a contagion model for vectors, inspired\nby the Susceptible–Infected (SI) model [2], and iii) a vital\ndynamics for vectors, which is modeled using a birth-death\nprocess [18]. We refer to the model obtained as the human-\nvector SIS (HV-SIS) epidemic model.\nIn addition to the formulation of the model, the main\ncontribution of this paper is twofold. First, by leveraging\nmonotone dynamical systems theory [19], we perform a\nthorough analysis of the asymptotic behavior of the HV-SIS\nmodel, characterizing two regimes: one where the epidemic\noutbreak is quickly eradicated, leading to global convergence\nto a disease-free equilibrium; and one where the disease\nbecomes endemic, and the system converges to a (unique) en-\ndemic equilibrium. Second, we introduce two control actions:\nnamely, vector control —which focuses on reducing the\nvector population of vectors (e.g., using pesticides) [20]—\nand the use of personal protection measures against conta-\ngion [21]. By studying the controlled HV-SIS model and\nformulating an optimization problem, we investigate the\noptimal control policies to prevent outbreaks of vector-borne\ndiseases, as a function of the model parameters and the cost\nassociated with implementing interventions.\nII. HUMAN-VECTOR SIS EPIDEMIC MODEL\nWe consider a large population of humans that interact\nwith a population of vectors. Similar to most epidemic\nmodels [2], we observe that the duration of an epidemic\noutbreak is typically negligible with respect to the life-span\nof humans. Hence, we approximate the size of the human\npopulation as constant. Moreover, being the population large,\narXiv:2510.14787v1 [eess.SY] 16 Oct 2025\n\nhuman dynamics\nvector dynamics\nS\nI\nN\nC\nγ\nµ\nω\nµ\nβh\nβv\nvector dynamics\nvector dynamics\nFig. 1: Schematic of the human-vector epidemic model. Solid arrows\nrepresent possible transitions of the state of humans (S and I for susceptible\nand infected, respectively) and vectors (N and C for non-carrier and carrier,\nrespectively). Dashed arrows are associated with the vital dynamics of\nvectors. Dotted colored arrows indicate transitions that are triggered by\ninteractions with humans or vectors with a specific state.\nwe approximate it as a continuum of individuals with total\nmass equal to 1 [2]. On the contrary, the life-span of a\nvector is typically comparable with the infection propagation\ndynamics [22]. Hence, we assume that the total quantity of\nvectors v(t) ≥0 (normalized with respect to the unit mass\nhuman population) evolves in continuous-time t ≥0 accord-\ning to a classical ODE associated with a birth-death process,\ntypically used in mathematical biological models [18]:\n˙v(t) = ω −µv(t),\n(1)\nwhere ω > 0 and µ > 0 are two constants representing the\nbirth and death rate, respectively.\nHumans can be healthy and susceptible to the disease or\ninfected with the disease. We assume that there is no natural\nimmunity: after recovery, individuals are again susceptible\nto the disease. This is a good proxy for many diseases, e.g.,\ndengue fever, for which natural immunity wanes quickly\nand it only protects against the virus serotype specific of\nthe previous infection [23]. We denote by x(t) ∈[0, 1]\nand s(t) ∈[0, 1] the fraction of infected individuals and\nsusceptible individuals at time t ≥0, respectively. Since\nthere is no immunity, it holds s(t) = 1 −x(t). Similarly,\nvectors can be either carriers of the pathogen or non-carriers.\nWe denote by y(t) ≥0 the number of non-carrier vectors\nand by z(t) ≥0 the quantity of carrier vectors. Being v(t)\nthe total quantity of vectors at time t, then y(t)+z(t) = v(t).\nWe assume that the two populations are well-mixed,\nand we define a human-vector compartmental model that\ndescribes the evolution of the fraction of susceptible and\ninfected individuals and the quantity of carriers and non-\ncarriers in the two populations. The compartmental model,\nillustrated in Fig. 1, yields the following 3-dimensional\nsystem of nonlinear ODEs:\n˙x(t) = −γx(t) + βh(1 −x(t))z(t)\n(2a)\n˙y(t) = ω −µy(t) −βvx(t)y(t)\n(2b)\n˙z(t) = βvx(t)y(t) −µz(t),\n(2c)\nwith initial condition in the domain D := {(x, y, z) :\nx, y, z ≥0, x ≤1}. In the following paragraphs, we\nextensively discuss these equations.\nIn Eq. (2a), the fraction of infected individuals evolves ac-\ncording to two contrasting mechanisms: the negative contri-\nbution −γx(t) accounts for infected individuals who recover\nat a rate γ > 0; the positive contribution βh(1 −x(t))z(t)\naccounts for new infections, whose number is proportional to\nthe quantity of susceptible humans, the quantity of carriers,\nand a parameter βh > 0 that captures the human contagion\nrate (i.e., the likelihood that the pathogen is transmitted from\na carrier vector to a human through a human-vector inter-\naction). This equation resembles the classical SIS epidemic\nmodel [2], but here new contagions, instead of being propor-\ntional to the quantity of infected humans, are proportional to\nthe quantity of carriers. For this reason, we shall refer to the\nmodel with dynamics in Eq. (2) and initial condition in D as\nthe human-vector SIS model, abbreviated as HV-SIS model.\nThe other two equations, Eqs. (2b)–(2c), govern the dy-\nnamics of vectors. In particular, the term βvx(t)y(t) captures\nnew carriers and gives a positive contribution to the dynamics\nof carriers and a negative contribution to non-carriers. This\nterm is proportional to the number of non-carrier vectors,\ninfected humans, and a parameter βv ≥0 that captures the\nvector contagion rate (i.e., the likeliness that the pathogen\nis transmitted from a human to a vector through a human-\nvector interaction). The other two terms come from Eq. (1):\nnew born vectors are not carriers of the pathogen (so the rate\nω appears in Eq. (2b)), while the death rate is independent of\nthe pathogen, since vectors are only carriers and not infected\nwith the disease, leading to the terms −µy(t) and −µz(t),\nrespectively.\nIII. MAIN RESULTS ON THE HV-SIS EPIDEMIC MODEL\nIn this section, we present our main results on the analysis\nof the HV-SIS epidemic model. First, we prove that the\nequations are well-defined.\nLemma\n1. The domain of the HV-SIS model D\n:=\n{(x, y, z) : x, y, z ≥0, x ≤1} can be split into two domains\nD1 := {(x, y, z) : x, y, z ≥0, x ≤1, y + z ≤\nω\nµ} and\nD2 := {(x, y, z) : x, y, z ≥0, x ≤1, y + z ≥ω\nµ}, which are\npositive invariant under Eq. (2).\nProof. The domain D is closed and convex and the vector\nfield in Eq. (2) is Lipschitz-continuous. Hence, Nagumo’s\nTheorem can be applied [24]. We need to verify that the\nvector field at the boundaries of the domain does not point\ntowards the boundary. We immediately observe that, if any of\nthe variables is equal to 0, then the corresponding derivative\nis always non-negative (hence, it does not point towards the\nboundary). Similarly, at x = 1, we get that Eq. (2a) is\nalways non-positive. Finally, when y+z = ω\nµ, from summing\nEq. (2b) and Eq. (2c) we get ˙y + ˙z = 0. Hence, the vector\nfield does not point towards any boundary, yielding the first\nclaim. The second claim follows the same arguments, where\nwe observe that it always holds that ˙y + ˙z ≤0 in D2.\nThen, we provide a complete characterization of the\nasymptotic behavior of the HV-SIS model, determining its\nequilibria. Specifically, we will prove that, depending on the\n\nmodel parameters, there is always one equilibrium that is\n(almost) globally asymptotically stable, characterizing two\ndistinct regimes: either the disease is eradicated and all\ntrajectories converge to a disease-free equilibrium (DFE), or\nthe disease becomes endemic and (almost) all trajectories\nconverge to an endemic equilibrium (EE), where a fraction\nof the population is infected (and a fraction of the vectors\nare carriers). The phase transition between these two regimes,\nwhich is a typical phenomenon of many epidemic models [2],\n[4], is shaped by the value of the model parameters that\ndetermine the so-called epidemic threshold [4]. We start\nour analysis by determining the equilibria of Eq. (2) and\ndetermining their (local) stability.\nProposition 1. The HV-SIS model in Eq. (2) has, at most\ntwo equilibria: i) The DFE\n(x∗, y∗, z∗) =\n\u0010\n0, ω\nµ, 0\n\u0011\n,\n(3)\nand ii) the EE\n(¯x, ¯y, ¯z) =\n\u0010 ωβhβv −µ2γ\nωβhβv + µγβv\n, γµ + βhω\nβh(βv + µ), ωβhβv −µ2γ\nµβhβv + µ2βh\n\u0011\n.\n(4)\nSpecifically, let us define the epidemic threshold\nσ0 := βhβvω\nγµ2 .\n(5)\nThe DFE in Eq. (3) always exists and is locally exponentially\nstable if σ0 < 1 and unstable if σ0 > 1. The EE in Eq. (4)\nexists and is distinct from the DFE if and only if σ0 > 1 and\n(if it exists) it is always locally exponentially stable.\nProof. First, we compute the equilibria of Eq. (2) by equat-\ning the right hand sides to 0, obtaining a system of three\nnonlinear equations, which yields the two solutions in Eq. (3)\nand Eq. (4). Then, we observe that the DFE is always in the\ndomain D. On the contrary, the EE is in the domain D if\nand only if the numerators of ¯x and ¯z are non-negative, i.e.,\nif ωβhβv −µ2γ ≥0, which yield the condition ωβhβv\nµ2γ\n≥1.\nFinally, we observe that, when ωβhβv\nµ2γ\n= 1, the DFE and the\nEE coincide, yielding the strict inequality for the existence\nof a second equilibrium of Eq. (2).\nAt this stage, we compute the Jacobian matrix of Eq. (2)\nin a generic point (x, y, z), that is,\nJ(x, y, z) =\n\n\n−γ −βhx\n0\nβh(1 −x)\n−βvy\n−µ −βvx\n0\nβvy\nβvx\n−µ\n\n.\n(6)\nBy evaluating Eq. (6) at the DFE in Eq. (3), we get\nJ(x∗, y∗, z∗) =\n\n\n−γ\n0\nβh\n−βv ω\nµ\n−µ\n0\nβv ω\nµ\n0\n−µ\n\n,\n(7)\nwhose eigenvalues are λ1 = −µ and λ2 =\n1\n2µ(−γµ −\nµ2 −√µ\np\nγ2µ −2γµ2 + µ3 + 4βhβvω), whose real parts\nare\nalways\nnegative,\nand\nλ3\n=\n1\n2µ(−γµ −µ2 +\np\nγ2µ2 −2γµ3 + µ4 + 4βhβvωµ), which is negative if and\nonly if γ2µ2 −2γµ3 + µ4 + 4βhβvωµ < (γµ + µ2)2, which\nsimplifies to the condition σ0 =\nβvβhω\nγµ2\n< 1. Hence the\nDFE is locally exponentially stable if σ0 < 1 and unstable\nif σ0 > 1.\nSimilarly, we evaluate the Jacobian matrix in Eq. (6) at\nthe EE in Eq. (4), obtaining\n\n\n−γµβv+ωβvβh\nµ(µ+βv)\n0\nµγβvβh+µ2γβh\nωβvβh+µγβv\n−βvγµ+βvβhω\nβh(βv+µ)\n−µ −ωβvβh−µ2γ\nωβh+µγ\n0\nβvγµ+βvβhω\nβh(βv+µ)\nωβvβh−µ2γ\nωβh+µγ\n−µ\n\n,\n(8)\nwhose eigenvalues are λ1 = −µ, which is always negative,\nand another pair of eigenvalues, which are not reported due\nto their cumbersome expression. Again, by imposing that\nthe largest of the two has negative real part, we obtain a\ncomplicated condition which can be simplified to σ0 > 1,\nwhere computations are omitted due to space constraints.\nRemark 1. From the expression of the epidemic threshold\nin Eq. (5), we observe that, as predictable, increasing the\ninfection rates βh and βv favors the spread of the disease.\nA similar effect is observed by increasing the vector birth\nrate ω. On the other hand, increasing the vector death rate\nµ and/or the human recovery rate γ favors the eradication\nof the disease. Interestingly, the vector death rate µ has a\nlarger impact, since it appears squared at the denominator,\nsuggesting that vector control is a potentially effective strat-\negy to avoid outbreaks of vector-borne diseases.\nProposition 1 characterizes the local behavior of Eq. (2)\nabout the two equilibria of the system. In order to prove\nglobal convergence, we now leverage monotone systems\ntheory [19]. However, since the Jacobian of Eq. (2) in Eq. (6)\nis evidently not a Metzler matrix, we cannot directly apply\nthe monotone systems theory to Eq. (2), and we need to\nintroduce a change of variables, as detailed in the proof of\nthe following result.\nTheorem 1. Let σ0 be the epidemic threshold from Eq. (5).\nIf σ0 ≤1, then all trajectories of the HV-SIS model in\nEq. (2) converge to the DFE in Eq. (3). If σ0 > 1, then\nall trajectories with initial condition such that x(0) ̸= 0 or\nz(0) ̸= 0 converge to the EE in Eq. (4).\nProof. We operate a change of variables, where we introduce\nan auxiliary 3-dimensional system formed by x(t), z(t), and\nv(t) = y(t) + z(t), which is governed by Eq. (1). Being\ny(t) = v(t) −z(t), we obtain\n˙x(t) = −γx(t) + βh(1 −x(t))z(t)\n(9a)\n˙z(t) = βvx(t)(v(t) −z(t)) −µz(t)\n(9b)\n˙v(t) = ω −µv(t).\n(9c)\nFirst, from Lemma 1, we derive that the two invariant sets,\nwritten in terms of the new variables are D1 := {(x, z, v) :\nx, z, v ≥0, x ≤1, v ≤ω\nµ, z ≤v} and D2 := {(x, z, v) :\nx, z, v ≥0, x ≤1, v ≥\nω\nµ, z ≤v}. Second, we prove\nthat all trajectories in D2 are bounded (those in D1 are\nnecessarily bounded, being D1 compact). From Eq. (9c),\nwe observe that ˙v(t) ≤0 for any v(t) ≥\nω\nµ, which\n\n0\n20\n40\n0\n0.02\n0.04\n0.06\n0.08\ntime, t\nPrevalence\nx(t)\n0\n20\n40\n0\n0.5\n1\ntime, t\ny(t)\nz(t)\n(a) µ = 0.2\n0\n50\n100\n0\n0.1\n0.2\ntime, t\nPrevalence\nx(t)\n0\n50\n100\n0\n1\n2\ntime, t\ny(t)\nz(t)\n(b) µ = 0.1\nFig. 2: Trajectories of the human-vector epidemic model. In (a) σ0 < 1\nand the trajectory converges to the DFE; in (b) σ0 > 1 and the trajectory\nconverges to the EE. The equilibrium predicted by Theorem 1 in the two\ncases is depicted with gray dashed horizontal lines. Common parameters\nare ω = βh = βv = 0.2, and γ = 0.4.\nimplies v(t) ≤max{v(0), ω/µ}, which in turn implies that\ntrajectories cannot diverge. Third, we compute the Jacobian\nmatrix of Eq. (9) at the generic point (x, z, v), obtaining\n˜J(x, z, v) =\n\n\n−γ −βhz\nβh(1 −x)\n0\nβv(v −z)\n−βvx −µ\nβvx\n0\n0\n−µ\n\n.\n(10)\nWe observe that the matrix in Eq. (10) is Metzler, since all\nits off-diagonal entries are non-negative for values of the\nparameters belonging to the two invariant sets. Hence, the\ndynamical system in Eq. (9) is monotone [19], which implies\nthat all its trajectories converge to a fixed point [19]. Fourth,\nsince v(t) and z(t) converge, also their difference y(t) =\nz(t) −x(t) necessarily converges to a fixed point, yielding\nthat also trajectories of Eq. (2) converge. Fifth, the analysis of\nthe local stability of the equilibria in Proposition 1 yields the\nclaim for σ0 < 1 and σ0 > 1. Finally, we observe that when\nσ0 = 1, Proposition 1 does not provide any information on\nthe stability of the equilibria, but it states that the system\nhas a unique equilibrium: the DFE. Combining this with\nthe system’s monotonicity (which implies convergence to an\nequilibrium), we obtain convergence to the DFE also in the\ncase σ0 = 1, yielding the claim.\nFigure 2 illustrates the results of Theorem 1. If σ0 <\n1, then the disease is quickly eradicated and the system\nconverges to the DFE. It is interesting to notice from Fig. 2a\nthat, unlike classical SIS-models [2], the convergence to the\nDFE may be non-monotone, with an initial increase in the\nepidemic prevalence x(t), until a peak is reached and the\nfraction of infected individuals starts decreasing to 0. On the\ncontrary, if we increase σ0 to reach a value larger than 1\n(e.g., by decreasing µ as in Fig. 2b), we enter the endemic\nregime and trajectories converge to the EE.\nIV. CONTROL OF THE HV-SIS MODEL\nIn this section, we consider two distinct control actions\nthat can be implemented in the prevention of vector-borne\ndiseases, and we encapsulate them within the dynamical\nsystem in Eq. (2) by introducing two additional terms that\ncapture these actions:\n• Vector control interventions, which focuses on reduc-\ning the vector population (e.g., using insecticide-based\ntools or integrated pest management) [20];\n• Incentives to adopt personal protection measures, such\nas promoting the use of insect repellent, wear long\nclothing, and limit outdoor activity [21],\nBoth these interventions have been proven effective in pre-\nventing vector-borne diseases [20], [21]. In the following,\nwe use the mathematical model developed in Section II to\nanalytically assess the effectiveness of these control actions\nand use them to design an optimal control strategy.\nIn order to incorporate vector control in the model, we\nintroduce a control parameter u1 ≥0 that captures the\nefficacy of this control action. In particular, we assume that\nthe death rate of the vector is increased by a parameter u1\nthanks to the impact of vector control actions. Then, we\nintroduce a control parameter u2 ∈[0, βh] that represents\nthe efficacy of personal protection measures in reducing\nhuman contagion rate. Hence, the controlled HV-SIS model\nis captured by the following system of ODEs:\n˙x(t) = −γx(t) + (βh −u2)(1 −x(t))z(t)\n(11a)\n˙y(t) = ω −(µ + u1)y(t) −βvx(t)y(t)\n(11b)\n˙z(t) = βvx(t)y(t) −(µ + u1)z(t).\n(11c)\nRepeating the same analysis performed for the uncon-\ntrolled model in Section III, we obtain the following result,\nwhose proof is a corollary of Proposition 1 and Theorem 1.\nCorollary 1. Let\nσc := (βh −u2)βvω\nγ(µ + u1)2 .\n(12)\nIf σc ≤1, then all trajectories of the controlled HV-SIS\nmodel in Eq. (11) converge to the DFE\n(x∗\nc, y∗\nc, z∗\nc) =\n\u0010\n0,\nω\nµ + u1\n, 0\n\u0011\n.\n(13)\nIf σc > 1, then all trajectories of the controlled HV-SIS\nmodel in Eq. (11) with initial condition such that x(0) ̸= 0\nor z(0) ̸= 0 converge to the EE\n¯xc = ωβv(βh −u2) −(µ + u1)2γ\nωβv(βh −u2) + (µ + u1)γβv\n(14a)\n¯yc = γ(µ + u1) + (βh −u2)ω\n(βh −u2)(βv + µ + u1)\n(14b)\n¯zc =\nωβv(βh −u2) −(µ + u1)2γ\n(µ + u1)(βh −u2)βv + (µ + u1)2(βh −u2). (14c)\nThe expressions derived in Corollary 1 can be used to\nassess the performance of the two different control strategies.\nIn Fig. 3, we report the value of the fraction of infected\n\n0\n0.02\n0.04\n0\n0.02\n0.04\n0.06\n0.08\n0.1\nvector control, u1\nprotection measures, u2\n0\n0.1\n0.2\n(a) Infected individuals, ¯xc\n0\n0.02\n0.04\n0\n0.02\n0.04\n0.06\n0.08\n0.1\nvector control, u1\n0\n0.2\n0.4\n0.6\n(b) Carrier vectors, ¯zc\nFig. 3: (a) Infected individuals and (b) carrier vectors at the EE for different\nvalues of the control inputs u1 and u2. Model parameters are ω = βh =\nβv = 0.2, γ = 0.4, and µ = 0.1.\nindividuals at the EE in Eq. (14) for different values of the\ncontrol inputs u1 and u2. The figure suggests that vector\ncontrol is more effective not only in reducing the epidemic\nthreshold (as observed in Remark 1), but also in reducing\nthe fraction of infected individuals at the EE. In fact, from\nFig. 3a, we observe that when u1 = 0.01, which means an\nincrease in the vector death rate by just 10%, the prevalence\nat the EE decreases by more than 25%. To obtain the same\nresults using only protective measures, one needs to have u2\nthat reduces the human infection rate by almost 20%. The\nsame relation is observed for the vectors in Fig. 3b.\nFrom Corollary 1 and the following discussion, a problem\nspontaneously arises. How can one design an optimal control\npolicy in terms of vector control and protection measures to\nachieve eradication of the outbreak, minimizing the cost of\nthe control strategy? Formally, we can define a cost function\nC(u1, u2), associated with implementing level u1 and u2\nof vector control and protection measures, respectively, for\nwhich it is reasonable to make the following assumptions.\nAssumption 1. The cost function C(u1, u2) : [0, ∞) ×\n[0, βh] →[0, ∞) is a non-negative differentiable function\nand it is monotonically increasing in u1 and u2.\nThen, we formulate the following optimization problem:\n(u∗\n1, u∗\n2) = arg min\nC(u1, u2)\nsubject to\n(βh −u2)βvω −γ(µ + u1)2 ≤0\nu1, u2 ≥0,\nu2 ≤βh,\n(15)\nwhere constraint (βh −u2)βvω −γ(µ+u1)2 ≤0 is obtained\nfrom Eq. (12), by imposing that the DFE is globally asymp-\ntotically stable, i.e., imposing σc ≤1. From the analysis of\nthe optimization problem in Eq. (15), we obtain the following\nresult, which provides an explicit way to compute the optimal\ncontrol policy for the HV-SIS epidemic model.\nTheorem 2. Under Assumption 1, the optimal solution\n(u∗\n1, u∗\n2) of Eq. (15) solves the following system of equations:\n∂\n∂u1 C(u1, u2) −2λγ(µ + u1) = 0\n∂\n∂u2 C(u1, u2) −λβvω = 0\n(βh −u2)βvω −γ(µ + u1)2 = 0.\n(16)\nProof. First, we observe that the problem is always feasible.\nIn fact, u1 = 0 and u2 = βh is a solution that satisfies all the\nconstraints. Then, we prove that the minimum of Eq. (15) is\nattained for values of the control inputs u1 and u2 that either\nare both equal to 0, or they satisfy the equality constraint\n(βh−u2)βvω−γ(µ+u1)2 = 0. To prove this statement, let us\ndefine g(u1, u2) = (βh−u2)βvω−γ(µ+u1)2. If g(0, 0) ≤0\n(which is equivalent to σ0 ≤1), then the monotonicity of C\nimplies that the minimum is attained at u∗\n1 = u∗\n2 = 0. If\ng(0, 0) > 0, assume that (u∗\n1, u∗\n2) is the optimal solution of\nEq. (15) and that g(u∗\n1, u∗\n2) < 0. The cost function at the\noptimal solution is equal to C(u∗\n1, u∗\n2). If u∗\n1 > 0, we can\ndefine ˜u∗\n1(ζ) = u∗\n1 −ζ. By continuity, being g(u∗\n1, u∗\n2) < 0\nthere exists ∆u > 0 such that g(˜u∗\n1(∆u), u∗\n2) ≤0. Clearly\n(˜u∗\n1(∆u), u∗\n2) is then a feasible solution of Eq. (15), and\nC(˜u∗\n1(∆u), u∗\n2) < C(u1, u∗\n2), due to the monotonicity of the\ncost function, which contradicts the assumption that (u∗\n1, u∗\n2)\nis the optimal solution of Eq. (15), yielding the claim. If\nu∗\n1 = 0, the same argument holds letting ˜u∗\n2(ζ) = u∗\n2 −ζ.\nOnce we know that the optimal solution is attained at\nthe boundary g(u1, u2) = 0, we use Lagrange multipliers\nto solve the optimization problem [25], by writing the\nLagrangian function\nL(u1, u2, λ) = C(u1, u2) + λg(u1, u2)\n= C(u1, u2) + λ((βh −u2)βvω −γ(µ + u1)2).\n(17)\nFinally, a necessary condition for optimality is that the solu-\ntion should solve the nonlinear system obtained by posing the\npartial derivatives of Eq. (17) to 0 [25], yielding Eq. (16)\nIn general, Eq. (16) can have multiple solutions, and thus\nbeing a solution of Eq. (16) is only a necessary condition for\noptimality. However, in the special case in which the cost\nfunction is linear, we can derive a close-form expression for\nthe optimal solution of the control problem in Eq. (15).\nCorollary 2. Assume that C(u1, u2) = c1u1 + c2u2, where\nc1 > 0 and c2 > are positive constants that weight the\ncost for implementing vector control and personal protection\nmeasures, respectively. Then, the optimal solution (u∗\n1, u∗\n2) of\nEq. (15) is given by\nu∗\n1 =\n( q\nβhβvω\nγ\n−µ\nif σ0 > 1 and (c1, c2) /∈C\n0\notherwise,\n(18a)\nu∗\n2 =\n(\nβhβvω−γµ2\nβvω\nif σ0 > 1 and (c1, c2) ∈C\n0\notherwise,\n(18b)\nwhere\nC :=\n\u001a\n(c1, c2) : c1\nc2\n> 2γµ\nβvω ,\n\u0010c1\nc2\nβvω −γµ\n\u00112\n≥βhβvγω\n\u001b\n.\n(19)\nProof. Eq. (18) is obtained as the unique solution of Eq. (16)\nfor C(u1, u2) = c1u1 + c2u2.\nThe results in Corollary 2 suggest that, if the cost for\nimplementing intervention policies grows linearly in the\neffectiveness of the intervention, then it is always beneficial\nto focus on implementing only one of the two types of\npolicies. Which policy to implement depends on the model\n\n0\n2\n4\n0\n2\n4\nvector control, c1\nprotections, c2\n(a) βh = 0.4, βv = 0.1\n0\n2\n4\n0\n2\n4\nvector control, c1\nprotections, c2\n(b) βh = 0.1, βv = 0.4\nFig. 4: In the blue area, vector control is preferable; in the yellow area,\npersonal protection measures are preferable, according to Corollary 2\nCommon model parameters are ω = 0.2, γ = 0.4, and µ = 0.1.\nparameters and on the ratio between the cost for imple-\nmenting the two control actions, as illustrated in Fig. 4.\nNote that, as the vector infection rate βv becomes larger,\nthe region in which incentivizing protection measures is\npreferable becomes larger. However, even when βv is four\ntimes larger than βh, the region in which vector control is\nmore effective is larger.\nV. CONCLUSION\nIn this paper, we have proposed and analyzed a novel\nmodel for the spread of vector-borne diseases. The model,\nbuilt using a system of ODEs, accounts for human and\nvector contagion, as well as for the vital dynamics of the\nvectors. Using systems theoretic tools, we have studied the\nasymptotic behavior of the system, characterizing a phase\ntransition between a regime where the DFE is globally\nasymptotically stable and a regime where the system con-\nverges to a (unique) EE. Then, by introducing two control\nactions in the equations, we have analytically assessed the\neffectiveness of vector control interventions and incentives\nfor humans to adopt protection measures.\nThese preliminary results pave the way for several lines\nof future research. First, the control strategies proposed in\nSection IV are assumed constant. Future research should\nfocus on designing dynamical control strategies, following\nthe approaches developed in [1], [4]. Moreover, despite\nTheorem 2 applies to quite general cost functions, we have\nfocused our discussion on the linear scenario, for which\na closed-form expression for the optimal control can be\neasily derived. Nonlinear cost functions that accounts, e.g.,\nfor diminishing returns in the effectiveness of interventions\nshould be investigated. Second, embedding the model on a\nnetwork structure is a key future step to gain insights into the\nimpact of geographical displacement of humans and vectors\non vector-borne diseases. Finally, the HV-SIS model can be\ncoupled with more realistic models of human behavior, e.g.,\nusing game theory [26]–[29], to develop a more realistic\nframework to study interventions.\nREFERENCES\n[1] C. Nowzari, V. M. Preciado, and G. J. 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Gupta, “Impacts of game-theoretic\nactivation on epidemic spread over dynamical networks,” SIAM J.\nControl Optim., vol. 60, no. 2, pp. S92–S118, 2022.\n[29] K. Paarporn and C. Eksin, “SIS epidemics coupled with evolutionary\nsocial distancing dynamics,” in 2023 Am. Control Conf., 2023, pp.\n4308–4313.\n"},"cleaned":{"kind":"string","value":"A Human-Vector Susceptible-Infected-Susceptible Model for Analyzing and Controlling the Spread of Vector-Borne Diseases Lorenzo Zino, Alessandro Casu, and Alessandro Rizzo Abstract- We propose an epidemic model for the spread of vector-borne diseases. The model, which is built extending the classical susceptible-infected-susceptible model, accounts for two populations -humans and vectors- and for crosscontagion between the two species, whereby humans become infected upon interaction with carrier vectors, and vectors become carriers after interaction with infected humans. We formulate the model as a system of ordinary differential equations and leverage monotone systems theory to rigorously characterize the epidemic dynamics. Specifically, we characterize the global asymptotic behavior of the disease, determining conditions for quick eradication of the disease (i.e., for which all trajectories converge to a disease-free equilibrium), or convergence to a (unique) endemic equilibrium. Then, we incorporate two control actions: namely, vector control and incentives to adopt protection measures. Using the derived mathematical tools, we assess the impact of these two control actions and determine the optimal control policy. I. INTRODUCTION In the last decade, mathematical models of epidemic diseases have gained traction within the systems and control community [1]-[5]. In fact, the development of increasingly refined models has allowed to accurately predict the course of an epidemic outbreak and, ultimately, to design and assess intervention policies [1], [4], [6]. In particular, the latest epidemiological threats, such as the outbreaks of Ebola, COVID-19, and seasonal flu have provided further motivation to pursue these studies, yielding tailored versions of these general epidemic models [7]-[11]. Typical modeling setups deal with human-to-human contagion mechanisms [1]-[5]. However, according to the World Health Organization, more than 17% of all infectious diseases are vector-borne [12]. This means that they are not transmitted through human-to-human interactions, but by arthropod vectors (such as mosquitoes, fleas, or ticks) that can carry pathogens and transmit them to humans [12]. Vector-borne diseases (including dengue, malaria, and West Nile fever) pose a significant threat to our society, being causing more than 700,000 deaths annually [12]. Moreover, the ongoing climate change crisis exacerbates concerns on the prevention of these diseases, as vectors adapt to new habitats [13]. This is the case, e.g., of Aedes aegypti (responsible for the transmission of several diseases, including dengue, Zika, and yellow fever), which is predicted to infest many regions of Europe if the temperature increases by 2°C [14]. The authors are with the - tions, Politecnico di Torino, Torino, Italy ( , , ). Numerous mathematical models of vector-borne diseases have been proposed and studied, particularly in response to the increasing concern for dengue fever, leading to a rich body of research [15], [16]. However, most of these models, developed by computational epidemiologists as complex simulation tools, offer limited analytical tractability [17]. Conversely, there is a scarcity of parsimonious models that efficiently balance accuracy and interpretability. Here, we fill in this gap by proposing a novel mathematical model for vector-borne diseases. Our model, grounded in dynamical systems theory, considers two interacting populations of humans and vectors. Through such interactions, the pathogen is transmitted from carrier vectors to susceptible humans and from infectious humans to vectors, establishing a positive feedback loop of contagion. Formally, we cast our model as a system of nonlinear ordinary differential equations (ODEs), in which we couple i) an epidemic model for humans, inspired by the Susceptible-Infected-Susceptible (SIS) model [2], ii) a contagion model for vectors, inspired by the Susceptible-Infected (SI) model [2], and iii) a vital dynamics for vectors, which is modeled using a birth-death process [18]. We refer to the model obtained as the humanvector SIS (HV-SIS) epidemic model. In addition to the formulation of the model, the main contribution of this paper is twofold. First, by leveraging monotone dynamical systems theory [19], we perform a thorough analysis of the asymptotic behavior of the HV-SIS model, characterizing two regimes: one where the epidemic outbreak is quickly eradicated, leading to global convergence to a disease-free equilibrium; and one where the disease becomes endemic, and the system converges to a (unique) endemic equilibrium. Second, we introduce two control actions: namely, vector control -which focuses on reducing the vector population of vectors (e.g., using pesticides) [20]- and the use of personal protection measures against contagion [21]. By studying the controlled HV-SIS model and formulating an optimization problem, we investigate the optimal control policies to prevent outbreaks of vector-borne diseases, as a function of the model parameters and the cost associated with implementing interventions. II. HUMAN-VECTOR SIS EPIDEMIC MODEL We consider a large population of humans that interact with a population of vectors. Similar to most epidemic models [2], we observe that the duration of an epidemic outbreak is typically negligible with respect to the life-span of humans. Hence, we approximate the size of the human population as constant. Moreover, being the population large, 16 Oct 2025 human dynamics vector dynamics S I N C γ μ ω μ βh βv vector dynamics vector dynamics Fig. 1: Schematic of the human-vector epidemic model. Solid arrows represent possible transitions of the state of humans (S and I for susceptible and infected, respectively) and vectors (N and C for non-carrier and carrier, respectively). Dashed arrows are associated with the vital dynamics of vectors. Dotted colored arrows indicate transitions that are triggered by interactions with humans or vectors with a specific state. we approximate it as a continuum of individuals with total mass equal to 1 [2]. On the contrary, the life-span of a vector is typically comparable with the infection propagation dynamics [22]. Hence, we assume that the total quantity of vectors v(t) ≥0 (normalized with respect to the unit mass human population) evolves in continuous-time t ≥0 according to a classical ODE associated with a birth-death process, typically used in mathematical biological models [18]: ̇v(t) = ω -μv(t), (1) where ω > 0 and μ > 0 are two constants representing the birth and death rate, respectively. Humans can be healthy and susceptible to the disease or infected with the disease. We assume that there is no natural immunity: after recovery, individuals are again susceptible to the disease. This is a good proxy for many diseases, e.g., dengue fever, for which natural immunity wanes quickly and it only protects against the virus serotype specific of the previous infection [23]. We denote by x(t) ∈[0, 1] and s(t) ∈[0, 1] the fraction of infected individuals and susceptible individuals at time t ≥0, respectively. Since there is no immunity, it holds s(t) = 1 -x(t). Similarly, vectors can be either carriers of the pathogen or non-carriers. We denote by y(t) ≥0 the number of non-carrier vectors and by z(t) ≥0 the quantity of carrier vectors. Being v(t) the total quantity of vectors at time t, then y(t)+z(t) = v(t). We assume that the two populations are well-mixed, and we define a human-vector compartmental model that describes the evolution of the fraction of susceptible and infected individuals and the quantity of carriers and noncarriers in the two populations. The compartmental model, illustrated in Fig. 1, yields the following 3-dimensional system of nonlinear ODEs: ̇x(t) = -γx(t) + βh(1 -x(t))z(t) (2a) ̇y(t) = ω -μy(t) -βvx(t)y(t) (2b) ̇z(t) = βvx(t)y(t) -μz(t), (2c) with initial condition in the domain D := {(x, y, z) : x, y, z ≥0, x ≤1}. In the following paragraphs, we extensively discuss these equations. In Eq. (2a), the fraction of infected individuals evolves according to two contrasting mechanisms: the negative contribution -γx(t) accounts for infected individuals who recover at a rate γ > 0; the positive contribution βh(1 -x(t))z(t) accounts for new infections, whose number is proportional to the quantity of susceptible humans, the quantity of carriers, and a parameter βh > 0 that captures the human contagion rate (i.e., the likelihood that the pathogen is transmitted from a carrier vector to a human through a human-vector interaction). This equation resembles the classical SIS epidemic model [2], but here new contagions, instead of being proportional to the quantity of infected humans, are proportional to the quantity of carriers. For this reason, we shall refer to the model with dynamics in Eq. (2) and initial condition in D as the human-vector SIS model, abbreviated as HV-SIS model. The other two equations, Eqs. (2b)-(2c), govern the dynamics of vectors. In particular, the term βvx(t)y(t) captures new carriers and gives a positive contribution to the dynamics of carriers and a negative contribution to non-carriers. This term is proportional to the number of non-carrier vectors, infected humans, and a parameter βv ≥0 that captures the vector contagion rate (i.e., the likeliness that the pathogen is transmitted from a human to a vector through a humanvector interaction). The other two terms come from Eq. (1): new born vectors are not carriers of the pathogen (so the rate ω appears in Eq. (2b)), while the death rate is independent of the pathogen, since vectors are only carriers and not infected with the disease, leading to the terms -μy(t) and -μz(t), respectively. III. MAIN RESULTS ON THE HV-SIS EPIDEMIC MODEL In this section, we present our main results on the analysis of the HV-SIS epidemic model. First, we prove that the equations are well-defined. Lemma 1. The domain of the HV-SIS model D := {(x, y, z) : x, y, z ≥0, x ≤1} can be split into two domains D1 := {(x, y, z) : x, y, z ≥0, x ≤1, y + z ≤ ω μ} and D2 := {(x, y, z) : x, y, z ≥0, x ≤1, y + z ≥ω μ}, which are positive invariant under Eq. (2). Proof. The domain D is closed and convex and the vector field in Eq. (2) is Lipschitz-continuous. Hence, Nagumo's Theorem can be applied [24]. We need to verify that the vector field at the boundaries of the domain does not point towards the boundary. We immediately observe that, if any of the variables is equal to 0, then the corresponding derivative is always non-negative (hence, it does not point towards the boundary). Similarly, at x = 1, we get that Eq. (2a) is always non-positive. Finally, when y+z = ω μ, from summing Eq. (2b) and Eq. (2c) we get ̇y + ̇z = 0. Hence, the vector field does not point towards any boundary, yielding the first claim. The second claim follows the same arguments, where we observe that it always holds that ̇y + ̇z ≤0 in D2. Then, we provide a complete characterization of the asymptotic behavior of the HV-SIS model, determining its equilibria. Specifically, we will prove that, depending on the model parameters, there is always one equilibrium that is (almost) globally asymptotically stable, characterizing two distinct regimes: either the disease is eradicated and all trajectories converge to a disease-free equilibrium (DFE), or the disease becomes endemic and (almost) all trajectories converge to an endemic equilibrium (EE), where a fraction of the population is infected (and a fraction of the vectors are carriers). The phase transition between these two regimes, which is a typical phenomenon of many epidemic models [2], [4], is shaped by the value of the model parameters that determine the so-called epidemic threshold [4]. We start our analysis by determining the equilibria of Eq. (2) and determining their (local) stability. Proposition 1. The HV-SIS model in Eq. (2) has, at most two equilibria: i) The DFE (x∗, y∗, z∗) = 0, ω μ, 0 , (3) and ii) the EE ( ̄x, ̄y, ̄z) = ωβhβv -μ2γ ωβhβv + μγβv , γμ + βhω βh(βv + μ), ωβhβv -μ2γ μβhβv + μ2βh . (4) Specifically, let us define the epidemic threshold σ0 := βhβvω γμ2 . (5) The DFE in Eq. (3) always exists and is locally exponentially stable if σ0 1. The EE in Eq. (4) exists and is distinct from the DFE if and only if σ0 > 1 and (if it exists) it is always locally exponentially stable. Proof. First, we compute the equilibria of Eq. (2) by equating the right hand sides to 0, obtaining a system of three nonlinear equations, which yields the two solutions in Eq. (3) and Eq. (4). Then, we observe that the DFE is always in the domain D. On the contrary, the EE is in the domain D if and only if the numerators of ̄x and ̄z are non-negative, i.e., if ωβhβv -μ2γ ≥0, which yield the condition ωβhβv μ2γ ≥1. Finally, we observe that, when ωβhβv μ2γ = 1, the DFE and the EE coincide, yielding the strict inequality for the existence of a second equilibrium of Eq. (2). At this stage, we compute the Jacobian matrix of Eq. (2) in a generic point (x, y, z), that is, J(x, y, z) = -γ -βhx 0 βh(1 -x) -βvy -μ -βvx 0 βvy βvx -μ . (6) By evaluating Eq. (6) at the DFE in Eq. (3), we get J(x∗, y∗, z∗) = -γ 0 βh -βv ω μ -μ 0 βv ω μ 0 -μ , (7) whose eigenvalues are λ1 = -μ and λ2 = 1 2μ(-γμ - μ2 -√μ p γ2μ -2γμ2 + μ3 + 4βhβvω), whose real parts are always negative, and λ3 = 1 2μ(-γμ -μ2 + p γ2μ2 -2γμ3 + μ4 + 4βhβvωμ), which is negative if and only if γ2μ2 -2γμ3 + μ4 + 4βhβvωμ 1. Similarly, we evaluate the Jacobian matrix in Eq. (6) at the EE in Eq. (4), obtaining -γμβv+ωβvβh μ(μ+βv) 0 μγβvβh+μ2γβh ωβvβh+μγβv -βvγμ+βvβhω βh(βv+μ) -μ -ωβvβh-μ2γ ωβh+μγ 0 βvγμ+βvβhω βh(βv+μ) ωβvβh-μ2γ ωβh+μγ -μ , (8) whose eigenvalues are λ1 = -μ, which is always negative, and another pair of eigenvalues, which are not reported due to their cumbersome expression. Again, by imposing that the largest of the two has negative real part, we obtain a complicated condition which can be simplified to σ0 > 1, where computations are omitted due to space constraints. Remark 1. From the expression of the epidemic threshold in Eq. (5), we observe that, as predictable, increasing the infection rates βh and βv favors the spread of the disease. A similar effect is observed by increasing the vector birth rate ω. On the other hand, increasing the vector death rate μ and/or the human recovery rate γ favors the eradication of the disease. Interestingly, the vector death rate μ has a larger impact, since it appears squared at the denominator, suggesting that vector control is a potentially effective strategy to avoid outbreaks of vector-borne diseases. Proposition 1 characterizes the local behavior of Eq. (2) about the two equilibria of the system. In order to prove global convergence, we now leverage monotone systems theory [19]. However, since the Jacobian of Eq. (2) in Eq. (6) is evidently not a Metzler matrix, we cannot directly apply the monotone systems theory to Eq. (2), and we need to introduce a change of variables, as detailed in the proof of the following result. Theorem 1. Let σ0 be the epidemic threshold from Eq. (5). If σ0 ≤1, then all trajectories of the HV-SIS model in Eq. (2) converge to the DFE in Eq. (3). If σ0 > 1, then all trajectories with initial condition such that x(0) ̸= 0 or z(0) ̸= 0 converge to the EE in Eq. (4). Proof. We operate a change of variables, where we introduce an auxiliary 3-dimensional system formed by x(t), z(t), and v(t) = y(t) + z(t), which is governed by Eq. (1). Being y(t) = v(t) -z(t), we obtain ̇x(t) = -γx(t) + βh(1 -x(t))z(t) (9a) ̇z(t) = βvx(t)(v(t) -z(t)) -μz(t) (9b) ̇v(t) = ω -μv(t). (9c) First, from Lemma 1, we derive that the two invariant sets, written in terms of the new variables are D1 := {(x, z, v) : x, z, v ≥0, x ≤1, v ≤ω μ, z ≤v} and D2 := {(x, z, v) : x, z, v ≥0, x ≤1, v ≥ ω μ, z ≤v}. Second, we prove that all trajectories in D2 are bounded (those in D1 are necessarily bounded, being D1 compact). From Eq. (9c), we observe that ̇v(t) ≤0 for any v(t) ≥ ω μ, which 0 20 40 0 0.02 0.04 0.06 0.08 time, t Prevalence x(t) 0 20 40 0 0.5 1 time, t y(t) z(t) (a) μ = 0.2 0 50 100 0 0.1 0.2 time, t Prevalence x(t) 0 50 100 0 1 2 time, t y(t) z(t) (b) μ = 0.1 Fig. 2: Trajectories of the human-vector epidemic model. In (a) σ0 1 and the trajectory converges to the EE. The equilibrium predicted by Theorem 1 in the two cases is depicted with gray dashed horizontal lines. Common parameters are ω = βh = βv = 0.2, and γ = 0.4. implies v(t) ≤max{v(0), ω/μ}, which in turn implies that trajectories cannot diverge. Third, we compute the Jacobian matrix of Eq. (9) at the generic point (x, z, v), obtaining ̃J(x, z, v) = -γ -βhz βh(1 -x) 0 βv(v -z) -βvx -μ βvx 0 0 -μ . (10) We observe that the matrix in Eq. (10) is Metzler, since all its off-diagonal entries are non-negative for values of the parameters belonging to the two invariant sets. Hence, the dynamical system in Eq. (9) is monotone [19], which implies that all its trajectories converge to a fixed point [19]. Fourth, since v(t) and z(t) converge, also their difference y(t) = z(t) -x(t) necessarily converges to a fixed point, yielding that also trajectories of Eq. (2) converge. Fifth, the analysis of the local stability of the equilibria in Proposition 1 yields the claim for σ0 1. Finally, we observe that when σ0 = 1, Proposition 1 does not provide any information on the stability of the equilibria, but it states that the system has a unique equilibrium: the DFE. Combining this with the system's monotonicity (which implies convergence to an equilibrium), we obtain convergence to the DFE also in the case σ0 = 1, yielding the claim. Figure 2 illustrates the results of Theorem 1. If σ0 1, then all trajectories of the controlled HV-SIS model in Eq. (11) with initial condition such that x(0) ̸= 0 or z(0) ̸= 0 converge to the EE ̄xc = ωβv(βh -u2) -(μ + u1)2γ ωβv(βh -u2) + (μ + u1)γβv (14a) ̄yc = γ(μ + u1) + (βh -u2)ω (βh -u2)(βv + μ + u1) (14b) ̄zc = ωβv(βh -u2) -(μ + u1)2γ (μ + u1)(βh -u2)βv + (μ + u1)2(βh -u2). (14c) The expressions derived in Corollary 1 can be used to assess the performance of the two different control strategies. In Fig. 3, we report the value of the fraction of infected 0 0.02 0.04 0 0.02 0.04 0.06 0.08 0.1 vector control, u1 protection measures, u2 0 0.1 0.2 (a) Infected individuals, ̄xc 0 0.02 0.04 0 0.02 0.04 0.06 0.08 0.1 vector control, u1 0 0.2 0.4 0.6 (b) Carrier vectors, ̄zc Fig. 3: (a) Infected individuals and (b) carrier vectors at the EE for different values of the control inputs u1 and u2. Model parameters are ω = βh = βv = 0.2, γ = 0.4, and μ = 0.1. individuals at the EE in Eq. (14) for different values of the control inputs u1 and u2. The figure suggests that vector control is more effective not only in reducing the epidemic threshold (as observed in Remark 1), but also in reducing the fraction of infected individuals at the EE. In fact, from Fig. 3a, we observe that when u1 = 0.01, which means an increase in the vector death rate by just 10%, the prevalence at the EE decreases by more than 25%. To obtain the same results using only protective measures, one needs to have u2 that reduces the human infection rate by almost 20%. The same relation is observed for the vectors in Fig. 3b. From Corollary 1 and the following discussion, a problem spontaneously arises. How can one design an optimal control policy in terms of vector control and protection measures to achieve eradication of the outbreak, minimizing the cost of the control strategy? Formally, we can define a cost function C(u1, u2), associated with implementing level u1 and u2 of vector control and protection measures, respectively, for which it is reasonable to make the following assumptions. Assumption 1. The cost function C(u1, u2) : [0, ∞) × [0, βh] →[0, ∞) is a non-negative differentiable function and it is monotonically increasing in u1 and u2. Then, we formulate the following optimization problem: (u∗ 1, u∗ 2) = arg min C(u1, u2) subject to (βh -u2)βvω -γ(μ + u1)2 ≤0 u1, u2 ≥0, u2 ≤βh, (15) where constraint (βh -u2)βvω -γ(μ+u1)2 ≤0 is obtained from Eq. (12), by imposing that the DFE is globally asymptotically stable, i.e., imposing σc ≤1. From the analysis of the optimization problem in Eq. (15), we obtain the following result, which provides an explicit way to compute the optimal control policy for the HV-SIS epidemic model. Theorem 2. Under Assumption 1, the optimal solution (u∗ 1, u∗ 2) of Eq. (15) solves the following system of equations: ∂ ∂u1 C(u1, u2) -2λγ(μ + u1) = 0 ∂ ∂u2 C(u1, u2) -λβvω = 0 (βh -u2)βvω -γ(μ + u1)2 = 0. (16) Proof. First, we observe that the problem is always feasible. In fact, u1 = 0 and u2 = βh is a solution that satisfies all the constraints. Then, we prove that the minimum of Eq. (15) is attained for values of the control inputs u1 and u2 that either are both equal to 0, or they satisfy the equality constraint (βh-u2)βvω-γ(μ+u1)2 = 0. To prove this statement, let us define g(u1, u2) = (βh-u2)βvω-γ(μ+u1)2. If g(0, 0) ≤0 (which is equivalent to σ0 ≤1), then the monotonicity of C implies that the minimum is attained at u∗ 1 = u∗ 2 = 0. If g(0, 0) > 0, assume that (u∗ 1, u∗ 2) is the optimal solution of Eq. (15) and that g(u∗ 1, u∗ 2) 0, we can define ̃u∗ 1(ζ) = u∗ 1 -ζ. By continuity, being g(u∗ 1, u∗ 2) 0 such that g( ̃u∗ 1(∆u), u∗ 2) ≤0. Clearly ( ̃u∗ 1(∆u), u∗ 2) is then a feasible solution of Eq. (15), and C( ̃u∗ 1(∆u), u∗ 2) 0 and c2 > are positive constants that weight the cost for implementing vector control and personal protection measures, respectively. Then, the optimal solution (u∗ 1, u∗ 2) of Eq. (15) is given by u∗ 1 = ( q βhβvω γ -μ if σ0 > 1 and (c1, c2) /∈C 0 otherwise, (18a) u∗ 2 = ( βhβvω-γμ2 βvω if σ0 > 1 and (c1, c2) ∈C 0 otherwise, (18b) where C := (c1, c2) : c1 c2 > 2γμ βvω , c1 c2 βvω -γμ 2 ≥βhβvγω . (19) Proof. Eq. (18) is obtained as the unique solution of Eq. (16) for C(u1, u2) = c1u1 + c2u2. The results in Corollary 2 suggest that, if the cost for implementing intervention policies grows linearly in the effectiveness of the intervention, then it is always beneficial to focus on implementing only one of the two types of policies. Which policy to implement depends on the model 0 2 4 0 2 4 vector control, c1 protections, c2 (a) βh = 0.4, βv = 0.1 0 2 4 0 2 4 vector control, c1 protections, c2 (b) βh = 0.1, βv = 0.4 Fig. 4: In the blue area, vector control is preferable; in the yellow area, personal protection measures are preferable, according to Corollary 2 Common model parameters are ω = 0.2, γ = 0.4, and μ = 0.1. parameters and on the ratio between the cost for implementing the two control actions, as illustrated in Fig. 4. Note that, as the vector infection rate βv becomes larger, the region in which incentivizing protection measures is preferable becomes larger. However, even when βv is four times larger than βh, the region in which vector control is more effective is larger. V. CONCLUSION In this paper, we have proposed and analyzed a novel model for the spread of vector-borne diseases. The model, built using a system of ODEs, accounts for human and vector contagion, as well as for the vital dynamics of the vectors. Using systems theoretic tools, we have studied the asymptotic behavior of the system, characterizing a phase transition between a regime where the DFE is globally asymptotically stable and a regime where the system converges to a (unique) EE. Then, by introducing two control actions in the equations, we have analytically assessed the effectiveness of vector control interventions and incentives for humans to adopt protection measures. These preliminary results pave the way for several lines of future research. First, the control strategies proposed in Section IV are assumed constant. Future research should focus on designing dynamical control strategies, following the approaches developed in [1], [4]. Moreover, despite Theorem 2 applies to quite general cost functions, we have focused our discussion on the linear scenario, for which a closed-form expression for the optimal control can be easily derived. Nonlinear cost functions that accounts, e.g., for diminishing returns in the effectiveness of interventions should be investigated. Second, embedding the model on a network structure is a key future step to gain insights into the impact of geographical displacement of humans and vectors on vector-borne diseases. Finally, the HV-SIS model can be coupled with more realistic models of human behavior, e.g., using game theory [26]-[29], to develop a more realistic framework to study interventions. REFERENCES [1] C. Nowzari, V. M. Preciado, and G. J. Pappas, \"Analysis and control of epidemics: a survey of spreading processes on complex networks,\" IEEE Control Syst. Mag., vol. 36, no. 1, pp. 26-46, 2016. [2] W. Mei, S. Mohagheghi, S. Zampieri, and F. 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Giordano et al., \"Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy,\" Nat. Med., vol. 26, pp. 855-860, 2020. [9] R. Carli, G. Cavone, N. Epicoco, P. Scarabaggio, and M. Dotoli, \"Model predictive control to mitigate the COVID-19 outbreak in a multi-region scenario,\" Annu. Rev. Control, vol. 50, pp. 373-393, 2020. [10] F. Parino, L. Zino, M. Porfiri, and A. Rizzo, \"Modelling and predicting the effect of social distancing and travel restrictions on COVID-19 spreading,\" J. R. Soc. Interface, vol. 18, no. 175, p. 20200875, 2021. [11] S. Fiandrino et al., \"Collaborative forecasting of influenza-like illness in Italy: The Influcast experience,\" Epidemics, vol. 50, p. 100819, 2025. [12] World Health Organization, \"Vector-borne diseases,\" https://www.who.int/news-room/fact-sheets/detail/vector-bornediseases, 2024. [13] J. Rockl ̈ov and R. Dubrow, \"Climate change: an enduring challenge for vector-borne disease prevention and control,\" Nat. 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Wilson et al., \"The importance of vector control for the control and elimination of vector-borne diseases,\" PLOS Negl. Trop. Dis., vol. 14, no. 1, p. e0007831, 2020. [21] J. D. Alpern, S. J. Dunlop, B. J. Dolan, W. M. Stauffer, and D. R. Boulware, \"Personal protection measures against mosquitoes, ticks, and other arthropods,\" Med. Clin. N. Am., vol. 100, no. 2, p. 303-316, 2016. [22] J. Lourenc ̧o and M. Recker, \"Viral and epidemiological determinants of the invasion dynamics of novel dengue genotypes,\" PLOS Negl. Trop. Dis., vol. 4, no. 11, p. e894, 2010. [23] World Health Organization, \"Dengue and severe dengue,\" https://www.who.int/health-topics/dengue-and-severe-dengue, 2024. [24] F. Blanchini, \"Set invariance in control,\" Automatica, vol. 35, no. 11, pp. 1747-1767, 1999. [25] D. P. Bertsekas, Nonlinear Programming. Belmont MA, US: Athena Scientific, 1995. [26] M. Ye, L. Zino, A. Rizzo, and M. Cao, \"Game-theoretic modeling of collective decision making during epidemics,\" Phys. Rev. E, vol. 104, no. 2, p. 024314, 2021. [27] K. Frieswijk, L. Zino, M. Ye, A. Rizzo, and M. Cao, \"A mean-field analysis of a network behavioral-epidemic model,\" IEEE Control Syst. Lett., vol. 6, pp. 2533-2538, 2022. [28] A. R. Hota, T. Sneh, and K. Gupta, \"Impacts of game-theoretic activation on epidemic spread over dynamical networks,\" SIAM J. Control Optim., vol. 60, no. 2, pp. S92-S118, 2022. [29] K. Paarporn and C. Eksin, \"SIS epidemics coupled with evolutionary social distancing dynamics,\" in 2023 Am. Control Conf., 2023, pp. 4308-4313."}}},{"rowIdx":194,"cells":{"id":{"kind":"string","value":"2510.14789"},"raw":{"kind":"string","value":"arXiv:2510.14789v1 [gr-qc] 16 Oct 2025\nQuantum confinement of scalar bosons in the Bonnor-Melvin spacetime: uniform magnetic\nfield and rainbow gravity effects\nOmar Mustafa∗\nDepartment of Physics, Eastern Mediterranean University,\n99628, G. Magusa, north Cyprus, Mersin 10 - T¨urkiye\nAbdullah Guvendi†\nDepartment of Basic Sciences, Erzurum Technical University, 25050, Erzurum, T¨urkiye\n(Dated: October 17, 2025)\nWe present an exact analytical study of Klein-Gordon (KG) scalar bosons and antibosons confined in the\nBonnor-Melvin (BM) spacetime under a uniform magnetic field, incorporating rainbow gravity (RG) correc-\ntions with a positive cosmological constant. The cosmological constant partitions spacetime into an infinite\nsequence of confinement domains bounded by impenetrable barriers. Within the first allowed domain, the\nKG equation reduces to a hypergeometric differential equation, yielding closed-form expressions for both the\nenergy spectra and the radial wavefunctions in terms of hypergeometric polynomials. Two representative RG\nmodels, inspired by the Magueijo-Smolin framework and loop quantum gravity (LQG), produce Planck-scale\nbounded, symmetric particle-antiparticle spectra. A distinctive feature of the curved magnetized geometry is\nthe collapse of all magnetic quantum states m ̸= 0 onto the m = 0 level for each radial excitation, a degener-\nacy absent in flat spacetime. Increasing the cosmological constant partially lifts this collapse, establishing a\ndirect link between the global spacetime curvature and the local quantum structure. Radial probability den-\nsity analysis further shows that stronger magnetic fields enhance spatial localization, confining bosons into\nstatic or rotating ring-like configurations with nodal architectures that evolve systematically with quantum\nnumbers. These findings reveal how gravitational confinement, topology, magnetic fields, and Planck-scale\ncorrections jointly govern the spectral and spatial properties of relativistic quantum fields in curved and\nmagnetized backgrounds.\nI.\nINTRODUCTION\nEinstein’s theory of general relativity (GR) [1] describes gravity as a\nmanifestation of the curvature of spacetime.\nThe attempt to unify\nGR with quantum mechanics, often referred to as quantum gravity\n(QG), offers fundamental insights into the behavior of quantum systems\nin curved backgrounds. This connection has motivated many studies\nexamining how spacetime curvature influences quantum spectroscopic\nproperties, especially for scalar and spinor particles described by the\nKG and Dirac equations, respectively [2]. The present study focuses\non KG particles in such settings.\nMagnetic fields are central to many astrophysical and cosmological phe-\nnomena, including stellar structures, accretion disks, galactic centers,\nmagnetars [3, 4], and heavy-ion collisions [5–7]. The simultaneous pres-\nence of strong gravitational and magnetic fields near compact objects\nhas led to considerable interest in general relativistic models that in-\ncorporate both effects.\nOne prominent example is the BM universe,\nan exact solution of the Einstein-Maxwell equations that describes a\nstatic spacetime with cylindrical symmetry, sourced by a magnetic field\naligned with the symmetry axis and self-consistently embedded in its\nown gravitational field [8–12]. In this model, the magnetic field con-\n∗omar.mustafa@emu.edu.tr (Corr. Author)\n† abdullah.guvendi@erzurum.edu.tr\ntributes to the energy-momentum tensor and thereby affects the geom-\netry of spacetime. A positive cosmological constant Λ > 0 is introduced\nto stabilize the spacetime and preserve its symmetry. The BM space-\ntime metric is given in [8] as\nds2 = −dt2 + dr2 + ˜σ2 sin2 \u0010√\n2Λr\n\u0011\ndϕ2 + dz2,\n(1)\nwhere z ∈[−∞, ∞], ϕ ∈[0, 2π], r ∈[0, ∞], and Λ in units of inverse\nlength squared. Following the coordinate transformation\n√\n2Λr →ρ\nand ˜σ\n√\n2Λϕ →φ, as in ˇZofka’s treatment [8], the line element (1)\nbecomes\nds2 = −dt2 + 1\n2Λ\n\u0000dρ2 + sin2 (ρ) dφ2\u0001 + dz2,\n(2)\nwhere H =\n1\n√\n2 sin(ρ) denotes the magnetic field [8].\nTopological defects such as domain walls are expected in various unified\nfield theories [13–16], and their effects on quantum dynamics have been\nexamined in [17–34]. The relationship between gravitational geometry\nand quantum systems remains a key motivation for studying such con-\nfigurations, and the BM spacetime provides a suitable model for this\npurpose [8–10, 35–38].\nIn the high-energy (ultraviolet) regime, the energy of a test particle can\ninfluence the structure of spacetime, leading to a modification of the\nstandard dispersion relation [39–45]:\nE2f0 (u)2 −p2f1(u)2 = m2\n◦.\n(3)\n\n2\nThis form appears in various QG theories, such as string theory, LQG,\nand non-commutative geometry [46–48], as well as in some recent re-\nsearch framed within RG, including: energy-momentum distributions\nof a cylindrical black hole in rainbow gravity [49]; generalized uncer-\ntainty principle corrections in Rastall-Rainbow Casimir wormholes [50];\nfermions moving around strong gravitational sources [51]; and the ex-\nploration of the physical properties of the strange star SAX J1808.4–\n3658 in rainbow gravity [52].\nConsequently, the BM metric (2) under RG becomes\nds2 = −\ndt2\nf0 (u)2 +\n1\nf1 (u)2\n\u0012 1\n2Λ\n\u0002dρ2 + sin2 (ρ) dφ2\u0003 + dz2\n\u0013\n,\n(4)\nwhere fi(u); i = 0, 1, are the so-called rainbow functions that obey the\ninfrared limit lim\nu→0 fi (u) = 1, and u = |E|/Ep satisfies 0 ≤u ≤1.\nThis ensures a consistent transition to the classical regime and an\nequal treatment of particles and antiparticles [45, 53–56]. The func-\ntions must preserve Ep, the Planck energy, as an invariant scale. We\nrefer to the metric in (4) as the BM-RG spacetime. In our recent stud-\nies [55, 56], we showed that the BM spacetime contains singularities\nat ρ =\n√\n2Λr = τ π, with τ = 0, 1, 2, · · ·.\nThese singularities corre-\nspond to infinite potential barriers, acting effectively as impenetrable\nwalls. The surfaces at r =\nτ π\n√\n2Λ do not constitute domain walls in the\nstrict field-theoretic sense, since a domain wall normally separates dis-\ntinct vacuum phases. Instead, they arise as geometric or topological\ndefects due to degeneration of the angular metric component gφφ. At\nthese radial locations, the angular component degenerates, producing\nsingularities in the spacetime structure.\nNotably, this array of im-\npenetrable surfaces imposes effective confinement for fields within each\nradial domain, resembling domain-wall-like behavior phenomenologi-\ncally, while their origin is purely geometric rather than associated with\nany phase transition of the field. The confinement mechanism in the\nBM spacetime is therefore emphasized. As a result, quantum particles\nare confined between two successive infinite walls, such as the region\nρ =\n√\n2Λr ∈[0, π].\nThis confinement agrees with ˇZofka’s result [8],\nconfirming the presence of a boundary at\n√\n2Λr = π, and makes any\ntreatment that assumes r →∞(ρ →∞) invalid [12, 57].\nIn [55],\nwe analyzed KG particles in a nonuniform magnetic field in the BM-\nRG background and showed that the resulting problem is conditionally\nexactly solvable using the general Heun function HG(a, q, α, β, γ, δ, z).\nIn the present study, we consider KG particles in a uniform external\nmagnetic field in the BM-RG background. In Section II, we formulate\nthe KG equation in this setting and show that it reduces to a hyperge-\nometric differential equation. This leads to exact solutions in terms of\nhypergeometric polynomials of order n. Unlike the case of a nonuniform\nmagnetic field in [55], where we were only able to find a conditionally\nexact solution using general Heun polynomials. We derive the energy\nspectrum and wave functions by analyzing the associated hypergeo-\nmetric series. Here, our analyses interestingly show that, contrary to\nstandard quantum mechanics, where one would expect a magnetic field\nto at least partially lift the degeneracy associated with the magnetic\nquantum number m = ±|m|, we observe no such effect. Moreover, we\nfind that as the magnetic field strength increases, all m-states appear\nto collapse onto the m = 0 state for a given radial quantum number n.\nNevertheless, the cosmological constant Λ is observed to lift the degen-\neracy associated with m, even within its standard range 0 < Λ ≪1. In\nour recent study [55] of KG particles in a nonuniform external magnetic\nfield within the BM RG framework, such surprising degeneracy behav-\niors were not observed. In Section III, we examine the effect of RG using\ntwo sets of rainbow functions. The first set, proposed by Magueijo and\nSmolin, is given by f0(u) = 1/(1 −˜ǫ|E|) and f1(u) = 1, with ˜ǫ = ǫ/Ep,\nmotivated by varying speed of light models [58]. The second set, in-\nspired by LQG [59, 60], uses f0(u) = 1 and f1(u) =\np\n1 −˜ǫ|E|ν, with\nν = 1, 2. Both sets comply with the principles of RG and maintain\nEp as an upper energy bound [53–56, 61]. Our concluding remarks are\npresented in Section IV.\nII.\nKG-PARTICLES IN A UNIFORM MAGNETIC\nFIELD IN BM-RG-SPACETIME BACKGROUND\nIn this section, we formulate a non-perturbative wave equation by in-\ncorporating the effects of the RG framework. For the BM-RG space-\ntime, the non-zero components of the contravariant metric tensor gµν\nassociated with the line element (4) are given by\ng00 = −f0(u)2, g11 = 2Λf1(u)2, g22 = 2Λf1(u)2\nsin(ρ)2 , g33 = f1(u)2. (5)\nThe determinant of the covariant metric tensor is\ndet(gµν) = g = −\nsin(ρ)2\n4Λ2f1(u)6f0(u)2 .\nWe proceed by considering a KG particle and antiparticle in the pres-\nence of an external magnetic field within the BM background.\nThe\ngoverning equation reads\n1\n√−g (∂µ −ieAµ) √−g gµν (∂ν −ieAν) ψ(t, ρ, φ, z) = m2\n◦ψ(t, ρ, φ, z),\n(6)\nwhere m◦denotes the rest mass energy (or rest energy) of the particle.\nWe consider the electromagnetic four-vector potential in the form Aµ =\n(0, 0, Aφ, 0), with\nAφ = −B◦\n2Λ cos(ρ) ⇒B = B◦f1(u)2 ˆz,\nas obtained from the electromagnetic field invariant FµνF µν/2 =\n||B||2 −||E||2/c2.\nIt is evident that in the absence of RG correc-\ntions, where f1(u) = 1, the magnetic field is strictly uniform. How-\never, the presence of RG effects modifies the field, rendering it energy-\ndependent.\nTo determine the influence of the BM spacetime on\nKG bosons under a uniform magnetic field, we employ the ansatz\nψ(t, ρ, φ, z) = ei (m φ+k z−E t) R(ρ) in Eq. (6), which yields the following\nradial differential equation:\nR′′ (ρ) +\n1\ntan ρR′ (ρ) +\n\n\nE −\nh\nm + ˜B cos ρ\ni2\nsin2 ρ\n\n\nR (ρ) = 0,\n(7)\nwhere\n˜B = eB◦\n2Λ ,\nE = f0 (u)2 E2 −m2\n◦\n2Λf1 (u)2\n−k2\n2Λ ,\n(8)\nand the integer m = 0, ±1, ±2, · · · corresponds to the magnetic quan-\ntum number. This expression can be further simplified into the form\nR′′ (ρ) +\n1\ntan ρR′ (ρ) +\n \n˜E −\nη2\nsin2 ρ −2 m ˜B cos ρ\nsin2 ρ\n!\nR (ρ) = 0,\n(9)\nwhere we have introduced ˜E = E + ˜B2 and η2 = (m2 + ˜B2) ⇒\nη =\n\f\f\f\f\nq\n(m2 + ˜B2)\n\f\f\f\f.\nTo recast this equation into a one-dimensional\nSchr¨odinger-like form, which facilitates the identification of the effective\n\n3\ngravitational potential, we define R(ρ) = U(ρ)/\np\nsin(ρ). Substituting\ninto Eq. (9) leads to\nU ′′(ρ) +\n\"\n˜λ −(η2 −1/4)\nsin2 ρ\n−2 m ˜B cos ρ\nsin2 ρ\n#\nU(ρ) = 0,\n(10)\nwith ˜λ = ˜E +1/4. Accordingly, the effective gravitational potential can\nbe identified as\nVeff(ρ) = (η2 −1/4)\nsin2 ρ\n+ 2 m ˜B cos ρ\nsin2 ρ\n.\n(11)\nIt should be noted that, due to the small but non-zero cosmological con-\nstant 0 < Λ ≪1, and the condition ˜B ̸= 0, the effective potential does\nnot impose restrictions on the values of the magnetic quantum number\nm. Instead, it introduces potential barriers at ρ = 0, π, 2π, . . ., con-\nfining the quantum particle/antiparticle within the interval ρ ∈[0, π].\nThis would suggest the boundary conditions U(0) = 0 = U(π) for the\nradial wave function.\nIn the subsequent section, we will derive the\nexact solution to the corresponding one-dimensional Schr¨odinger-type\nequation given in (10).\nA.\nGeometric Confinement of Klein-Gordon Particles\nWe begin by analyzing the KG test particles in a uniform magnetic\nfield within BM spacetime induced confinement and described by the\nmetric in (10), using the substitutions x = cos ρ →ρ = arccos(x), and\nU(x) = (x −1)α (x + 1)β F (x),\n(12)\nwhere\nα = 1\n4\n\u0010p\n4 η2 + 4 γ + 1 + 1\n\u0011\n,\nβ = 1\n4\n\u0010p\n4 η2 −4 γ + 1 + 1\n\u0011\n,\nγ = 2 m ˜B, and η2 = m2 + ˜B2. Note that for x = 1 (ρ = 0) and x = −1\n(ρ = π), the function U(x) = 0, analogous to the standard textbook\ninfinite potential well, represented by the first two impenetrable walls\nin the BM spacetime scenario. Next, we apply the change of variables\ny = (x + 1)/2, leading to\n(y2 −y)F ′′(x) + [(a + b + 1)y −c] F ′(y) + ab F (y) = 0,\n(13)\nwhere\na = α + β +\np\n˜λ,\nb = α + β −\np\n˜λ,\nc = 2β + 1\n2.\n(14)\nIt is evident that equation (13) is the hypergeometric differential equa-\ntion, which admits the solution F (y) =\n2F1(a, b, c, y), where a = −n\nand/or b = −n, would ensure that the hypergeometric power series\ntruncates to a polynomial of order n ≥0.\nFurthermore, note that\nα, β ≥1/2, as can be seen from\nα = 1\n4\n\u0012q\n4(m + ˜B)2 + 1 + 1\n\u0013\n,\nβ = 1\n4\n\u0012q\n4(m −˜B)2 + 1 + 1\n\u0013\n.\n(15)\nIt is important to recognize that the term α + β in a and b of (14)\nremains unchanged for m = ±|m|, where α|±|m| = β|∓|m|. To verify\nthe uniqueness of F (y) = 2F1(a, b, c, y) as a solution of (13), we consider\na power series expansion of the form\nF (y) =\n∞\nX\nj=0\nCj yj+σ,\n(16)\nwhich yields\n∞\nX\nj=0\nCj [ab + (j + σ)(j + σ + a + b)] yj+σ\n=\n∞\nX\nj=0\nCj+1 [(j + σ + 1)(j + σ + c)] yj+σ + C0 σ [σ + c −1] yσ−1.\n(17)\nSince C0 ̸= 0, it follows that σ [σ + c −1] = 0. Consequently, σ = 0 or\nσ = −c + 1 = −2β + 1/2. The latter case would cause F (y) →∞as\ny →0 (i.e., ρ = π ∈[0, π]), and must therefore be excluded. We thus\nadopt σ = 0, leading to the recurrence relation\nCj+1 [(j + 1)(j + c)] = Cj [ab + j(j + a + b)] .\n(18)\nTo truncate the power series into a polynomial of degree n ≥0, we im-\npose the condition Cn+1 = 0 while ensuring Cn ̸= 0. This requirement\nyields\nab + n(n + a + b) = 0 ⇒a = −n and/or b = −n.\nAs a result, we obtain ˜λ = (n + α + β)2. Consequently, the energy\nexpression becomes\nf0(u)2E2 = f1(u)2Gnm + m2\n◦;\nGnm = 2Λ\n\u0014\n(n + α + β)2 −˜B2 −1\n4\n\u0015\n+ k2.\n(19)\nWe observe that the invariance of α + β under the transformation\nm = ±|m| is reflected in the spectroscopic structure of KG particles\nand antiparticles in (19). Specifically, the spectrum exhibits degener-\nacy with respect to the magnetic quantum number m = ±|m|, aris-\ning from the identity α|±|m| = β|∓|m|.\nFurthermore, since F (y) =\n2F1(a, b, c, y) = Pn\nj=0 Cj yj, the full solution reads\nUnm(x) ∼(x −1)α (x + 1)β\nn\nX\nj=0\nCj\n\u0012x + 1\n2\n\u0013j\n⇒\nUnm(ρ) = N (cos(ρ) −1)α (cos(ρ) + 1)β\nn\nX\nj=0\nCj\n\u0012\ncos\n\u0012ρ\n2\n\u0013\u00132j\nRnm(ρ) = N (cos(ρ) −1)α−1/4 (cos(ρ) + 1)β−1/4\nn\nX\nj=0\nCj\n\u0012\ncos\n\u0012ρ\n2\n\u0013\u00132j\n.\n(20)\nHere, the coefficients Cj are given by the recurrence relation in (18),\nwith α and β defined in (15). This result will be used in the follow-\ning analysis to examine the effects of RG corrections on the spectro-\nscopic structure of KG particles and antiparticles in BM spacetime\nsubjected to a uniform external magnetic field. It is clear that both\nradial functions R(ρ) and U(ρ) vanish at the surfaces ρ = 0 and ρ = π,\nin agreement with the standard behavior of quantum wave functions\nfor particles confined within an infinite potential well of width π.\nIII.\nRAINBOW GRAVITY EFFECTS\nIn this section, we investigate the influence of RG on KG particles and\nantiparticles by employing two distinct classes of rainbow functions.\nThe first class, originally proposed within the framework of the varying\nspeed of light hypothesis [58], is defined by\nf0(u) =\n1\n1 −ǫu =\n1\n1 −˜ǫ|E|,\nf1(u) = 1,\n\n4\nFIG. 1. The figure displays the energy levels of KG particles and antiparticles given by (21). Specifically, we plot: (a) E versus ˜ǫ for n = 0,\nm = 0, 1, 2, 3, 4, Λ = 0.5, and B◦= 1 = e; (b) E versus B◦for n = 0, m = 0, 1, 2, 3, 4, ˜ǫ = 0.5, and Λ = 0.1; (c) E versus B◦for n = 2,\nm = 0, 1, 2, 3, 4, ˜ǫ = 0.5, and Λ = 0.1; (d) E versus the cosmological constant Λ for n = 2, m = 0, 1, 2, 3, 4, ˜ǫ = 0.5, and B◦= 4; (e) E versus Λ for\nn = 0, 1, 2, 3, 4, m = 2, ˜ǫ = 0.5, and B◦= 4; and (f) E versus Λ for n = 2, m = 0, 1, 2, 3, 4, ˜ǫ = 0.5, and B◦= 1.\nwhere u = |E|/Ep denotes a dimensionless energy scale normalized by\nthe Planck energy Ep, and ˜ǫ = ǫ/Ep is the corresponding dimensionless\nrainbow parameter. The second class, inspired by the considerations of\nLQG [59, 60], is given by\nf0(u) = 1,\nf1(u) =\nq\n1 −˜ǫ|E|ν,\nν = 1, 2,\nwith the admissible energy range limited to 0 ≤u = |E|/Ep ≤1.\nThese functional forms introduce energy-dependent deformations to\nthe spacetime geometry, encapsulating potential quantum gravitational\ncorrections at trans-Planckian scales.\nA.\nMagueijo-Smolin Rainbow Functions\nThis set of rainbow functions, f0 (u) = 1/ (1 −˜ǫ |E|), f1 (u) = 1 (with\n˜ǫ = ǫ/Ep), is motivated by the hypothesis of varying the speed of light\n[58]. Substituting this set into the result (19) leads to\nE2 = ˜Gnm (1 −˜ǫ |E|)2 ; ˜Gnm = Gnm + m2\n◦.\n(21)\nIt is important to recall that |E| = E+ is used for test particles, while\n|E| = −E−is used for antiparticles.\nThis distinction leads to two\nseparate equations:\nE2\n+\n\u0010\n1 −˜ǫ2 ˜Gnm\n\u0011\n+ 2˜ǫ ˜Gnm E+ −˜Gnm = 0,\n(22)\nfor particles, and\nE2\n−\n\u0010\n1 −˜ǫ2 ˜Gnm\n\u0011\n−2˜ǫ ˜GnmE−−˜Gnm = 0,\n(23)\nfor antiparticles. These can be combined to yield the general result:\nEn,m = ±\np ˜Gnm\n1 + ˜ǫ\np ˜Gnm\n.\n(24)\nThis result suggests a clear symmetrization of the KG particle and\nantiparticle energies with respect to E = 0, as also observed in Figure\n1. However, it should be noted that the asymptotic behavior of |E±|\nas B◦→∞approaches a limiting value |E±| ∼2 = 1/˜ǫ = Ep/ǫ,\nas seen in Figs. 1(b) and 1(c). Consequently, it follows that |E±| ≤\nEp =⇒|E±|max = Ep when ǫ = 1.\nAlthough it is known that the\ncosmological constant typically satisfies 0 < Λ ≪1 ( i.e., Λ = 1.4657 ×\n10−52m−2 = 3.827 × 10−123l−2\np , where lp is the Planck length), we\nhave extended this range and considered larger values of Λ purely for\nhypothetical exploration. Although it is hypothetical, it allows us to\nprobe theoretical possibilities and emphasizes that the Planck energy\nEp is the maximum possible energy for particles and antiparticles (a\nnatural consequence of the RG proposal) and is yet another invariant,\nalongside the speed of light, as clearly illustrated in Figs. 1(d), 1(e),\nand 1(f). According to standard quantum mechanics, one would expect\nthat a magnetic field lifts the degeneracy associated with the magnetic\nquantum number m = ±|m|, at least partially. Interestingly, contrary\nto this expectation, we observe no such effect. In fact, as the magnetic\nfield strength increases, all m-states appear to collapse into the m = 0\nstate for a given n.\nHowever, the cosmological constant Λ lifts the\ndegeneracy associated with m, even within its standard range 0 < Λ <\n1, as clearly shown in Figs. 1(d), 1(e) and 1(f). We attribute this to\nthe uniform nature of the magnetic field employed here. In our recent\nstudy [55] of KG particles in a non-uniform external magnetic field\nwithin the BM RG framework, such surprising degeneracy behaviors\nwere not observed.\nB.\nRainbow functions pair f0(u) = 1,\nf1 (u) =\np\n1 −˜ǫ |E|ν ; ν = 1, 2\nThis pair of rainbow functions is motivated by LQG [59, 60], which has\nbeen shown to be fully consistent with the RG principles, thus ensuring\ninvariance of the Planck energy Ep. This energy scale represents the\nmaximum attainable energy for both particles and antiparticles (see,\ne.g., [53, 54, 61]). Accordingly, it is of particular interest to explore\n\n5\nFIG. 2. The figure illustrates the energy levels of KG particles and antiparticles as given by Eq. (25). Specifically, the plots show: (a) E as a\nfunction of ˜ǫ for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.5, and B◦= 1 = e; (b) E as a function of B◦for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.1, and ˜ǫ = 0.5; (c)\nE as a function of B◦for n = 1, m = 0, 1, 2, 3, 4, Λ = 0.1, and ˜ǫ = 0.5; (d) E as a function of the cosmological constant Λ for n = 0, m = 0, 1, 2,\nB◦= 4, and ˜ǫ = 0.5.\nFIG. 3. The figure shows the energy levels for KG particles and antiparticles given by (28), and we plot (a) E against ˜ǫ for n = 0, m = 0, 1, 2, 3, 4,\nΛ = 0.5, and B◦= 1 = e, (b) E against B◦for n = 0 and m = 0, 1, 2, 3, 4, Λ = 0.1, and ˜ǫ = 0.5, (c) E against B◦for n = 1 and m = 0, 1, 2, 3, 4,\nΛ = 0.1, and ˜ǫ = 0.5, and (d) E against Λ for n = 0 and m = 0, 1, 2, B◦= 4, and ˜ǫ = 0.5, (d) E against .\nthe behavior of such rainbow function pairs when applied to KG par-\nticles and antiparticles in a magnetized BM spacetime. We begin by\nconsidering the case υ = 1, which, through (19), leads to the following\nexpression:\nE2 + ˜ǫ Gnm|E| −˜Gnm = 0.\n(25)\nThe corresponding energy solutions are given by:\nE+ = −˜ǫ Gnm\n2\n+ 1\n2\nq\n˜ǫ2 G2\nnm + 4 ˜Gnm,\n(26)\nand\nE−= ˜ǫ Gnm\n2\n−1\n2\nq\n˜ǫ2 G2\nnm + 4 ˜Gnm.\n(27)\nThese energy levels are illustrated in Figure 2. It is evident that, as\nthe cosmological constant Λ →∞, equation (25) implies that the max-\nimum energy, |E|max, tends to 1/˜ǫ = 2. This indicates an asymptotic\nsaturation of the energy levels of KG particles and antiparticles in the\nBM spacetime, as clearly observed in Figure 2.\nNext, we consider the case υ = 2, which yields the following result:\nE2 =\n˜Gnm\n1 + ˜ǫ Gnm\n⇒\nE± = ±\ns\n˜Gnm\n1 + ˜ǫ Gnm\n.\n(28)\nThe corresponding energy levels are shown in Figure 3. This plot fur-\nther confirms the asymptotic behavior of the energy spectrum as the\ncosmological constant Λ →∞. As indicated by (28), the maximum\nenergy in this case is given by |E|max ∼\np\n1/˜ǫ =\n√\n2 ≈1.41.\nThis\nbehavior is in full agreement with the analytical expression and is visu-\nally represented in Figure 3. Notably, the same behavior observed for\nthe Magueijo-Smolin rainbow functions in Figure 1 reemerges in Fig-\nures 2 and 3. Remarkably, an increase in the uniform magnetic field\nstrength causes all m states to collapse into the m = 0 state for a given\nradial quantum number n. However, as the cosmological constant Λ\n(of the magnetized BM spacetime) increases, the m-related degenera-\ncies are partially lifted, although the degeneracy between states with\nm = ±|m| remains preserved as Λ →1. Interestingly, such behavior\nis absent in the case of scalar KG bosonic particles in a non-uniform\nmagnetic field within BM spacetime, as discussed in [55]. The three\nsets of rainbow functions examined here, a Magueijo-Smolin set and\ntwo sets motivated by LQG, are all found to be consistent with the\nRG-inherited Planck energy invariance.\nIV.\nCONCLUDING REMARKS\nIn this comprehensive investigation, we have conducted an exact ana-\nlytical study of KG bosons and antibosons subjected to an external uni-\nform magnetic field in the background of the BM spacetime, modified\nby the framework of RG and endowed with a positive cosmological con-\nstant. This work builds upon recent advances in quantum field theory\nin curved spacetimes, providing a detailed exploration of how gravita-\ntional confinement, topological structures, magnetic interactions, and\nQG-inspired modifications affect the energy-momentum dispersion re-\nlations. A key aspect of our analysis is that the geometry of the BM\nspacetime naturally induces an infinite sequence of infinite potential\nbarriers, appearing at discrete surfaces determined by the expression\nρ =\n√\n2Λr = τπ, with τ = 0, π, 2π, . . . . These potential barriers act\nas impenetrable boundaries, dividing spacetime into an infinite number\nof confinement regions, each similar to an ideal infinite quantum well\nwith rigid boundaries. Within the first of these confinement domains,\n\n6\n0\n0.5\n1\n1.5\n2\n2.5\n3\n0\n0.5\n1\n1.5\n0\n0.5\n1\n1.5\n2\n2.5\n3\n-0.5\n0\n0.5\n0\n0.5\n1\n1.5\n2\n2.5\n3\n-0.2\n0\n0.2\n0.4\n0\n0.5\n1\n1.5\n2\n2.5\n3\n0\n0.5\n1\n0\n0.5\n1\n1.5\n2\n2.5\n3\n-0.1\n-0.05\n0\n0.05\n0.1\n0\n0.5\n1\n1.5\n2\n2.5\n3\n-0.01\n-0.005\n0\n0.005\n0.01\nFIG. 4. Radial wave functions Rn,m(ρ) in (20) for quantum numbers n = 0, 1, 2 and m = 0, 1, 2 under two magnetic field strengths, B◦= 1 (left\ncolumn) and B◦= 10 (right column). Each subplot depicts the behavior of Rn,m(ρ) over the domain ρ ∈[0, π], illustrating how the magnetic field\nintensity affects the spatial distribution of scalar field modes in the magnetized BM universe.\nFIG. 5. Radial probability density distributions Pn,m(ρ) are presented for quantum states with n = 0, 1, 2 and m = 0, 1, 2, where Pn,m(ρ) =\nR ρ\n0 |Rn,m(ρ′)|2 ρ′ dρ′. The wavefunctions Rn,m(ρ) are computed under a transverse magnetic field with effective coupling ˜B = eB◦\n2Λ , using e = 1,\nB◦= 1, and Λ = 0.1. The surface plots are mapped to Cartesian coordinates via x = ρ cos φ, y = ρ sin φ, where both the vertical axis and color\nscale represent the magnitude of Pn,m(ρ).\nwe have obtained exact, closed-form solutions to the KG equation, ex-\npressed using hypergeometric polynomials for the radial wave functions.\nThis framework allowed us to precisely determine the energy spectrum\nand examine the effects of external fields and RG corrections.\nOur results reveal several notable features regarding the spectroscopic\nstructure of the system.\nThe inclusion of RG effects, through both\nMagueijo-Smolin rainbow functions and those inspired by LQG, sig-\nnificantly influences the energy spectrum. In particular, the uniform\nexternal magnetic field, which normally lifts the degeneracies associ-\nated with the magnetic quantum number m, behaves in an unexpected\nway within this curved spacetime setting. Specifically, the combined\neffect of the magnetic field and the BM spacetime geometry causes all\nmagnetic quantum states m to collapse onto the m = 0 state for a\nfixed radial quantum number n.\nThis behavior differs from what is\nseen in flat spacetime quantum mechanics and highlights how space-\ntime curvature and topology can dominate the spectral properties of\nquantum fields. Furthermore, our analysis shows that the cosmological\nconstant Λ, which is usually important at large (cosmological) scales,\nalso has a direct effect on the local quantum behavior of the system.\nWhen Λ increases beyond physically realistic values, a partial lifting\nof the degeneracies associated with the magnetic quantum number is\nobserved. This indicates a subtle connection between the global geom-\netry of spacetime and the local quantum characteristics of relativistic\nparticles.\nThe study also confirms the consistency of the rainbow functions used\nwith the RG principles. In both the Magueijo-Smolin and LQG-inspired\ncases, the Planck energy Ep appears clearly as the highest possible\nenergy for particles and antiparticles. This maintains the theoretical\n\n7\nFIG. 6. Radial probability density distributions Pn,m(ρ) are shown for quantum states labeled by (n, m), representing the eigenstates of a charged\nparticle confined to a curved surface under a transverse magnetic field.\nThe system parameters are fixed as e = 1, Λ = 0.1, and B◦= 10,\ncorresponding to a strong dimensionless magnetic coupling ˜B = eB◦\n2Λ ≫1. The plots display the spatial distributions of Pn,m(ρ) for discrete states\nwith n = 0, 1, 2 and m = 0, 1, 2, rendered as three-dimensional surfaces in Cartesian coordinates via the polar mapping x = ρ cos φ, y = ρ sin φ,\nwith height and colormap indicating the magnitude of the probability density.\nframework of QG phenomenology and keeps the symmetry of the par-\nticle and antiparticle energy spectra around E = 0. It is important\nto note that the unusual degeneracy behavior caused by the uniform\nmagnetic field is not observed in cases with non-uniform magnetic fields\nin the same BM-RG spacetime [55]. This difference emphasizes that\nboth the magnetic field configuration and the geometry of spacetime\ninfluence the spectral and confinement properties of quantum fields.\nWith reference to the radial wave functions given in equation (20),\nwe plot in Figure 4 the radial wave functions Rn,m(ρ) for quantum\nnumbers n = 0, 1, 2 and m = 0, 1, 2 at two magnetic field strengths\nB◦= 1 and 10. As the magnetic field strength increases, the radial\nwave functions, representing KG particles and antiparticles, become\nincreasingly localized. For B◦= 1, the particles are mainly distributed\nbetween ρ ∼0.5 and ρ ∼3, while for B◦= 10, they are restricted to the\nregion ρ ∼1 to ρ ∼2. This localization effect is further illustrated in\nthe corresponding radial probability densities shown in Figures 5 and\n6. Radial probability densities are defined as\nPn,m(ρ) =\nZ ρ\n0\n|Rn,m(ρ′)|2 ρ′ dρ′,\nand are visualized in two dimensions via x = ρ cos φ and y = ρ sin φ,\nwhere the color intensity represents the magnitude of Pn,m. The pa-\nrameter ˜B =\ne B◦\n2 Λ\ncharacterizes the effect of the external magnetic\nfield on the quantum states (B◦= 1 in Figure 5 and B◦= 10 in Fig-\nure 6). As the radial quantum number n increases, the radial prob-\nability distributions develop additional nodal rings, corresponding to\nhigher energy states with more radial nodes. For n = 0, the density\nis mainly concentrated near the origin, while for n = 1, 2, more com-\nplex ring-like structures emerge. For fixed n, increasing the azimuthal\nquantum number m pushes the probability density outward due to the\ncentrifugal barrier, broadening the spatial extent of the wave function.\nImportantly, increasing the magnetic field strength B◦enhances the\nmagnetic confinement, effectively compressing the probability densities\ntoward smaller radial distances and strengthening localization near the\norigin. The magnetic field acts as a confining potential that counter-\nacts the outward push caused by larger m values, resulting in sharper\nand more localized radial peaks. Stronger magnetic fields also slightly\nshift the nodal positions inward, modifying the spatial probability dis-\ntribution. These results illustrate how tuning the magnetic field allows\ncontrol over quantum confinement and the spatial characteristics of\nthe system’s eigenstates. Throughout all states, rotational symmetry\nis preserved, consistent with conservation of angular momentum. The\nobserved evolution of probability densities with increasing n, m, and\nB◦provides a quantitative understanding of how quantum numbers and\nmagnetic field strength shape spatial quantum states under magnetic\nconfinement.\nThis work provides an exact and detailed description of the KG boson\ndynamics in the BM-RG spacetime, showing how gravitational con-\nfinement, topological features, magnetic fields, and RG corrections in-\nfluence the energy spectrum and wave functions.\nThese results con-\ntribute to the theoretical understanding of quantum fields in curved\nand magnetized backgrounds and suggest potential areas for further\nstudy. Future work could include extending the analysis to fermionic\nor higher-spin fields, exploring analogous phenomena in optical or con-\ndensed matter systems (in principle), and examining how these effects\nmight appear in astrophysical or cosmological contexts. 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Magusa, north Cyprus, Mersin 10 - T ̈urkiye Abdullah Guvendi† 25050, Erzurum, T ̈urkiye (Dated: October 17, 2025) We present an exact analytical study of Klein-Gordon (KG) scalar bosons and antibosons confined in the Bonnor-Melvin (BM) spacetime under a uniform magnetic field, incorporating rainbow gravity (RG) corrections with a positive cosmological constant. The cosmological constant partitions spacetime into an infinite sequence of confinement domains bounded by impenetrable barriers. Within the first allowed domain, the KG equation reduces to a hypergeometric differential equation, yielding closed-form expressions for both the energy spectra and the radial wavefunctions in terms of hypergeometric polynomials. Two representative RG models, inspired by the Magueijo-Smolin framework and loop quantum gravity (LQG), produce Planck-scale bounded, symmetric particle-antiparticle spectra. A distinctive feature of the curved magnetized geometry is the collapse of all magnetic quantum states m ̸= 0 onto the m = 0 level for each radial excitation, a degeneracy absent in flat spacetime. Increasing the cosmological constant partially lifts this collapse, establishing a direct link between the global spacetime curvature and the local quantum structure. Radial probability density analysis further shows that stronger magnetic fields enhance spatial localization, confining bosons into static or rotating ring-like configurations with nodal architectures that evolve systematically with quantum numbers. These findings reveal how gravitational confinement, topology, magnetic fields, and Planck-scale corrections jointly govern the spectral and spatial properties of relativistic quantum fields in curved and magnetized backgrounds. I. INTRODUCTION Einstein's theory of general relativity (GR) [1] describes gravity as a manifestation of the curvature of spacetime. The attempt to unify GR with quantum mechanics, often referred to as quantum gravity (QG), offers fundamental insights into the behavior of quantum systems in curved backgrounds. This connection has motivated many studies examining how spacetime curvature influences quantum spectroscopic properties, especially for scalar and spinor particles described by the KG and Dirac equations, respectively [2]. The present study focuses on KG particles in such settings. Magnetic fields are central to many astrophysical and cosmological phenomena, including stellar structures, accretion disks, galactic centers, magnetars [3, 4], and heavy-ion collisions [5-7]. The simultaneous presence of strong gravitational and magnetic fields near compact objects has led to considerable interest in general relativistic models that incorporate both effects. One prominent example is the BM universe, an exact solution of the Einstein-Maxwell equations that describes a static spacetime with cylindrical symmetry, sourced by a magnetic field aligned with the symmetry axis and self-consistently embedded in its own gravitational field [8-12]. In this model, the magnetic field con- ∗ (Corr. Author) † tributes to the energy-momentum tensor and thereby affects the geometry of spacetime. A positive cosmological constant Λ > 0 is introduced to stabilize the spacetime and preserve its symmetry. The BM spacetime metric is given in [8] as ds2 = -dt2 + dr2 + ̃σ2 sin2 √ 2Λr dφ2 + dz2, (1) where z ∈[-∞, ∞], φ ∈[0, 2π], r ∈[0, ∞], and Λ in units of inverse length squared. Following the coordinate transformation √ 2Λr →ρ and ̃σ √ 2Λφ →φ, as in ˇZofka's treatment [8], the line element (1) becomes ds2 = -dt2 + 1 2Λ dρ2 + sin2 (ρ) dφ2 + dz2, (2) where H = 1 √ 2 sin(ρ) denotes the magnetic field [8]. Topological defects such as domain walls are expected in various unified field theories [13-16], and their effects on quantum dynamics have been examined in [17-34]. The relationship between gravitational geometry and quantum systems remains a key motivation for studying such configurations, and the BM spacetime provides a suitable model for this purpose [8-10, 35-38]. In the high-energy (ultraviolet) regime, the energy of a test particle can influence the structure of spacetime, leading to a modification of the standard dispersion relation [39-45]: E2f0 (u)2 -p2f1(u)2 = m2 ◦. (3) 2 This form appears in various QG theories, such as string theory, LQG, and non-commutative geometry [46-48], as well as in some recent research framed within RG, including: energy-momentum distributions of a cylindrical black hole in rainbow gravity [49]; generalized uncertainty principle corrections in Rastall-Rainbow Casimir wormholes [50]; fermions moving around strong gravitational sources [51]; and the exploration of the physical properties of the strange star SAX J1808.43658 in rainbow gravity [52]. Consequently, the BM metric (2) under RG becomes ds2 = - dt2 f0 (u)2 + 1 f1 (u)2 1 2Λ dρ2 + sin2 (ρ) dφ2 + dz2 , (4) where fi(u); i = 0, 1, are the so-called rainbow functions that obey the infrared limit lim u→0 fi (u) = 1, and u = |E|/Ep satisfies 0 ≤u ≤1. This ensures a consistent transition to the classical regime and an equal treatment of particles and antiparticles [45, 53-56]. The functions must preserve Ep, the Planck energy, as an invariant scale. We refer to the metric in (4) as the BM-RG spacetime. In our recent studies [55, 56], we showed that the BM spacetime contains singularities at ρ = √ 2Λr = τ π, with τ = 0, 1, 2, · · ·. These singularities correspond to infinite potential barriers, acting effectively as impenetrable walls. The surfaces at r = τ π √ 2Λ do not constitute domain walls in the strict field-theoretic sense, since a domain wall normally separates distinct vacuum phases. Instead, they arise as geometric or topological defects due to degeneration of the angular metric component gφφ. At these radial locations, the angular component degenerates, producing singularities in the spacetime structure. Notably, this array of impenetrable surfaces imposes effective confinement for fields within each radial domain, resembling domain-wall-like behavior phenomenologically, while their origin is purely geometric rather than associated with any phase transition of the field. The confinement mechanism in the BM spacetime is therefore emphasized. As a result, quantum particles are confined between two successive infinite walls, such as the region ρ = √ 2Λr ∈[0, π]. This confinement agrees with ˇZofka's result [8], confirming the presence of a boundary at √ 2Λr = π, and makes any treatment that assumes r →∞(ρ →∞) invalid [12, 57]. In [55], we analyzed KG particles in a nonuniform magnetic field in the BMRG background and showed that the resulting problem is conditionally exactly solvable using the general Heun function HG(a, q, α, β, γ, δ, z). In the present study, we consider KG particles in a uniform external magnetic field in the BM-RG background. In Section II, we formulate the KG equation in this setting and show that it reduces to a hypergeometric differential equation. This leads to exact solutions in terms of hypergeometric polynomials of order n. Unlike the case of a nonuniform magnetic field in [55], where we were only able to find a conditionally exact solution using general Heun polynomials. We derive the energy spectrum and wave functions by analyzing the associated hypergeometric series. Here, our analyses interestingly show that, contrary to standard quantum mechanics, where one would expect a magnetic field to at least partially lift the degeneracy associated with the magnetic quantum number m = ±|m|, we observe no such effect. Moreover, we find that as the magnetic field strength increases, all m-states appear to collapse onto the m = 0 state for a given radial quantum number n. Nevertheless, the cosmological constant Λ is observed to lift the degeneracy associated with m, even within its standard range 0 < Λ ≪1. In our recent study [55] of KG particles in a nonuniform external magnetic field within the BM RG framework, such surprising degeneracy behaviors were not observed. In Section III, we examine the effect of RG using two sets of rainbow functions. The first set, proposed by Magueijo and Smolin, is given by f0(u) = 1/(1 - ̃ǫ|E|) and f1(u) = 1, with ̃ǫ = ǫ/Ep, motivated by varying speed of light models [58]. The second set, inspired by LQG [59, 60], uses f0(u) = 1 and f1(u) = p 1 - ̃ǫ|E|ν, with ν = 1, 2. Both sets comply with the principles of RG and maintain Ep as an upper energy bound [53-56, 61]. Our concluding remarks are presented in Section IV. II. KG-PARTICLES IN A UNIFORM MAGNETIC FIELD IN BM-RG-SPACETIME BACKGROUND In this section, we formulate a non-perturbative wave equation by incorporating the effects of the RG framework. For the BM-RG spacetime, the non-zero components of the contravariant metric tensor gμν associated with the line element (4) are given by g00 = -f0(u)2, g11 = 2Λf1(u)2, g22 = 2Λf1(u)2 sin(ρ)2 , g33 = f1(u)2. (5) The determinant of the covariant metric tensor is det(gμν) = g = - sin(ρ)2 4Λ2f1(u)6f0(u)2 . We proceed by considering a KG particle and antiparticle in the presence of an external magnetic field within the BM background. The governing equation reads 1 √-g (∂μ -ieAμ) √-g gμν (∂ν -ieAν) ψ(t, ρ, φ, z) = m2 ◦ψ(t, ρ, φ, z), (6) where m◦denotes the rest mass energy (or rest energy) of the particle. We consider the electromagnetic four-vector potential in the form Aμ = (0, 0, Aφ, 0), with Aφ = -B◦ 2Λ cos(ρ) ⇒B = B◦f1(u)2 ˆz, as obtained from the electromagnetic field invariant FμνF μν/2 = ||B||2 -||E||2/c2. It is evident that in the absence of RG corrections, where f1(u) = 1, the magnetic field is strictly uniform. However, the presence of RG effects modifies the field, rendering it energydependent. To determine the influence of the BM spacetime on KG bosons under a uniform magnetic field, we employ the ansatz ψ(t, ρ, φ, z) = ei (m φ+k z-E t) R(ρ) in Eq. (6), which yields the following radial differential equation: R′′ (ρ) + 1 tan ρR′ (ρ) + E - h m + ̃B cos ρ i2 sin2 ρ R (ρ) = 0, (7) where ̃B = eB◦ 2Λ , E = f0 (u)2 E2 -m2 ◦ 2Λf1 (u)2 -k2 2Λ , (8) and the integer m = 0, ±1, ±2, · · · corresponds to the magnetic quantum number. This expression can be further simplified into the form R′′ (ρ) + 1 tan ρR′ (ρ) + ̃E - η2 sin2 ρ -2 m ̃B cos ρ sin2 ρ ! R (ρ) = 0, (9) where we have introduced ̃E = E + ̃B2 and η2 = (m2 + ̃B2) ⇒ η = q (m2 + ̃B2) . To recast this equation into a one-dimensional Schr ̈odinger-like form, which facilitates the identification of the effective 3 gravitational potential, we define R(ρ) = U(ρ)/ p sin(ρ). Substituting into Eq. (9) leads to U ′′(ρ) + \" ̃λ -(η2 -1/4) sin2 ρ -2 m ̃B cos ρ sin2 ρ # U(ρ) = 0, (10) with ̃λ = ̃E +1/4. Accordingly, the effective gravitational potential can be identified as Veff(ρ) = (η2 -1/4) sin2 ρ + 2 m ̃B cos ρ sin2 ρ . (11) It should be noted that, due to the small but non-zero cosmological constant 0 < Λ ≪1, and the condition ̃B ̸= 0, the effective potential does not impose restrictions on the values of the magnetic quantum number m. Instead, it introduces potential barriers at ρ = 0, π, 2π, . . ., confining the quantum particle/antiparticle within the interval ρ ∈[0, π]. This would suggest the boundary conditions U(0) = 0 = U(π) for the radial wave function. In the subsequent section, we will derive the exact solution to the corresponding one-dimensional Schr ̈odinger-type equation given in (10). A. Geometric Confinement of Klein-Gordon Particles We begin by analyzing the KG test particles in a uniform magnetic field within BM spacetime induced confinement and described by the metric in (10), using the substitutions x = cos ρ →ρ = arccos(x), and U(x) = (x -1)α (x + 1)β F (x), (12) where α = 1 4 p 4 η2 + 4 γ + 1 + 1 , β = 1 4 p 4 η2 -4 γ + 1 + 1 , γ = 2 m ̃B, and η2 = m2 + ̃B2. Note that for x = 1 (ρ = 0) and x = -1 (ρ = π), the function U(x) = 0, analogous to the standard textbook infinite potential well, represented by the first two impenetrable walls in the BM spacetime scenario. Next, we apply the change of variables y = (x + 1)/2, leading to (y2 -y)F ′′(x) + [(a + b + 1)y -c] F ′(y) + ab F (y) = 0, (13) where a = α + β + p ̃λ, b = α + β - p ̃λ, c = 2β + 1 2. (14) It is evident that equation (13) is the hypergeometric differential equation, which admits the solution F (y) = 2F1(a, b, c, y), where a = -n and/or b = -n, would ensure that the hypergeometric power series truncates to a polynomial of order n ≥0. Furthermore, note that α, β ≥1/2, as can be seen from α = 1 4 q 4(m + ̃B)2 + 1 + 1 , β = 1 4 q 4(m - ̃B)2 + 1 + 1 . (15) It is important to recognize that the term α + β in a and b of (14) remains unchanged for m = ±|m|, where α|±|m| = β|∓|m|. To verify the uniqueness of F (y) = 2F1(a, b, c, y) as a solution of (13), we consider a power series expansion of the form F (y) = ∞ X j=0 Cj yj+σ, (16) which yields ∞ X j=0 Cj [ab + (j + σ)(j + σ + a + b)] yj+σ = ∞ X j=0 Cj+1 [(j + σ + 1)(j + σ + c)] yj+σ + C0 σ [σ + c -1] yσ-1. (17) Since C0 ̸= 0, it follows that σ [σ + c -1] = 0. Consequently, σ = 0 or σ = -c + 1 = -2β + 1/2. The latter case would cause F (y) →∞as y →0 (i.e., ρ = π ∈[0, π]), and must therefore be excluded. We thus adopt σ = 0, leading to the recurrence relation Cj+1 [(j + 1)(j + c)] = Cj [ab + j(j + a + b)] . (18) To truncate the power series into a polynomial of degree n ≥0, we impose the condition Cn+1 = 0 while ensuring Cn ̸= 0. This requirement yields ab + n(n + a + b) = 0 ⇒a = -n and/or b = -n. As a result, we obtain ̃λ = (n + α + β)2. Consequently, the energy expression becomes f0(u)2E2 = f1(u)2Gnm + m2 ◦; Gnm = 2Λ (n + α + β)2 - ̃B2 -1 4 + k2. (19) We observe that the invariance of α + β under the transformation m = ±|m| is reflected in the spectroscopic structure of KG particles and antiparticles in (19). Specifically, the spectrum exhibits degeneracy with respect to the magnetic quantum number m = ±|m|, arising from the identity α|±|m| = β|∓|m|. Furthermore, since F (y) = 2F1(a, b, c, y) = Pn j=0 Cj yj, the full solution reads Unm(x) ∼(x -1)α (x + 1)β n X j=0 Cj x + 1 2 j ⇒ Unm(ρ) = N (cos(ρ) -1)α (cos(ρ) + 1)β n X j=0 Cj cos ρ 2 2j Rnm(ρ) = N (cos(ρ) -1)α-1/4 (cos(ρ) + 1)β-1/4 n X j=0 Cj cos ρ 2 2j . (20) Here, the coefficients Cj are given by the recurrence relation in (18), with α and β defined in (15). This result will be used in the following analysis to examine the effects of RG corrections on the spectroscopic structure of KG particles and antiparticles in BM spacetime subjected to a uniform external magnetic field. It is clear that both radial functions R(ρ) and U(ρ) vanish at the surfaces ρ = 0 and ρ = π, in agreement with the standard behavior of quantum wave functions for particles confined within an infinite potential well of width π. III. RAINBOW GRAVITY EFFECTS In this section, we investigate the influence of RG on KG particles and antiparticles by employing two distinct classes of rainbow functions. The first class, originally proposed within the framework of the varying speed of light hypothesis [58], is defined by f0(u) = 1 1 -ǫu = 1 1 - ̃ǫ|E|, f1(u) = 1, 4 FIG. 1. The figure displays the energy levels of KG particles and antiparticles given by (21). Specifically, we plot: (a) E versus ̃ǫ for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.5, and B◦= 1 = e; (b) E versus B◦for n = 0, m = 0, 1, 2, 3, 4, ̃ǫ = 0.5, and Λ = 0.1; (c) E versus B◦for n = 2, m = 0, 1, 2, 3, 4, ̃ǫ = 0.5, and Λ = 0.1; (d) E versus the cosmological constant Λ for n = 2, m = 0, 1, 2, 3, 4, ̃ǫ = 0.5, and B◦= 4; (e) E versus Λ for n = 0, 1, 2, 3, 4, m = 2, ̃ǫ = 0.5, and B◦= 4; and (f) E versus Λ for n = 2, m = 0, 1, 2, 3, 4, ̃ǫ = 0.5, and B◦= 1. where u = |E|/Ep denotes a dimensionless energy scale normalized by the Planck energy Ep, and ̃ǫ = ǫ/Ep is the corresponding dimensionless rainbow parameter. The second class, inspired by the considerations of LQG [59, 60], is given by f0(u) = 1, f1(u) = q 1 - ̃ǫ|E|ν, ν = 1, 2, with the admissible energy range limited to 0 ≤u = |E|/Ep ≤1. These functional forms introduce energy-dependent deformations to the spacetime geometry, encapsulating potential quantum gravitational corrections at trans-Planckian scales. A. Magueijo-Smolin Rainbow Functions This set of rainbow functions, f0 (u) = 1/ (1 - ̃ǫ |E|), f1 (u) = 1 (with ̃ǫ = ǫ/Ep), is motivated by the hypothesis of varying the speed of light [58]. Substituting this set into the result (19) leads to E2 = ̃Gnm (1 - ̃ǫ |E|)2 ; ̃Gnm = Gnm + m2 ◦. (21) It is important to recall that |E| = E+ is used for test particles, while |E| = -E-is used for antiparticles. This distinction leads to two separate equations: E2 + 1 - ̃ǫ2 ̃Gnm + 2 ̃ǫ ̃Gnm E+ - ̃Gnm = 0, (22) for particles, and E2 - 1 - ̃ǫ2 ̃Gnm -2 ̃ǫ ̃GnmE-- ̃Gnm = 0, (23) for antiparticles. These can be combined to yield the general result: En,m = ± p ̃Gnm 1 + ̃ǫ p ̃Gnm . (24) This result suggests a clear symmetrization of the KG particle and antiparticle energies with respect to E = 0, as also observed in Figure 1. However, it should be noted that the asymptotic behavior of |E±| as B◦→∞approaches a limiting value |E±| ∼2 = 1/ ̃ǫ = Ep/ǫ, as seen in Figs. 1(b) and 1(c). Consequently, it follows that |E±| ≤ Ep =⇒|E±|max = Ep when ǫ = 1. Although it is known that the cosmological constant typically satisfies 0 < Λ ≪1 ( i.e., Λ = 1.4657 × 10-52m-2 = 3.827 × 10-123l-2 p , where lp is the Planck length), we have extended this range and considered larger values of Λ purely for hypothetical exploration. Although it is hypothetical, it allows us to probe theoretical possibilities and emphasizes that the Planck energy Ep is the maximum possible energy for particles and antiparticles (a natural consequence of the RG proposal) and is yet another invariant, alongside the speed of light, as clearly illustrated in Figs. 1(d), 1(e), and 1(f). According to standard quantum mechanics, one would expect that a magnetic field lifts the degeneracy associated with the magnetic quantum number m = ±|m|, at least partially. Interestingly, contrary to this expectation, we observe no such effect. In fact, as the magnetic field strength increases, all m-states appear to collapse into the m = 0 state for a given n. However, the cosmological constant Λ lifts the degeneracy associated with m, even within its standard range 0 < Λ < 1, as clearly shown in Figs. 1(d), 1(e) and 1(f). We attribute this to the uniform nature of the magnetic field employed here. In our recent study [55] of KG particles in a non-uniform external magnetic field within the BM RG framework, such surprising degeneracy behaviors were not observed. B. Rainbow functions pair f0(u) = 1, f1 (u) = p 1 - ̃ǫ |E|ν ; ν = 1, 2 This pair of rainbow functions is motivated by LQG [59, 60], which has been shown to be fully consistent with the RG principles, thus ensuring invariance of the Planck energy Ep. This energy scale represents the maximum attainable energy for both particles and antiparticles (see, e.g., [53, 54, 61]). Accordingly, it is of particular interest to explore 5 FIG. 2. The figure illustrates the energy levels of KG particles and antiparticles as given by Eq. (25). Specifically, the plots show: (a) E as a function of ̃ǫ for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.5, and B◦= 1 = e; (b) E as a function of B◦for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.1, and ̃ǫ = 0.5; (c) E as a function of B◦for n = 1, m = 0, 1, 2, 3, 4, Λ = 0.1, and ̃ǫ = 0.5; (d) E as a function of the cosmological constant Λ for n = 0, m = 0, 1, 2, B◦= 4, and ̃ǫ = 0.5. FIG. 3. The figure shows the energy levels for KG particles and antiparticles given by (28), and we plot (a) E against ̃ǫ for n = 0, m = 0, 1, 2, 3, 4, Λ = 0.5, and B◦= 1 = e, (b) E against B◦for n = 0 and m = 0, 1, 2, 3, 4, Λ = 0.1, and ̃ǫ = 0.5, (c) E against B◦for n = 1 and m = 0, 1, 2, 3, 4, Λ = 0.1, and ̃ǫ = 0.5, and (d) E against Λ for n = 0 and m = 0, 1, 2, B◦= 4, and ̃ǫ = 0.5, (d) E against . the behavior of such rainbow function pairs when applied to KG particles and antiparticles in a magnetized BM spacetime. We begin by considering the case υ = 1, which, through (19), leads to the following expression: E2 + ̃ǫ Gnm|E| - ̃Gnm = 0. (25) The corresponding energy solutions are given by: E+ = - ̃ǫ Gnm 2 + 1 2 q ̃ǫ2 G2 nm + 4 ̃Gnm, (26) and E-= ̃ǫ Gnm 2 -1 2 q ̃ǫ2 G2 nm + 4 ̃Gnm. (27) These energy levels are illustrated in Figure 2. It is evident that, as the cosmological constant Λ →∞, equation (25) implies that the maximum energy, |E|max, tends to 1/ ̃ǫ = 2. This indicates an asymptotic saturation of the energy levels of KG particles and antiparticles in the BM spacetime, as clearly observed in Figure 2. Next, we consider the case υ = 2, which yields the following result: E2 = ̃Gnm 1 + ̃ǫ Gnm ⇒ E± = ± s ̃Gnm 1 + ̃ǫ Gnm . (28) The corresponding energy levels are shown in Figure 3. This plot further confirms the asymptotic behavior of the energy spectrum as the cosmological constant Λ →∞. As indicated by (28), the maximum energy in this case is given by |E|max ∼ p 1/ ̃ǫ = √ 2 ≈1.41. This behavior is in full agreement with the analytical expression and is visually represented in Figure 3. Notably, the same behavior observed for the Magueijo-Smolin rainbow functions in Figure 1 reemerges in Figures 2 and 3. Remarkably, an increase in the uniform magnetic field strength causes all m states to collapse into the m = 0 state for a given radial quantum number n. However, as the cosmological constant Λ (of the magnetized BM spacetime) increases, the m-related degeneracies are partially lifted, although the degeneracy between states with m = ±|m| remains preserved as Λ →1. Interestingly, such behavior is absent in the case of scalar KG bosonic particles in a non-uniform magnetic field within BM spacetime, as discussed in [55]. The three sets of rainbow functions examined here, a Magueijo-Smolin set and two sets motivated by LQG, are all found to be consistent with the RG-inherited Planck energy invariance. IV. CONCLUDING REMARKS In this comprehensive investigation, we have conducted an exact analytical study of KG bosons and antibosons subjected to an external uniform magnetic field in the background of the BM spacetime, modified by the framework of RG and endowed with a positive cosmological constant. This work builds upon recent advances in quantum field theory in curved spacetimes, providing a detailed exploration of how gravitational confinement, topological structures, magnetic interactions, and QG-inspired modifications affect the energy-momentum dispersion relations. A key aspect of our analysis is that the geometry of the BM spacetime naturally induces an infinite sequence of infinite potential barriers, appearing at discrete surfaces determined by the expression ρ = √ 2Λr = τπ, with τ = 0, π, 2π, . . . . These potential barriers act as impenetrable boundaries, dividing spacetime into an infinite number of confinement regions, each similar to an ideal infinite quantum well with rigid boundaries. Within the first of these confinement domains, 6 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 -0.5 0 0.5 0 0.5 1 1.5 2 2.5 3 -0.2 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 0 0.5 1 1.5 2 2.5 3 -0.1 -0.05 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 -0.01 -0.005 0 0.005 0.01 FIG. 4. Radial wave functions Rn,m(ρ) in (20) for quantum numbers n = 0, 1, 2 and m = 0, 1, 2 under two magnetic field strengths, B◦= 1 (left column) and B◦= 10 (right column). Each subplot depicts the behavior of Rn,m(ρ) over the domain ρ ∈[0, π], illustrating how the magnetic field intensity affects the spatial distribution of scalar field modes in the magnetized BM universe. FIG. 5. Radial probability density distributions Pn,m(ρ) are presented for quantum states with n = 0, 1, 2 and m = 0, 1, 2, where Pn,m(ρ) = R ρ 0 |Rn,m(ρ′)|2 ρ′ dρ′. The wavefunctions Rn,m(ρ) are computed under a transverse magnetic field with effective coupling ̃B = eB◦ 2Λ , using e = 1, B◦= 1, and Λ = 0.1. The surface plots are mapped to Cartesian coordinates via x = ρ cos φ, y = ρ sin φ, where both the vertical axis and color scale represent the magnitude of Pn,m(ρ). we have obtained exact, closed-form solutions to the KG equation, expressed using hypergeometric polynomials for the radial wave functions. This framework allowed us to precisely determine the energy spectrum and examine the effects of external fields and RG corrections. Our results reveal several notable features regarding the spectroscopic structure of the system. The inclusion of RG effects, through both Magueijo-Smolin rainbow functions and those inspired by LQG, significantly influences the energy spectrum. In particular, the uniform external magnetic field, which normally lifts the degeneracies associated with the magnetic quantum number m, behaves in an unexpected way within this curved spacetime setting. Specifically, the combined effect of the magnetic field and the BM spacetime geometry causes all magnetic quantum states m to collapse onto the m = 0 state for a fixed radial quantum number n. This behavior differs from what is seen in flat spacetime quantum mechanics and highlights how spacetime curvature and topology can dominate the spectral properties of quantum fields. Furthermore, our analysis shows that the cosmological constant Λ, which is usually important at large (cosmological) scales, also has a direct effect on the local quantum behavior of the system. When Λ increases beyond physically realistic values, a partial lifting of the degeneracies associated with the magnetic quantum number is observed. This indicates a subtle connection between the global geometry of spacetime and the local quantum characteristics of relativistic particles. The study also confirms the consistency of the rainbow functions used with the RG principles. In both the Magueijo-Smolin and LQG-inspired cases, the Planck energy Ep appears clearly as the highest possible energy for particles and antiparticles. This maintains the theoretical 7 FIG. 6. Radial probability density distributions Pn,m(ρ) are shown for quantum states labeled by (n, m), representing the eigenstates of a charged particle confined to a curved surface under a transverse magnetic field. The system parameters are fixed as e = 1, Λ = 0.1, and B◦= 10, corresponding to a strong dimensionless magnetic coupling ̃B = eB◦ 2Λ ≫1. The plots display the spatial distributions of Pn,m(ρ) for discrete states with n = 0, 1, 2 and m = 0, 1, 2, rendered as three-dimensional surfaces in Cartesian coordinates via the polar mapping x = ρ cos φ, y = ρ sin φ, with height and colormap indicating the magnitude of the probability density. framework of QG phenomenology and keeps the symmetry of the particle and antiparticle energy spectra around E = 0. It is important to note that the unusual degeneracy behavior caused by the uniform magnetic field is not observed in cases with non-uniform magnetic fields in the same BM-RG spacetime [55]. This difference emphasizes that both the magnetic field configuration and the geometry of spacetime influence the spectral and confinement properties of quantum fields. With reference to the radial wave functions given in equation (20), we plot in Figure 4 the radial wave functions Rn,m(ρ) for quantum numbers n = 0, 1, 2 and m = 0, 1, 2 at two magnetic field strengths B◦= 1 and 10. As the magnetic field strength increases, the radial wave functions, representing KG particles and antiparticles, become increasingly localized. For B◦= 1, the particles are mainly distributed between ρ ∼0.5 and ρ ∼3, while for B◦= 10, they are restricted to the region ρ ∼1 to ρ ∼2. This localization effect is further illustrated in the corresponding radial probability densities shown in Figures 5 and 6. Radial probability densities are defined as Pn,m(ρ) = Z ρ 0 |Rn,m(ρ′)|2 ρ′ dρ′, and are visualized in two dimensions via x = ρ cos φ and y = ρ sin φ, where the color intensity represents the magnitude of Pn,m. The parameter ̃B = e B◦ 2 Λ characterizes the effect of the external magnetic field on the quantum states (B◦= 1 in Figure 5 and B◦= 10 in Figure 6). As the radial quantum number n increases, the radial probability distributions develop additional nodal rings, corresponding to higher energy states with more radial nodes. For n = 0, the density is mainly concentrated near the origin, while for n = 1, 2, more complex ring-like structures emerge. For fixed n, increasing the azimuthal quantum number m pushes the probability density outward due to the centrifugal barrier, broadening the spatial extent of the wave function. Importantly, increasing the magnetic field strength B◦enhances the magnetic confinement, effectively compressing the probability densities toward smaller radial distances and strengthening localization near the origin. The magnetic field acts as a confining potential that counteracts the outward push caused by larger m values, resulting in sharper and more localized radial peaks. Stronger magnetic fields also slightly shift the nodal positions inward, modifying the spatial probability distribution. These results illustrate how tuning the magnetic field allows control over quantum confinement and the spatial characteristics of the system's eigenstates. Throughout all states, rotational symmetry is preserved, consistent with conservation of angular momentum. The observed evolution of probability densities with increasing n, m, and B◦provides a quantitative understanding of how quantum numbers and magnetic field strength shape spatial quantum states under magnetic confinement. This work provides an exact and detailed description of the KG boson dynamics in the BM-RG spacetime, showing how gravitational confinement, topological features, magnetic fields, and RG corrections influence the energy spectrum and wave functions. These results contribute to the theoretical understanding of quantum fields in curved and magnetized backgrounds and suggest potential areas for further study. Future work could include extending the analysis to fermionic or higher-spin fields, exploring analogous phenomena in optical or condensed matter systems (in principle), and examining how these effects might appear in astrophysical or cosmological contexts. 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The two-dimensional system exhibits unconventional f-wave magnetism and a specific\ntopological metal state characterized by a spin-textured, one-sheeted Fermi surface. The observable\ntransport signatures of the topological metal and its potential experimental realization are briefly\ndiscussed.\nThe topology of electronic band structures has been a\ncentral topic in solid-state physics in recent decades.[1, 2]\nResearch in this area has largely focused on topological\ninsulators and their edge states,[2, 3] as well as on sys-\ntems with topological defects, such as Dirac and Weyl\nsemimetals.[4] It has been shown that nontrivial topo-\nlogical properties give rise to observable transport phe-\nnomena, including highly mobile surface charge carriers,\nanomalous and nonlinear Hall effects.[5, 6] The Berry (ge-\nometric) phase [7] and derived quantities, such as the\nBerry vector potential and curvature, are key theoretical\ntools for describing these phenomena.\nCombined with\nsymmetry analysis, they enable the derivation of topo-\nlogical invariants and the comprehensive classification of\ntopological phases.[8]\nMagnetic topological insulators and semimetals, in-\ncluding helimagnets, have also been studied.[9, 10] At the\nsame time, specific nontrivial band structures in metallic\nhelimagnets have been identified that are distinct from\nthese established classes. In this Letter, we demonstrate\nthat they are unrelated to the Berry phase or topological\ndefects, and we illustrate this finding with simple exam-\nples of helimagnetic systems.\nThe electronic band structure of metallic helimagnets\nhas been studied extensively.[11–16] Both approximate\nmethods [11, 16, 17] and exact solution [18] have estab-\nlished that an effective helical magnetic field produces a\ncharacteristic gapless multiband structure.\nHelical magnetic systems typically comprise two peri-\nodic structures. Let t be a primitive vector of the crys-\ntal lattice such that a translation by this vector rotates\nthe magnetic (spin) system by an angle of αm = 2π/m\n(m > 1). Here and below, we discuss only commensu-\nrate magnetic structures, i.e., m ∈N.\nThe primitive\ntranslation vectors of the system as a whole are then de-\ntermined by the magnetic superstructure (e.g., T = mt).\nBloch’s theorem allows for the construction of a magnetic\nBrillouin zone and yields periodic dispersion relations in\nreciprocal space.\nHowever, within the framework of spin space group\ntheory,[12] a translation by vector t combined with a ro-\n∗yu kudasov@yahoo.com\ntation of the magnetic system by an angle αm, being a\nsymmetry operator (ˆtˆrm), leads to a generalized Bloch\ntheorem [19] and an extended Brillouin zone. As shown\nbelow, the presence of the two commensurate periodic\nstructures has profound consequences for the topology of\nthe band structure.\nIt has been proved recently that the electron dispersion\nin commensurate helimagnets has a symmetry related to\ntime reversal:[20]\nεk,⟨σ⟩= ε−k,−⟨σ⟩\n(1)\nwhere k is the wave vector and ⟨σ⟩is the expectation\nvalue of the spin projection. A Kramers-like degeneracy\nexists throughout the Brillouin zone, except at special\npoints defined by the following conditions: exp\n\u0000ikT\n\u0001\n=\n−1 for even m and exp\n\u00002ikT\n\u0001\n= 1 for odd m.[20]\nThe realization that spin-orbit-free compensated mag-\nnets with strong band spin splitting – such as altermag-\nnets in collinear structures [21, 22] and unconventional\nmagnets in noncollinear ones [23, 24] – constitute a new,\npromising class of functional materials has renewed inter-\nest in these systems.[25, 26] The helimagnets with even\nm possess the symmetry operator ˆT1/2ˆθ, where ˆT1/2 is\nthe translation by T/2 = mt/2 and ˆθ is the time-reversal\noperator.[20] This symmetry gives rise to unconventional\nmagnetism (e.g., of a p-wave type [23]).\nLet us consider the states of a single electron in a crys-\ntal lattice, i.e., in a periodic potential U(r + t) = U(r),\nwhere t is a primitive vector of the Bravais lattice. The\npotential U(r) is assumed to be a scalar (real) or spinor\nfunction. The Schr¨odinger equation in this case has the\nform\nˆHϕ(r) =\n\u0014\n−∆\n2 + U(r)\n\u0015\nϕ(r) = εϕ(r).\n(2)\nEigenvalues εi(k) and eigenvectors ϕi(k) of the Hamil-\ntonian are defined by the wave vectors k. In the limit\nof an infinite crystal, these functions are continuous and,\naccording to Bloch’s theorem, periodic in k-space:[27]\nεi(k + K) = εi(k),\n(3)\nϕi(k + K) = ϕi(k)\n(4)\nwhere K is a vectors of the reciprocal lattice. Here, the\nindex i enumerates the solutions of Eq. (2).\narXiv:2510.14784v1 [cond-mat.other] 16 Oct 2025\n\n2\nTaking into account the boundary conditions Eqs. (3)\nand (4), the Brillouin zone can be represented as a closed\nsmooth manifold B, which is topologically equivalent to\na torus T n with n = 1, 2, 3 for 1D, 2D, and 3D systems,\nrespectively.\nThe Bloch wave function can be expanded as follows:\nϕi(k) = Pn\nj=1 cij(k)ψij where ψij form an orthonormal\nbasis in the coordinate space, cij(k) are the smooth com-\nplex functions of k, and n > 1. Assuming a finite num-\nber of the terms in the sum and taking into account the\nnormalization condition Pn\nj=1 |cij|2 = 1, one can asso-\nciate ϕi(k) at fixed k and i with a point on a (2n −1)-\nsphere (S2n−1).\nIn the case of an infinite number of\nthe terms, S2n−1 is replaced by the infinite-dimensional\nHilbert space ℓ2. Consequently, ϕi(k) defines a closed\nsmooth manifold ϕ in a space X ∼= B ×G, where B ∼= T n\nand G is topologically equivalent to S2n−1 or ℓ2. ϕ and\nB are obviously of the same dimension.\nThe\neigenvectors ϕi(k)\nare\nmutually\northogonal.\nTherefore, for any given k, they correspond to distinct\npoints in the fiber G, ensuring that the sheets ϕi and\nϕj (i ̸= j) do not intersect. This allows us to assume\nthat the map f : ϕ →B is regular, i.e., has a nowhere\nvanishing Jacobian:[28]\ndet\n\u0012∂kα\n∂xβ\n\u0013\n̸= 0,\n(5)\nwhere kα and xβ are the local coordinates defined in B\n(the Brillouin zone) and ϕ, respectively.\nThe energy dispersion εi(k) can also be represented\nas a manifold ε in the space B × I, where I is the unit\ninterval [0, 1] (under the assumption that εi(k) bounded).\nHowever, the map f : ε →B can be singular because the\neigenvalues can be degenerate.\nUnder the conditions stated above, the map f : ϕ →B\nis a covering map. In solid-state physics, usually only\ntrivial coverings are considered; that is, ϕ ∼= B×F where\nF is a countable or finite discrete topological space. In\nthis case, each sheet εi(k) is a closed manifold topologi-\ncally equivalent to the base B. Consequently, the index\ni in Eqs. (3) and (4) specifies the indecomposable cover-\nings, as shown in Fig. 1a.\nThe complete inverse image f −1(k) (the fiber of the\ncovering) of any point k ∈B consists of m points. This\nnumber is referred to as the number of sheets. An inde-\ncomposable covering with m > 1 is nontrivial. Since the\nmap f is regular, the number of sheets for an indecom-\nposable covering must be finite.[29]\nLet γ be a loop in B starting and ending at k0, i.e.,\nk0 →k0 + K in terms of k-space, and let {x1, ..., xm}\nbe the fiber over k0 for an indecomposable covering. For\neach starting point xi in this fiber, there is a unique cov-\nering path µ(xi) in ϕ that covers γ.\nThe endpoint of\nµ(xi) is some xj in the same fiber; however, µ(xi) is not\nnecessarily a loop (if i ̸= j). A schematic 1D example\nof the dispersion curve corresponding to nontrivial ϕ is\nshown in Fig. 1b.\na\nb\n0\n0\n0\n0\np\np\np\np\n-p\n-p\ne\ne\nFIG. 1. 1D band structures and their topological classifica-\ntion. The right panels provide a schematic topological repre-\nsentation, depicting the Brillouin zone as a circle (S1).\nSince any loop within the Brillouin zone B is homo-\ntopic to a null path (i.e., contractible to a point), the\nchoice of the starting point is inessential. For this rea-\nson, we omit the starting point in the notation, where\npossible.\nR2 is the universal covering space of the torus, there-\nfore it is convenient to use a plane representation for 2D\nsystems, as shown in Fig.2a. The horizontal and inclined\nlines in this figure schematically denote the boundaries\nof the Brillouin zone. Gluing them together in the usual\nway forms a torus.[30] The path from the center of any\ncell to that of an adjacent one is a non-contractible loop\non the torus. A pair of such paths, which are generators\nof the fundamental group of the torus, are shown (P and\nQ) in Fig.2a. As an example, consider the three-sheeted\ncovering of the torus, which corresponds to the model\ndiscussed below. The different sheets are indicated by\ncolors. It is clear that the lifts of paths P and Q are not\nclosed, because there is a transition to another sheet. It\nshould be emphasized that the dashed lines and color-\ning in Fig.2a,b are guides for the eye; in effect, there are\nno singular points or lines. The plane representation of\nthe covering is also periodic. The paths PQ and P 2Q−1\nare generators of the fundamental group of the covering\nspace.[31] This group is isomorphic to the fundamental\ngroup of the torus, i.e., π1(ϕ) ≈π1(B).\nR3 is the universal covering space of T 3. Let the 3D\nBrillouin zone be a parallelepiped. A schematic repre-\nsentation of a three-sheeted covering over this base is\nshown in Fig. 2c, where coloring again denotes the dif-\nferent sheets of the covering space. The edges of the unit\nparallelepiped correspond to the generators of π1(B), and\nthe grey arrows indicate the generators of π1(ϕ).\nThe curve in the right panel of Fig. 1b resembles\na knot, particularly considering that ϕ has no self-\nintersections.\nIt is well-known that embedding of a\ncircle into the R3 as a knot leads to the question its\nnontriviality.[32] Therefore, the notion of topological\n\n3\nP\nQ\nP\nQ\na\nP\nQ\nb\nc\nFIG. 2.\nSchematic views of 2D and 3D coverings: (a) plane representation of torus (base), (b) plane representation of a\n3-sheeted covering over T 2, and (c) solid representation of a 3-sheeted covering over T 3. The sheets of covering spaces are\ndistinguished by color and texture.\nequivalence for the coverings discussed above requires\nprecise clarification.\nProposition. Let ϕ be a subspace of X ∼= B × G,\nwhere B and G are path-connected topological spaces,\nand let the restriction of the projection π : X →B be a\nregular covering map p : ϕ →B. Then, if ˜ϕ is ambient\nisotopic to ϕ within X, ˜ϕ is also a covering space (under\nthe restriction of π) which is equivalent to ϕ.\nProof. The ambient isotopy is a continuous mapping\nF : X × I →X such that Ft is a homeomorphism for\nevery t ∈I with F0 = idX, and F1(ϕ) = ˜ϕ.[30] Define\nthe map ˜p : ˜ϕ →B as the composition\n˜p = p ◦¯F1| ˜ϕ,\n(6)\nwhere ¯F is the inverse ambient isotopy, i.e., ¯FtFt = idX.\nA composition of a homeomorphism and a covering map\nis again a covering map. Therefore, ˜p is a covering map.\nFurthermore, ˜p and p are equivalent coverings [33] by\ndefinition Eq. (6).■\nCorollary 1. Monodromy representations of p and ˜p\nare isomorphic.\nCorollary 2.\nSince the covering map p induces a\nmonomorphism p∗: π1(ϕ, x0) →π1(B, k0), the indecom-\nposable covering can be nontrivial only if the fundamen-\ntal group of the base π1(B, k0) is nontrivial. This condi-\ntion is met in case of the Brillouin zone: π1(T n) ≈Zn.\nThus, the topology of the base space, e.g. B = T n, is the\nsource of the nontriviality of the covering map.\nLet us translate the above results into band struc-\nture theory. If ϕ(k) is a m-sheeted regular covering over\nthe Brillouin zone, an individual dispersion sheet is non-\nperiodic within the Brillouin zone, and the conditions\nEqs. (3) and (4) are satisfied due to permutations of the\ndispersion sheets, that is, by monodromy, as schemati-\ncally shown in Fig. 1b. The number of sheets m, as well\nas the sequence of the sheet permutations are topologi-\ncal invariants. Since m is finite, a superstructure must\nexist in the system, i.e., the covering space itself forms a\nperiodic structure in the k-space with a period commen-\nsurate with that of the Brillouin zone.\nThe nontrivial band structure discussed above can be\ndemonstrated with a tight-binding model of a helimag-\nnetic metal. The Hamiltonian has the same general form\nfor 1D, 2D, and 3D lattices:\nˆH3sl = −\nX\n⟨i,j⟩,σ\n\u0010\nˆa†\niσˆajσ + h.c.\n\u0011\n−hi\nX\ni,σ,σ′\nˆa†\niσ ˆσˆaiσ′\n(7)\nwhere ˆa†\niσ(ˆaiσ) is the creation (annihilation) operator for\nan electron with spin projection σ =↑, ↓at the i-th site,\nˆσ are the Pauli matrices, and hi is the (effective) mag-\nnetic field at the i-th site. The notation ⟨. . .⟩denotes the\nsum over nearest-neighbor pairs. We consider a three-\nsublattice model with a 120◦magnetic order.\nIn this\nconfiguration, all hi vectors are coplanar with constant\nmagnitude |hi| = h0 and the angle between the vectors on\ndifferent sublattices is ±2π/3. Furthermore, all nearest-\nneighbor sites belong to different sublattices. Since spin-\norbit coupling is absent in the model, the magnetic plane,\nin which the effective field lies, can be chosen arbitrarily.\nThe 120◦magnetic order is schematically shown in Fig. 3\nfor 1D chain, 2D hexagonal, and 3D simple hexagonal\nlattices. A primitive crystallographic translation rotates\nthe entire magnetic structure by ±2π/3. An explicit ma-\ntrix form of the Hamiltonian Eq. (7) is presented in the\nSupplemental material.\nThe dispersion curves obtained from the 1D tight-\nbinding model are shown in Fig. 4a within the ordi-\nnary (magnetic) (k ∈[−π, π]), and extended (crystallo-\ngraphic) Brillouin zones (k ∈[−3π, 3π]). The lower band\nmarked by the thick line unambiguously corresponds to a\ncontinuous ϕi(k). It is periodic within the extended Bril-\nlouin zone. Band folding results in three non-periodic dis-\npersion sheets within the magnetic (ordinary) one. The\nmagnetic structure is shown by color and arrows within\nthe magnetic Brillouin zone. It should be noted that the\ncommensurate helical magnetic field lifts the spin degen-\neracy of the band structure but conserves the Kramers-\nlike symmetry Eq. (1).\nIf the Fermi level falls into the energy range between\nthe horizontal dashed lines in Fig. 4a, a special state\n\n4\na\nb\nc\nFIG. 3. Helical structures with the 120◦-order on (a) 1D, (b)\n2D, and (c) 3D lattices.\nThe magnetic field at the sites is\nindicated by color and arrows.\n0\n-2\n-1\n0\n1\n2\na\nE\n-\n3\n-3\n-6\n-4\n-2\n0\n2\n4\nb\nM\nK\nM'\nM\nK\nM\nE\n1\nE\n2\nFIG. 4. The band structure in the tight-binding model for (a)\n1D chain (h0 = 0.25) and (b) 2D hexagonal (h0 = 1) lattices.\nThe average spin projection on the axis perpendicular to the\nmagnetic plane is indicated by color and arrows: red and blue\nif |⟨ˆσz⟩| > 1/2, green otherwise.\nG\nM’\nM\nK\na\nG\nM’\nM\nK\nb\nFIG. 5. Fermi surface in the 2D tight-binding model for Fermi\nlevels corresponding to (a) E1 and (b) E2 in Fig. 4b. The\naverage spin projection is indicated in the same manner as in\nFig. 4.\nemerges: a single non-periodic band crosses the Fermi\nlevel.\nWe refer to this as a topological metal because\nsuch a structure is a consequence of a nontrivial covering\nover the Brillouin zone. One can see that a backward\nscattering without spin-flip is forbidden and that a per-\nsistent spin current exists in this case.[16]\nThe band structure of the tight-binding 2D model is\nshown in Fig. 4b.\nThe lower bands also exhibit a 3-\nsheeted structure. Along the M’-Γ-M direction, the lower\ndispersion curves are similar to those of the 1D model\n(Fig. 4a). The shapes of the Fermi surface for the Fermi\nlevel at E1 and E2 are presented in panels (a) and (b)\nof Fig. 5, respectively. The spin splitting of the Fermi\nsurface in Fig. 5a leads to the 2D f-wave magnet.[24]\nIf the Fermi level lies at E2 (Fig. 5b), a single spin-\ntextured Fermi surface appears, with the texture obeying\nEq. (1).\nThis contrasts with altermagnets and uncon-\nventional magnets (e.g. of p- and f-wave type) with the\ntrivial topology, where the spin splitting leads to pairs\nof bands and a Fermi surface with an even number of\nsheets.[22, 24] We refer to this state as a 2D topological\nmetal. It demonstrates unusual transport behavior: the\nsuppression of both backscattering without spin-flip and\numklapp electron-phonon scattering leads to a drastic in-\ncrease in conductivity.[16]\nThe 3D tight-binding model also features nontrivial\nbands. The band structure along the a∗direction cor-\nresponding to M’-Γ-M in the 2D model [34] is similar to\nFig. 4b (see Supplemental Material). However, a detailed\ndiscussion of the 3D model lies beyond the scope of the\npresent work.\nTo practically implement the concept of a topological\nmetal in a real material, several conditions must be met.\nFor instance, in a layered crystal, highly conductive lay-\ners must alternate with magnetic layers possessing 120◦\nordering to induce the effective magnetic field. Further-\nmore, the Fermi level must be located in an energy region\nwith a single non-periodic band. PdCrO2, a metallic de-\nlafossite with a complex magnetic order and anomalously\nhigh conductivity,[35] has been considered a candidate\nfor the topological metal. Non-reciprocal transport [36]\n\n5\nand an unusual anomalous Hall effect [37] have also been\nobserved in this compound.\nEvidence of the topologi-\ncal metal state in this substance could be provided by\nobserving the spin texture of the reconstructed Fermi\nsurface, for example using spin-resolved angle-resolved\nphotoemission spectroscopy.[38] Another way to realize\nthe topological metal is by creating an artificial struc-\nture with alternating magnetic and conductive layers, for\nexample, in van der Waals systems.[39]\nWhile a nontrivial band structure can also exist in non-\nmagnetic systems (e.g., in crystals with helical symmetry\n[40]), the topological metal state appears only in helimag-\nnets due to the combination of lifted spin degeneracy and\nthe Kramers-like symmetry Eq. (1).\nSUPPLEMENTARY MATERIAL\nAn explicit form of the tight-binding models used in\nthe text and the band structure for the 3D model are\nprovided.\nACKNOWLEDGMENTS\nThis work was supported by the National Center for\nPhysics and Mathematics (Project No. 7 “Investigations\nin high and ultrahigh magnetic fields”).\n[1] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88,\n021004 (2016).\n[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[3] G. Tkachev, Topological Insulators, The Physics of Spin\nHelicity in Quantum Transport (Taylor & Francis Group,\nLLC, 2016).\n[4] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.\nMod. Phys. 90, 015001 (2018).\n[5] Z. Du, C. M. Wang, S. Li, H.-Z. Lu, and X. Xie, Nat.\nCommun. 10, 3047 (2024).\n[6] S. Nandy and I. Sodemann, Phys. Rev. B 100, 195117\n(2019).\n[7] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).\n[8] C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Rev.\nMod. Phys. 88, 035005 (2016).\n[9] B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature\n603, 41 (2022).\n[10] X. Yao, J. Gaudet, R. Verma, D. E. Graf, H.-Y. Yang,\nF. Bahrami, R. Zhang, A. A. Aczel, S. Subedi, D. H.\nTorchinsky, J. Sun, A. Bansil, S.-M. Huang, B. Singh,\nP. Blaha, P. Nikoli´c, and F. Tafti, Rev. Rev. X 13, 011035\n(2023).\n[11] I. E. Dzyaloshinskii, JETP 20, 223 (1965).\n[12] W. Brinkman and R. J. Elliott, Proc. Roy. Soc. A 294,\n343 (1966).\n[13] M. D. R. Fr´esard and P. Wolfle, Europhys. Lett. 15, 325\n(1991).\n[14] A. A. Fraerman and O. G. Udalov, Phys. Rev. B 77,\n094401 (2008).\n[15] J. Kishine and A. Ovchinnikov, Theory of monoaxial chi-\nral helimagnet, in Solid State Physics, Vol. 66, edited by\nR. E. Camley and R. L. Stamps (Academic Press, 2015)\npp. 1–130.\n[16] Y. B. Kudasov, JETP Lett. 113, 155 (2021).\n[17] M. Calvo, Phys. Rev. B 19, 5507 (1979).\n[18] M. Calvo, Phys. Rev. B 18, 5073 (1978).\n[19] L. M. Sandratskii, J. Phys.: Cond. Matt. 3, 8565 (1991).\n[20] Y. B. Kudasov, Phy. Rev. B 109, L140402 (2024).\n[21] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and\nA. Zunger, Phys. Rev. B 102, 014422 (2020).\n[22] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X\n12, 040501 (2022).\n[23] B. Brekke, P. Sukhachov, H. G. Giil, A. Brataas, and\nJ. Linder, Phys. Rev. Lett. 133, 236703 (2024).\n[24] M. Ezawa, Phys. Rev. B 111, 125420 (2025).\n[25] Q. Song, S. Stavri´c, P. Barone, A. Droghetti, D. S. An-\ntonenko, J. W. F. Venderbos, C. A. Occhialini, B. Ilyas,\nE. Erge¸cen, N. Gedik, S.-W. Cheong, R. M. Fernandes,\nS. Picozzi, and R. Comin, Nature 642, 64 (2025).\n[26] J. Sears, J. Yao, Z. Hu, W. Tian, N. Aryal, W. Yin, Q. Li,\nand J. M. Tranquada, Phys. Rev. B 112, 094455 (2025).\n[27] N. W. Ashcroft and N. D. Mermin, Solid state physics\n(Harcourt lnc., 1976).\n[28] This is almost always true (apart from special cases, e.g.\ntopological defects in the Brillouin zone like in Dirac and\nWeyl systems).\n[29] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,\nModern Geometry – Methods and Applications, Vol. 2\n(Springer-Verlag, 1985).\n[30] S. Kalajdzievski, An illustrated introduction to topology\nand homotopy (Taylor & Francis, 2015).\n[31] Y. B. Kudasov, JETP Lett. 120, 416 (2024).\n[32] A. Hatcher, Algebraic topology (Cambridge University\nPress, 2002).\n[33] M. A. Armstrong, Basic Topology\n(Springer-Verlag,\n1983).\n[34] L. S. Hart, J. L. Webb, S. Dale, S. J. Bending, M. Mucha-\nKruczynski, D. Wolverson, C. Chen, J. Avila, and M. C.\nAsensio, Sci. Rep. 7, 5145 (2017).\n[35] A. P. Mackenzie, Rep. Prog. Phys. 80, 032501 (2017).\n[36] M. Akaike, Y. Nii, H. Masuda, and Y. Onose, Phys. Rev.\nB 103, 184428 (2021).\n[37] H. Takatsu, S. Yonezawa, S. Fujimoto, and Y. Maeno,\nPhys. Rev. Lett. 105, 137201 (2010).\n[38] J. H. Dil, Phys.: Condens. Matter 21, 403001 (2009).\n[39] C.-Z. Chang, Nat. Mater. 19, 484 (2020).\n[40] M. Damnjanovi´c and I. Miloˇsevi´c, Line Groups in Physics\n(Springer, 2010).\n"},"cleaned":{"kind":"string","value":"Topological bands in metals Yu. B. Kudasov1, ∗ 1Sarov Physics and Technology Institute NRNU \"MEPhI\", 6, str. Dukhov, Sarov, 607186, Russia In crystalline systems with a superstructure, the electron dispersion can form a nontrivial covering of the Brillouin zone. It is proved that the number of sheets in this covering and its monodromy are topological invariants under ambient isotopy. As a concrete manifestation of this nontrivial topology, we analyze three-sublattice models for 120◦-ordered helimagnets in one, two, and three dimensions. The two-dimensional system exhibits unconventional f-wave magnetism and a specific topological metal state characterized by a spin-textured, one-sheeted Fermi surface. The observable transport signatures of the topological metal and its potential experimental realization are briefly discussed. The topology of electronic band structures has been a central topic in solid-state physics in recent decades.[1, 2] Research in this area has largely focused on topological insulators and their edge states,[2, 3] as well as on systems with topological defects, such as Dirac and Weyl semimetals.[4] It has been shown that nontrivial topological properties give rise to observable transport phenomena, including highly mobile surface charge carriers, anomalous and nonlinear Hall effects.[5, 6] The Berry (geometric) phase [7] and derived quantities, such as the Berry vector potential and curvature, are key theoretical tools for describing these phenomena. Combined with symmetry analysis, they enable the derivation of topological invariants and the comprehensive classification of topological phases.[8] Magnetic topological insulators and semimetals, including helimagnets, have also been studied.[9, 10] At the same time, specific nontrivial band structures in metallic helimagnets have been identified that are distinct from these established classes. In this Letter, we demonstrate that they are unrelated to the Berry phase or topological defects, and we illustrate this finding with simple examples of helimagnetic systems. The electronic band structure of metallic helimagnets has been studied extensively.[11-16] Both approximate methods [11, 16, 17] and exact solution [18] have established that an effective helical magnetic field produces a characteristic gapless multiband structure. Helical magnetic systems typically comprise two periodic structures. Let t be a primitive vector of the crystal lattice such that a translation by this vector rotates the magnetic (spin) system by an angle of αm = 2π/m (m > 1). Here and below, we discuss only commensurate magnetic structures, i.e., m ∈N. The primitive translation vectors of the system as a whole are then determined by the magnetic superstructure (e.g., T = mt). Bloch's theorem allows for the construction of a magnetic Brillouin zone and yields periodic dispersion relations in reciprocal space. However, within the framework of spin space group theory,[12] a translation by vector t combined with a ro- ∗yu tation of the magnetic system by an angle αm, being a symmetry operator (ˆtˆrm), leads to a generalized Bloch theorem [19] and an extended Brillouin zone. As shown below, the presence of the two commensurate periodic structures has profound consequences for the topology of the band structure. It has been proved recently that the electron dispersion in commensurate helimagnets has a symmetry related to time reversal:[20] εk,⟨σ⟩= ε-k,-⟨σ⟩ (1) where k is the wave vector and ⟨σ⟩is the expectation value of the spin projection. A Kramers-like degeneracy exists throughout the Brillouin zone, except at special points defined by the following conditions: exp ikT = -1 for even m and exp 2ikT = 1 for odd m.[20] The realization that spin-orbit-free compensated magnets with strong band spin splitting - such as altermagnets in collinear structures [21, 22] and unconventional magnets in noncollinear ones [23, 24] - constitute a new, promising class of functional materials has renewed interest in these systems.[25, 26] The helimagnets with even m possess the symmetry operator ˆT1/2ˆθ, where ˆT1/2 is the translation by T/2 = mt/2 and ˆθ is the time-reversal operator.[20] This symmetry gives rise to unconventional magnetism (e.g., of a p-wave type [23]). Let us consider the states of a single electron in a crystal lattice, i.e., in a periodic potential U(r + t) = U(r), where t is a primitive vector of the Bravais lattice. The potential U(r) is assumed to be a scalar (real) or spinor function. The Schr ̈odinger equation in this case has the form ˆHφ(r) = -∆ 2 + U(r) φ(r) = εφ(r). (2) Eigenvalues εi(k) and eigenvectors φi(k) of the Hamiltonian are defined by the wave vectors k. In the limit of an infinite crystal, these functions are continuous and, according to Bloch's theorem, periodic in k-space:[27] εi(k + K) = εi(k), (3) φi(k + K) = φi(k) (4) where K is a vectors of the reciprocal lattice. Here, the index i enumerates the solutions of Eq. (2). 16 Oct 2025 2 Taking into account the boundary conditions Eqs. (3) and (4), the Brillouin zone can be represented as a closed smooth manifold B, which is topologically equivalent to a torus T n with n = 1, 2, 3 for 1D, 2D, and 3D systems, respectively. The Bloch wave function can be expanded as follows: φi(k) = Pn j=1 cij(k)ψij where ψij form an orthonormal basis in the coordinate space, cij(k) are the smooth complex functions of k, and n > 1. Assuming a finite number of the terms in the sum and taking into account the normalization condition Pn j=1 |cij|2 = 1, one can associate φi(k) at fixed k and i with a point on a (2n -1)- sphere (S2n-1). In the case of an infinite number of the terms, S2n-1 is replaced by the infinite-dimensional Hilbert space l2. Consequently, φi(k) defines a closed smooth manifold φ in a space X ∼= B ×G, where B ∼= T n and G is topologically equivalent to S2n-1 or l2. φ and B are obviously of the same dimension. The eigenvectors φi(k) are mutually orthogonal. Therefore, for any given k, they correspond to distinct points in the fiber G, ensuring that the sheets φi and φj (i ̸= j) do not intersect. This allows us to assume that the map f : φ →B is regular, i.e., has a nowhere vanishing Jacobian:[28] det ∂kα ∂xβ ̸= 0, (5) where kα and xβ are the local coordinates defined in B (the Brillouin zone) and φ, respectively. The energy dispersion εi(k) can also be represented as a manifold ε in the space B × I, where I is the unit interval [0, 1] (under the assumption that εi(k) bounded). However, the map f : ε →B can be singular because the eigenvalues can be degenerate. Under the conditions stated above, the map f : φ →B is a covering map. In solid-state physics, usually only trivial coverings are considered; that is, φ ∼= B×F where F is a countable or finite discrete topological space. In this case, each sheet εi(k) is a closed manifold topologically equivalent to the base B. Consequently, the index i in Eqs. (3) and (4) specifies the indecomposable coverings, as shown in Fig. 1a. The complete inverse image f -1(k) (the fiber of the covering) of any point k ∈B consists of m points. This number is referred to as the number of sheets. An indecomposable covering with m > 1 is nontrivial. Since the map f is regular, the number of sheets for an indecomposable covering must be finite.[29] Let γ be a loop in B starting and ending at k0, i.e., k0 →k0 + K in terms of k-space, and let {x1, ..., xm} be the fiber over k0 for an indecomposable covering. For each starting point xi in this fiber, there is a unique covering path μ(xi) in φ that covers γ. The endpoint of μ(xi) is some xj in the same fiber; however, μ(xi) is not necessarily a loop (if i ̸= j). A schematic 1D example of the dispersion curve corresponding to nontrivial φ is shown in Fig. 1b. a b 0 0 0 0 p p p p -p -p e e FIG. 1. 1D band structures and their topological classification. The right panels provide a schematic topological representation, depicting the Brillouin zone as a circle (S1). Since any loop within the Brillouin zone B is homotopic to a null path (i.e., contractible to a point), the choice of the starting point is inessential. For this reason, we omit the starting point in the notation, where possible. R2 is the universal covering space of the torus, therefore it is convenient to use a plane representation for 2D systems, as shown in Fig.2a. The horizontal and inclined lines in this figure schematically denote the boundaries of the Brillouin zone. Gluing them together in the usual way forms a torus.[30] The path from the center of any cell to that of an adjacent one is a non-contractible loop on the torus. A pair of such paths, which are generators of the fundamental group of the torus, are shown (P and Q) in Fig.2a. As an example, consider the three-sheeted covering of the torus, which corresponds to the model discussed below. The different sheets are indicated by colors. It is clear that the lifts of paths P and Q are not closed, because there is a transition to another sheet. It should be emphasized that the dashed lines and coloring in Fig.2a,b are guides for the eye; in effect, there are no singular points or lines. The plane representation of the covering is also periodic. The paths PQ and P 2Q-1 are generators of the fundamental group of the covering space.[31] This group is isomorphic to the fundamental group of the torus, i.e., π1(φ) ≈π1(B). R3 is the universal covering space of T 3. Let the 3D Brillouin zone be a parallelepiped. A schematic representation of a three-sheeted covering over this base is shown in Fig. 2c, where coloring again denotes the different sheets of the covering space. The edges of the unit parallelepiped correspond to the generators of π1(B), and the grey arrows indicate the generators of π1(φ). The curve in the right panel of Fig. 1b resembles a knot, particularly considering that φ has no selfintersections. It is well-known that embedding of a circle into the R3 as a knot leads to the question its nontriviality.[32] Therefore, the notion of topological 3 P Q P Q a P Q b c FIG. 2. Schematic views of 2D and 3D coverings: (a) plane representation of torus (base), (b) plane representation of a 3-sheeted covering over T 2, and (c) solid representation of a 3-sheeted covering over T 3. The sheets of covering spaces are distinguished by color and texture. equivalence for the coverings discussed above requires precise clarification. Proposition. Let φ be a subspace of X ∼= B × G, where B and G are path-connected topological spaces, and let the restriction of the projection π : X →B be a regular covering map p : φ →B. Then, if ̃φ is ambient isotopic to φ within X, ̃φ is also a covering space (under the restriction of π) which is equivalent to φ. Proof. The ambient isotopy is a continuous mapping F : X × I →X such that Ft is a homeomorphism for every t ∈I with F0 = idX, and F1(φ) = ̃φ.[30] Define the map ̃p : ̃φ →B as the composition ̃p = p ◦ ̄F1| ̃φ, (6) where ̄F is the inverse ambient isotopy, i.e., ̄FtFt = idX. A composition of a homeomorphism and a covering map is again a covering map. Therefore, ̃p is a covering map. Furthermore, ̃p and p are equivalent coverings [33] by definition Eq. (6).■ Corollary 1. Monodromy representations of p and ̃p are isomorphic. Corollary 2. Since the covering map p induces a monomorphism p∗: π1(φ, x0) →π1(B, k0), the indecomposable covering can be nontrivial only if the fundamental group of the base π1(B, k0) is nontrivial. This condition is met in case of the Brillouin zone: π1(T n) ≈Zn. Thus, the topology of the base space, e.g. B = T n, is the source of the nontriviality of the covering map. Let us translate the above results into band structure theory. If φ(k) is a m-sheeted regular covering over the Brillouin zone, an individual dispersion sheet is nonperiodic within the Brillouin zone, and the conditions Eqs. (3) and (4) are satisfied due to permutations of the dispersion sheets, that is, by monodromy, as schematically shown in Fig. 1b. The number of sheets m, as well as the sequence of the sheet permutations are topological invariants. Since m is finite, a superstructure must exist in the system, i.e., the covering space itself forms a periodic structure in the k-space with a period commensurate with that of the Brillouin zone. The nontrivial band structure discussed above can be demonstrated with a tight-binding model of a helimagnetic metal. The Hamiltonian has the same general form for 1D, 2D, and 3D lattices: ˆH3sl = - X ⟨i,j⟩,σ ˆa† iσˆajσ + h.c. -hi X i,σ,σ′ ˆa† iσ ˆσˆaiσ′ (7) where ˆa† iσ(ˆaiσ) is the creation (annihilation) operator for an electron with spin projection σ =↑, ↓at the i-th site, ˆσ are the Pauli matrices, and hi is the (effective) magnetic field at the i-th site. The notation ⟨. . .⟩denotes the sum over nearest-neighbor pairs. We consider a threesublattice model with a 120◦magnetic order. In this configuration, all hi vectors are coplanar with constant magnitude |hi| = h0 and the angle between the vectors on different sublattices is ±2π/3. Furthermore, all nearestneighbor sites belong to different sublattices. Since spinorbit coupling is absent in the model, the magnetic plane, in which the effective field lies, can be chosen arbitrarily. The 120◦magnetic order is schematically shown in Fig. 3 for 1D chain, 2D hexagonal, and 3D simple hexagonal lattices. A primitive crystallographic translation rotates the entire magnetic structure by ±2π/3. An explicit matrix form of the Hamiltonian Eq. (7) is presented in the Supplemental material. The dispersion curves obtained from the 1D tightbinding model are shown in Fig. 4a within the ordinary (magnetic) (k ∈[-π, π]), and extended (crystallographic) Brillouin zones (k ∈[-3π, 3π]). The lower band marked by the thick line unambiguously corresponds to a continuous φi(k). It is periodic within the extended Brillouin zone. Band folding results in three non-periodic dispersion sheets within the magnetic (ordinary) one. The magnetic structure is shown by color and arrows within the magnetic Brillouin zone. It should be noted that the commensurate helical magnetic field lifts the spin degeneracy of the band structure but conserves the Kramerslike symmetry Eq. (1). If the Fermi level falls into the energy range between the horizontal dashed lines in Fig. 4a, a special state 4 a b c FIG. 3. Helical structures with the 120◦-order on (a) 1D, (b) 2D, and (c) 3D lattices. The magnetic field at the sites is indicated by color and arrows. 0 -2 -1 0 1 2 a E - 3 -3 -6 -4 -2 0 2 4 b M K M' M K M E 1 E 2 FIG. 4. The band structure in the tight-binding model for (a) 1D chain (h0 = 0.25) and (b) 2D hexagonal (h0 = 1) lattices. The average spin projection on the axis perpendicular to the magnetic plane is indicated by color and arrows: red and blue if |⟨ˆσz⟩| > 1/2, green otherwise. G M' M K a G M' M K b FIG. 5. Fermi surface in the 2D tight-binding model for Fermi levels corresponding to (a) E1 and (b) E2 in Fig. 4b. The average spin projection is indicated in the same manner as in Fig. 4. emerges: a single non-periodic band crosses the Fermi level. We refer to this as a topological metal because such a structure is a consequence of a nontrivial covering over the Brillouin zone. One can see that a backward scattering without spin-flip is forbidden and that a persistent spin current exists in this case.[16] The band structure of the tight-binding 2D model is shown in Fig. 4b. The lower bands also exhibit a 3sheeted structure. Along the M'-Γ-M direction, the lower dispersion curves are similar to those of the 1D model (Fig. 4a). The shapes of the Fermi surface for the Fermi level at E1 and E2 are presented in panels (a) and (b) of Fig. 5, respectively. The spin splitting of the Fermi surface in Fig. 5a leads to the 2D f-wave magnet.[24] If the Fermi level lies at E2 (Fig. 5b), a single spintextured Fermi surface appears, with the texture obeying Eq. (1). This contrasts with altermagnets and unconventional magnets (e.g. of p- and f-wave type) with the trivial topology, where the spin splitting leads to pairs of bands and a Fermi surface with an even number of sheets.[22, 24] We refer to this state as a 2D topological metal. It demonstrates unusual transport behavior: the suppression of both backscattering without spin-flip and umklapp electron-phonon scattering leads to a drastic increase in conductivity.[16] The 3D tight-binding model also features nontrivial bands. The band structure along the a∗direction corresponding to M'-Γ-M in the 2D model [34] is similar to Fig. 4b (see Supplemental Material). However, a detailed discussion of the 3D model lies beyond the scope of the present work. To practically implement the concept of a topological metal in a real material, several conditions must be met. For instance, in a layered crystal, highly conductive layers must alternate with magnetic layers possessing 120◦ ordering to induce the effective magnetic field. Furthermore, the Fermi level must be located in an energy region with a single non-periodic band. PdCrO2, a metallic delafossite with a complex magnetic order and anomalously high conductivity,[35] has been considered a candidate for the topological metal. Non-reciprocal transport [36] 5 and an unusual anomalous Hall effect [37] have also been observed in this compound. Evidence of the topological metal state in this substance could be provided by observing the spin texture of the reconstructed Fermi surface, for example using spin-resolved angle-resolved photoemission spectroscopy.[38] Another way to realize the topological metal is by creating an artificial structure with alternating magnetic and conductive layers, for example, in van der Waals systems.[39] While a nontrivial band structure can also exist in nonmagnetic systems (e.g., in crystals with helical symmetry [40]), the topological metal state appears only in helimagnets due to the combination of lifted spin degeneracy and the Kramers-like symmetry Eq. (1). SUPPLEMENTARY MATERIAL An explicit form of the tight-binding models used in the text and the band structure for the 3D model are provided. ACKNOWLEDGMENTS This work was supported by the National Center for Physics and Mathematics (Project No. 7 \"Investigations in high and ultrahigh magnetic fields\"). [1] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88, 021004 (2016). [2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [3] G. Tkachev, Topological Insulators, The Physics of Spin Helicity in Quantum Transport (Taylor & Francis Group, LLC, 2016). [4] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018). [5] Z. Du, C. M. Wang, S. Li, H.-Z. Lu, and X. Xie, Nat. Commun. 10, 3047 (2024). [6] S. Nandy and I. Sodemann, Phys. Rev. B 100, 195117 (2019). [7] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). [8] C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016). [9] B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 603, 41 (2022). [10] X. Yao, J. Gaudet, R. Verma, D. E. Graf, H.-Y. Yang, F. Bahrami, R. Zhang, A. A. Aczel, S. Subedi, D. H. Torchinsky, J. Sun, A. Bansil, S.-M. Huang, B. Singh, P. Blaha, P. Nikoli ́c, and F. Tafti, Rev. Rev. X 13, 011035 (2023). [11] I. E. Dzyaloshinskii, JETP 20, 223 (1965). [12] W. Brinkman and R. J. Elliott, Proc. Roy. Soc. A 294, 343 (1966). [13] M. D. R. Fr ́esard and P. Wolfle, Europhys. Lett. 15, 325 (1991). [14] A. A. Fraerman and O. G. Udalov, Phys. Rev. B 77, 094401 (2008). [15] J. Kishine and A. Ovchinnikov, Theory of monoaxial chiral helimagnet, in Solid State Physics, Vol. 66, edited by R. E. Camley and R. L. Stamps (Academic Press, 2015) pp. 1-130. [16] Y. B. Kudasov, JETP Lett. 113, 155 (2021). [17] M. Calvo, Phys. Rev. B 19, 5507 (1979). [18] M. Calvo, Phys. Rev. B 18, 5073 (1978). [19] L. M. Sandratskii, J. Phys.: Cond. Matt. 3, 8565 (1991). [20] Y. B. Kudasov, Phy. Rev. B 109, L140402 (2024). [21] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020). [22] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022). [23] B. Brekke, P. Sukhachov, H. G. Giil, A. Brataas, and J. Linder, Phys. Rev. Lett. 133, 236703 (2024). [24] M. Ezawa, Phys. Rev. B 111, 125420 (2025). [25] Q. Song, S. Stavri ́c, P. Barone, A. Droghetti, D. S. Antonenko, J. W. F. Venderbos, C. A. Occhialini, B. Ilyas, E. Erge ̧cen, N. Gedik, S.-W. Cheong, R. M. Fernandes, S. Picozzi, and R. Comin, Nature 642, 64 (2025). [26] J. Sears, J. Yao, Z. Hu, W. Tian, N. Aryal, W. Yin, Q. Li, and J. M. Tranquada, Phys. Rev. B 112, 094455 (2025). [27] N. W. Ashcroft and N. D. Mermin, Solid state physics (Harcourt lnc., 1976). [28] This is almost always true (apart from special cases, e.g. topological defects in the Brillouin zone like in Dirac and Weyl systems). [29] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications, Vol. 2 (Springer-Verlag, 1985). [30] S. Kalajdzievski, An illustrated introduction to topology and homotopy (Taylor & Francis, 2015). [31] Y. B. Kudasov, JETP Lett. 120, 416 (2024). [32] A. Hatcher, Algebraic topology (Cambridge University Press, 2002). [33] M. A. Armstrong, Basic Topology (Springer-Verlag, 1983). [34] L. S. Hart, J. L. Webb, S. Dale, S. J. Bending, M. MuchaKruczynski, D. Wolverson, C. Chen, J. Avila, and M. C. Asensio, Sci. Rep. 7, 5145 (2017). [35] A. P. Mackenzie, Rep. Prog. Phys. 80, 032501 (2017). [36] M. Akaike, Y. Nii, H. Masuda, and Y. Onose, Phys. Rev. B 103, 184428 (2021). [37] H. Takatsu, S. Yonezawa, S. Fujimoto, and Y. Maeno, Phys. Rev. Lett. 105, 137201 (2010). [38] J. H. Dil, Phys.: Condens. Matter 21, 403001 (2009). [39] C.-Z. Chang, Nat. Mater. 19, 484 (2020). [40] M. Damnjanovi ́c and I. Miloˇsevi ́c, Line Groups in Physics (Springer, 2010)."}}},{"rowIdx":196,"cells":{"id":{"kind":"string","value":"2510.14786"},"raw":{"kind":"string","value":"SCALING LIMITS FOR THE CRITICAL LEVEL-SET PERCOLATION\nOF THE GAUSSIAN FREE FIELD ON REGULAR TREES\nJIˇR´I ˇCERN´Y, RAMON LOCHER\nAbstract. We continue the study of the level-set percolation of the discrete Gaussian\nfree field (GFF) on regular trees in the critical regime, initiated in [ˇCL25]. First, we\nderive a sharp asymptotic estimate for the probability that the connected component of\nthe critical level set containing the root of the tree reaches generation n. In particular, we\nshow that the one-arm exponent satisfies ρ = 1. Next, we establish a Yaglom-type limit\ntheorem for the values of the GFF at generation n within this component. Finally, we\nshow that, after a correct rescaling, this component conditioned on reaching generation\nn converges, as n →∞, to Aldous’ continuum random tree.\n1. Introduction\nThe Gaussian free field’s level-set percolation, especially on Zd, is a significant model\nin percolation theory that is characterised by its long-range dependencies. Initial investi-\ngations into this model trace back to the 1980s, with pioneering studies [BLM87, MS83,\nLS86]. Over the last decade, renewed interest has been ignited by the findings in [RS13],\ndemonstrating that on Zd, the model undergoes a distinctive percolative phase transition\nat a critical threshold h∗= h∗(d) for any dimension d ≥3. Follow-up research, including\npapers [DRS14, PR15, DPR18, Szn19, CN20, GRS22, PS22], has provided a comprehen-\nsive understanding of the model’s behaviour in both subcritical and supercritical phases,\noften making use of additional natural critical points in order to work in strongly sub-\n/super-critical regime.\nNotably, [DCGRS23] confirmed the alignment of these critical\npoints with h∗, indicating a precise phase transition.\nIn this paper, we continue the study of level-set percolation of the discrete Gaussian\nfree field (GFF) on regular trees in the critical regime, building on the work initiated\nin [ˇCL25]. Our main contributions are threefold: First, we derive a sharp asymptotic\nestimate for the one-arm probability. Second, we establish a Yaglom-type limit theorem\nfor the field at vertices located at distance n from the root. Finally, we show that the\nconnected component of the critical level set, conditioned to be large, converges to Aldous’\nContinuum Random Tree. These results are stated precisely in Theorems 2.1, 2.2, and 2.3,\nrespectively.\nThe considered model was initially studied in [Szn16] where the critical value h∗was\nidentified in terms of the largest eigenvalue of a specific integral operator. Additionally, a\ncomparison with random interlacements was employed to establish bounds on h∗, notably\ndemonstrating that 0 < h∗< ∞. Subsequently, in [AˇC20], sub- and supercritical phases of\nthe model were studied in detail. The main results of this paper include the continuity of\nthe percolation probability outside the critical level h∗, and accurate estimates on the size\nof connected components of the level sets in both phases. Later, in our previous paper\n[ˇCL25], properties of the critical and the near-critical model were investigated. It was\nproved there that there is no percolation at the critical point h∗, and that the percolation\nprobability is continuous also at this point, with a precise asymptotics formula for this\n1\narXiv:2510.14786v1 [math.PR] 16 Oct 2025\n\nSCALING LIMITS FOR THE CRITICAL GFF\n2\nprobability in the regime h ↑h∗. Further, we provided rather precise estimates on the tail\nof the size of the connected component at criticality. Here, we complement these findings\nwith further results concerning the model at the critical level h∗.\nAs in [Szn16, AˇC20, ˇCL25], we will strongly rely on the fact that the model admits a\nrepresentation as a branching process with an uncountable and unbounded type space.\nFor the critical single-type Galton-Watson process, analogous versions of Theorems 2.1\nand 2.2 go back to Kolmogorov [Kol38] and Yaglom [Yag47]. In the more general case of\nbranching processes with a finite type space, similar results are also well known. General\nscaling limits in the spirit of Theorem 2.3 were first introduced for critical Galton-Watson\nprocesses in [Ald91], and later extended to branching processes with finite or countably\ninfinite types [Mie08, dR17]. More recently, [Pow19] shows a similar scaling limit result\nin the setting of critical branching diffusion on bounded domains.\nRecent advances in branching process theory have extended these classical results in\nseveral important directions. [CTJP24] obtained refined convergence rates for Yaglom\nlimits in varying environments, providing Wasserstein metric bounds that may also be\nuseful for analyses in our unbounded type-space setting. [BDIM23] established Yaglom-\ntype theorems for branching processes in sparse random environments, an intermediate\nframework that connects the classical Galton-Watson model with fully random environ-\nments. [BFRS24] proved a Yaglom-type theorem for near-critical branching processes\nin random environments and further showed that, under survival conditioning, the ge-\nnealogical structure of the population at a fixed time horizon converges to a time-changed\nBrownian coalescent point process.\nDue to the nature of the branching process appearing in our model (in particular\nbecause its type space is uncountable and unbounded), no previous results are directly\napplicable to our setting. In this article, we adapt and extend the strategy of [Pow19],\naddressing the fundamental challenge of unbounded type spaces.\n2. Model and results\nWe start with the definition of the model. Let T be the infinite (d + 1)-regular tree,\nd ≥2, rooted at an arbitrary fixed vertex o ∈T, endowed with the usual graph distance\nd(·, ·). On T, we consider the Gaussian free field φ = (φv)v∈T which is a centred Gaussian\nprocess whose covariance function agrees with the Green function of the simple random\nwalk on T (see (3.2) for the precise definition). We use P to denote the law of this process\non RT. For x ∈R, we write Px for the conditional distribution of φ given that φo = x,\n(2.1)\nPx[ · ] := P[ · | φo = x].\n(For an explicit construction of Px, see (3.4) and the paragraph below it.) Furthermore,\nlet ¯o ∈T be an arbitrary fixed neighbour of the root o, and define the forward tree T+ by\n(2.2)\nT+ := {v ∈T : ¯o is not contained in the geodesic path from o to v}.\nWe analyse the percolation properties of the (super-)level sets of φ above level h ∈R,\n(2.3)\nEh\nφ := {v ∈T : φv ≥h}.\nIn particular, we are interested in the connected component of this set containing the\nroot o,\n(2.4)\nCh\no := {v ∈T : v is connected to o in Eh\nφ},\nh ∈R.\nThe critical height h∗of the level-set percolation is defined by\n(2.5)\nh∗= h∗(d) := inf\n\b\nh ∈R : P[|Ch\no | = ∞] = 0\n\t\n.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n3\nIt is well known that h∗is non-trivial and strictly positive (see [Szn16, Corollary 4.5]).\nMoreover, as proved in [Szn16], h∗can be characterized with the help of the operator\nnorms of a certain family of non-negative operators (Lh)h∈R acting on the space L2(ν),\nwhere ν is a centred Gaussian measure with variance σ2\nν = d/(d −1). We provide more\ndetails on this characterization in Section 3 below. Here, we only define λh to be the\nlargest eigenvalue of Lh and χh the corresponding normed eigenfunction, and recall that\nh∗is the unique solution to\n(2.6)\nλh∗= 1.\nThe scalar product on L2(ν) will be denoted by ⟨·, ·⟩.\nWe use N h\nn to denote the set of vertices in Ch\no that are at a distance n from the root,\n(2.7)\nN h\nn = {v ∈Ch\no : d(v, o) = n}.\nand set\n(2.8)\nN h,+\nn\n= N h\nn ∩T+.\nSince we almost exclusively deal with the critical case, we abbreviate χ := χh∗, Co = Ch∗\no ,\nL := Lh∗, Nn := N h∗\nn\nand N +\nn := N h∗,+\nn\n.\nOur first result describes the exact asymptotic behaviour of the probabilities (condi-\ntional and unconditional) that Nn and N +\nn are non-empty, that is, that Co has diameter\nat least n.\nTheorem 2.1. For every x ≥h∗, as n →∞,\nPx[N +\nn ̸= ∅] = C1χ(x)n−1(1 + o(1)),\n(2.9)\nP[N +\nn ̸= ∅] = C1⟨1, χ⟩n−1(1 + o(1)),\n(2.10)\nwhere\n(2.11)\nC1 =\n2d\nd −1\n1\n⟨χ2, χ⟩.\nIf the event {N +\nn ̸= ∅} is replaced by {Nn ̸= ∅}, the same results hold with C1 replaced by\neC1 = C1(d + 1)/d.\nIn particular, Theorem 2.1 proves that the one-arm exponent of the critical level set,\ndefined by ρ = −limn→∞log n/ log(P[Nn ̸= ∅]) (see, e.g., [Gri99, Section 9.1]) satisfies\n(2.12)\nρ = 1.\nThis complements the values of two other important critical exponents for our model\ngiven in [ˇCL25, (2.22)], where it was shown that δ = 2 and β = 1.\nThe one-arm probability and the associated critical exponent is an actively studied\nquantity in several prominent percolation models. Notably, in the context of level-set\npercolation of the GFF on the metric graph of Zd, recent work has led to a detailed un-\nderstanding of the one-arm probability across all dimensions. This includes the derivation\nof bounds for d = 3 in [DPR25], for d > 6 in [CD25], and for the intermediate regime\n3 ≤d ≤6 in [CD24].\nOur remaining main results consider the critical component conditioned on being large,\nmore precisely conditioned on the rare event {N +\nn ̸= ∅}. The first such result is a Yaglom-\ntype limit theorem for the GFF restricted to N +\nn .\n\nSCALING LIMITS FOR THE CRITICAL GFF\n4\nTheorem 2.2. For f ∈L2(ν), n ≥1, and x ≥h∗, let Zf,x\nn , resp. Zf\nn, be a random\nvariable distributed as n−1 P\nv∈N+\nn f(φv) under the conditional measure Px[ · |N +\nn ̸= ∅],\nresp. P[ · |N +\nn ̸= ∅]. Let further Z be an exponential random variable with mean one.\nThen, with C1 as in (2.11),\n(2.13)\nlim\nn→∞Zf,x\nn\n= lim\nn→∞Zf\nn = C−1\n1 ⟨χ, f⟩Z\nin distribution.\nIn the case of the critical single-type Galton-Watson process, results analogous to The-\norem 2.2 trace back to the classical work of Yaglom [Yag47]. For more general branching\nprocesses with a finite type space, similar theorems are also well established; see, for\ninstance, [Mod71, Theorem 10.1] and other foundational texts in the branching process\nliterature. In recent years, Yaglom-type limits have been proven in various extended set-\ntings. These include critical non-local branching Markov processes [HHKW22], branching\nBrownian motion with absorption [MS22], and branching diffusions in bounded domains\n[Pow19]. Additionally, [GLLP22] establishes a Yaglom-type result for critical branching\nprocesses in a random Markovian environment with finite state space.\nOur third result concerns a scaling limit for the critical component Co ∩T+, viewed as\na metric space, under the conditional law Px[ · |N +\nn ̸= ∅]. To this end, let (Tn,x, dn,x) be\na random compact metric space whose law coincides with that of (Co ∩T+, n−1d) under\nPx[ · |N +\nn ̸= ∅] (recall that d denotes the distance on T). We show that the sequence\n(Tn,x, dn,x) converges in distribution to a conditioned Brownian continuum random tree\n(Te, de), whose contour function e is a Brownian excursion conditioned to reach height at\nleast 1.\nTheorem 2.3. For every x ≥h∗, as n →∞,\n(2.14)\n(Tn,x, dn,x) →(Te, de)\nin distribution, with respect to the Gromov-Hausdorff topology.\nGeneral scaling limits of critical Galton-Watson processes, in the spirit of Theorem 2.3,\nwere first introduced in [Ald91]. A corresponding result for critical multi-type processes\nwith finitely many types was established in [Mie08], and later extended to processes\nwith a countably infinite type space in [dR17]. More recently, [CKKM24] extended the\nclassical continuous random tree convergence to Galton-Watson trees evolving in a random\nenvironment, where each generation has a random offspring distribution with mean one\nand finite expected variance. In the context of critical branching diffusions, [Pow19] proves\nan invariance principle under the assumptions of a bounded domain, finite second moment\nof the offspring distribution, and an elliptic diffusion generator. However, for branching\ndiffusions in general (unbounded) domains, analogous results are not yet available (see\n[Pow19, Question 1.8]).\nRemark 2.4. Theorems 2.2 and 2.3 hold without any further change if the conditioning\ntherein is changed from Px[ · |N +\nn ̸= ∅] to Px[ · |Nn ̸= ∅]. For the sake of brevity, we refrain\nfrom providing detailed proofs of these results.\nWe briefly discuss the organisation of this article. In Section 3, we collect relevant back-\nground on the GFF on regular trees along with the framework of branching processes with\nspines. Section 4 presents the proof of Theorem 2.1. In Section 5, we prove Theorem 2.2\nand, in a dedicated subsection, establish additional results for the model conditioned on\nthe event {N +\nn ̸= ∅}. Section 6 introduces an auxiliary martingale Sn and establishes\nscaling limit results for this martingale and some related processes. Section 7 focuses on\n\nSCALING LIMITS FOR THE CRITICAL GFF\n5\nthe “height process” Hn, defined via the distance to the origin in a depth-first traversal of\nCo ∩T+, and investigates its connection to the martingale Sn. Finally, Section 8 concludes\nthe article with the proof of Theorem 2.3, which combines topological arguments with the\nresults from Sections 6 and 7.\n3. Notation and useful results\nIn this section we introduce the notation used throughout the paper and recall some\nknown facts about the level set percolation of the Gaussian free field on trees. We then\nbriefly present the formalism of branching processes with spines and apply it to our model.\nAs already stated in the introduction, we use T to denote the (d+1)-regular tree, d ≥2,\nthat is an infinite tree whose every vertex has exactly d + 1 neighbours. For two vertices\nv, w ∈T we use d(v, w) to denote the usual graph distance. The tree is rooted at an\narbitrary fixed vertex o ∈T, ¯o ∈T denotes a fixed neighbour of o, and T+ stands for the\nforward tree, see (2.2). We set |v| = d(o, v) and write\n(3.1)\nSn = {v ∈T : |v| = n},\nS+\nn = Sn ∩T+\nfor the spheres with radius n centred at o. For every v ∈T \\ {o} we use p(v) to denote its\nparent in T, that is the only vertex on the geodesic path from v to o with |p(v)| = |v| −1.\nWe write desc(v) for the set of direct descendants of v, and sib(v) = desc(p(v)) for the\nset of its siblings, including itself. Finally, if w is an ancestor of v, that is w lies on the\ngeodesics from o to v, we write w ⪯v.\nThroughout the paper we use the usual notation for the asymptotic relation of two\nfunctions f and g: We will write f(s) ∼g(s) as s →∞if lims→∞f(s)/g(s) = 1,\nf(s) = o(g(s)) as s →∞if lims→∞|f(s)|/g(s) = 0, and f(s) = O(g(s)) as s →∞if\nlim sups→∞|f(s)|/g(s) < ∞. We use c, c′, c1, . . . to denote finite positive constants whose\nvalue may change from place to place and which can only depend on d. The dependence\nof these constants on additional parameters is explicitly mentioned.\n3.1. Properties of the GFF. We consider the Gaussian free field φ = (φv)v∈T which is\nthe centred Gaussian process on T whose covariance function is the Green function of the\nsimple random walk on T,\n(3.2)\nE[φvφw] = g(v, w) :=\n1\nd + 1Ev\nh\n∞\nX\nk=0\n1Xk=w\ni\n,\nv, w ∈T,\nwhere Ev stands for the expectation with respect to the simple random walk (Xk)k≥0 on\nT starting at v ∈T.\nWe frequently use the fact that the Gaussian free field on T can be viewed as a multi-\ntype branching process with a continuous type space (see [Szn16, Section 3] and [AˇC20,\nSection 2.1]). To this end, we define\n(3.3)\nσ2\nν :=\nd\nd −1\nand\nσ2\nY := d + 1\nd\n,\nand let (Yv)v∈T be a collection of independent centred Gaussian random variables on some\nauxiliary probability space such that Yo ∼N(0, σ2\nν) and Yv ∼N(0, σ2\nY ) for v ̸= o. We\nthen define another field eφ on T by\n(a)\neφo := Yo,\n(b)\nfor v ̸= o we recursively set eφv := d−1 eφp(v) + Yv.\n(3.4)\n\nSCALING LIMITS FOR THE CRITICAL GFF\n6\nAs explained, e.g., in [AˇC20, (2.9)], the law of (eφv)v∈T agrees with the law P of the\nGaussian free field φ. Therefore, we will always assume that the considered Gaussian free\nfield is constructed in this way and will not distinguish between φ, and eφ from now on.\nRepresentation (3.4) of φ can be used to give a concrete construction for the conditional\nprobability Px introduced in (2.1): It suffices to replace (a) in (3.4) by eφo = x.\nIn\naddition, (3.4) can directly be used to construct a monotone coupling of Px and Py. As a\nconsequence:\n(3.5)\nIf x < y, then Py stochastically dominates Px,\nthat is Ex[f(φ)] ≤Ey[f(φ)] for every bounded increasing function f : RT →R.\nFrom the construction (3.4) it follows that the GFF on T can be viewed as a multi-type\nbranching process where the type corresponds to the value of the field φ. The type of\nthe initial individual o of this branching process is distributed as Yo. Every individual\nv in this branching process then has d descendants (d + 1 if v = o) whose types are\nindependently given by d−1φv + Y , with Y ∼N(0, σ2\nY ). The branching process point of\nview can be adapted to the connected component Ch\no (defined in (2.4)) by considering\nthe same multi-type branching process but killing immediately all individuals with type\nsmaller than h (and not allowing them to have descendants themselves).\nWe now recall in more detail the spectral machinery introduced in [Szn16] in order to\ncharacterise the critical value h∗. Let ν be a centred Gaussian measure on R with variance\nσ2\nν (as defined in (3.3)), and let Y be a centred Gaussian random variable with variance\nσ2\nY . The expectation with respect to this random variable is denoted by EY . We consider\nthe Hilbert space L2(ν) := L2(R, B(R), ν), and for every h ∈R, define the operator Lh\nacting on L2(ν) by\nLh[f](x) := 1[h,∞)(x) d EY\nh\n1[h,∞)\n\u0010\nY + x\nd\n\u0011\nf\n\u0010\nY + x\nd\n\u0011i\n= 1[h,∞)(x) d\nZ\n[h,∞)\nf(y)ρY\n\u0010\ny −x\nd\n\u0011\ndy,\n(3.6)\nwhere ρY denotes the density of Y . From the branching process construction (3.4) of φ,\nit follows that\n(3.7)\nLh[f](x) = Ex\n\u0014 X\nv∈Nh,+\n1\nf(φv)\n\u0015\n,\nwhere N h,+\nn\nis defined in (2.8). Iterating this expression one also obtains\n(3.8)\nLn\nh[f](x) = Ex\n\u0014 X\nv∈Nh,+\nn\nf(φv)\n\u0015\n,\nn ∈N.\nFinally, we let λh stand for the operator norm of Lh in L2(ν),\n(3.9)\nλh := ∥Lh∥L2(ν)→L2(ν).\nThe following proposition summarises some known properties of the operator Lh as well\nas the connection between Lh and the critical height h∗.\nProposition 3.1 ([Szn16] Propositions 3.1, 3.3, Corollary 4.5). For all h ∈R, Lh is a\nself-adjoint non-negative Hilbert-Schmidt operator on L2(ν), λh is a simple eigenvalue of\nLh, and there exists a unique χh ≥0 with unit L2(ν)-norm, which is continuous, strictly\npositive on [h, ∞), vanishing on (−∞, h), and such that\n(3.10)\nLh[χh] = λhχh.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n7\nAdditionally, the map h 7→λh is a decreasing homeomorphism from R to (0, d) and h∗is\nthe unique value in R such that λh∗= 1. Finally, for every d ≥2,\n(3.11)\n0 < h∗< ∞.\nCombining Proposition 3.1 with (3.8) gives that for every n ∈N,\n(3.12)\nEx\n\u0014 X\nw∈Nh,+\nn\nχh(φw)\n\u0015\n= λn\nhχh(x).\nWe will require estimates on the norms of Lh[f], which follow from the hypercontrac-\ntivity of the Ornstein-Uhlenbeck semigroup. For part (a) of the following proposition, we\nrefer to (3.14) in [Szn16] or (2.14) in [AˇC20]. Part (b) then follows directly from part (a)\nin combination with generalized H¨older’s inequality.\nProposition 3.2. (a) For every f ∈L2(ν), h ∈R, 1 < p < ∞and q ≤(p −1)d2 + 1,\n(3.13)\n∥Lh[f]∥Lq(ν) ≤d∥f∥Lp(ν).\n(b) For every 1 ≤k ≤(d2 + 1)/2 and f1, . . . , fk ∈L2(ν)\n(3.14)\n\r\r\r\nk\nY\ni=1\nLh[fi]\n\r\r\r\nL2(ν) ≤dk\nk\nY\ni=1\n∥fi∥L2(ν).\nThe next proposition recalls known properties of the critical component Co from [ˇCL25].\nProposition 3.3 (Theorem 2.1 and Theorem 2.3 in [ˇCL25]). There is C ∈(0, ∞) such\nthat for every x ∈R, as n →∞,\n(3.15)\nPx[|Co ∩T+| > n] = Cχ(x)n−1/2(1 + o(1)).\nIn particular, for every x ∈R,\n(3.16)\nPx[|Co ∩T+| = ∞] = 0.\nWe will also need the following three properties of χ, the first one is proved in Re-\nmark 2.5 of [ˇCL25], the second is a direct consequence of Proposition 3.1 in [AˇC20], and\nthe last one is proved in Appendix A:\nx 7→χ(x) is non-decreasing,\n(3.17)\nc1x ≤χ(x) ≤c2x\nfor all x ≥h∗and some c1, c2 ∈(0, ∞),\n(3.18)\nx 7→χ(x) is Lipschitz on [h∗, ∞).\n(3.19)\n3.2. Branching processes with spines. We now recall the machinery of branching\nprocesses with spines which is frequently used in the theory of branching processes, and\nspecialize it to our model, in order to study the critical component Co. We then state\nmany-to-few formulas that express certain moments for the original branching process in\nterms of the dynamics along the spines, see Proposition 3.5 below. Later, in Section 5.1,\nwe will see that the processes with spines can be used to describe the distribution of Co\nconditioned on being infinite. The content of this section is mostly based on [HR17].\nThe main idea of the machinery is to designate several lines of descent in the branching\nprocess, called spines. These spines are then used to introduce a certain change of measure,\nunder which the vertices on the spine exhibit modified branching behaviour, while the\nnon-spine vertices behave as in the original process.\nFor the construction, we need some notation. For x ∈R and k ∈N, we first introduce\nmeasures P k\nx under which the behaviour of the field is the same as under Px, but in\n\nSCALING LIMITS FOR THE CRITICAL GFF\n8\naddition there are k distinguished spines. Formally, the measure P k\nx is the distribution\nof a (R ∪{†}) × {0, 1, . . . , k}-valued stochastic process (φv, lv)v∈T+. This process assigns\nto every v ∈T+ a field value φv ∈R ∪{†} (where † is a cemetery state) and a number\nlv ∈{0, . . . , k} that represents the number of spine marks on v. Under P k\nx , the law of\n(φv)v∈T+ is a straightforward modification of the original measure Px, the only difference\nis that we set φv = † for every v ̸∈Co. The random variables (lv)v∈T+ are independent of\nthe field values (φv)v∈T+. The root node o has exactly k marks, lo = k. The remaining\nrandom variables (lv)v̸=o are constructed recursively as follows: If a node v ∈T+ carries j\nmarks, then each of its j marks ‘moves’ to one of its d direct descendants independently\nuniformly at random.\n(Note that nodes in the cemetery state can carry marks.)\nAs\nconsequence, under P k\nx , in every generation n, there are exactly k marks present, that is\nP\nv∈S+\nn lv = k. We use P k\nx also for the corresponding expectations.\nFor i = 1, . . . , k, we denote by σi\nn the node that carries the i-th mark in generation\nn (that is |σi\nn| = n), and set ξi\nn = φσin to be its type. We also define skel(n) = {v ∈\nT+ : |v| ≤n, lv ≥1} to be the set of nodes of generation at most n having at least one\nmark. We let Fn stand for the natural filtration of the branching process, and Fk\nn for the\nfiltration containing in addition the information about the k spine marks,\n(3.20)\nFn = σ(φv : v ∈T+, |v| ≤n)\nand\nFk\nn = σ(φv, lv : v ∈T+, |v| ≤n).\nAny f : R →R is extended to R ∪{†} by setting f(†) = 0. Then, by definition of P k\nx ,\n(3.21)\nEx\n\u0014 X\nv∈N+\nn\nf(φv)\n\u0015\n= P k\nx\n\u0014 X\nv∈S+\nn\nf(φv)\n\u0015\n.\nWe now define another measure Qk\nx, where the nodes without a spine mark behave as\nunder P k\nx but the nodes with a mark have a modified branching behaviour: Under Qk\nx\nthe movement of the marks, and thus the distribution of (lv)v∈T+, is exactly the same as\nunder P k\nx : If a node v carries k marks, each of the marks is given to one of its d direct\ndescendants independently uniformly at random. To describe the distribution of the field\nφ under Qk\nx, we first define a transition kernel (recall ρY from (3.6))\n(3.22)\nK(x, dy) = dχ(y)\nχ(x)ρY\n\u0010\ny −x\nd\n\u0011\ndy,\nx ≥h∗, y ∈R.\nNote that, by (3.6) and Proposition 3.1, for every x ≥h∗, K(x, ·) is a probability measure\nwith support [h∗, ∞). Conditionally on the marks (lv)v∈T+, the field (φv)v∈T+ under Qk\nx\nis recursively constructed by\n(3.23)\n(a) φo := x,\n(b) If v ̸= o and lv = 0, then φv = d−1φp(v) + Yv (as under P k\nx ).\n(c) If v ̸= o and lv ≥1, then φv is K(φp(v), ·)-distributed, independently of\nprevious randomness.\nTo simplify notation, we write Qx instead of Q1\nx; in this case we also set σn = σ1\nn and\nξn = ξ1\nn.\nNote that, unlike under P k\nx , under the measure Qk\nx the nodes in the cemetery state\ncannot carry any mark. Consequently, Qk\nx-a.s., there are nodes not in the cemetery state\nin every generation.\nBy construction, under Qk\nx, the process (ξi\nn)n∈N recording the value of the field along\nthe i-th spine is a Markov chain with transition kernel K. This chain never enters the\ncemetery state. The following lemma determines its invariant distribution.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n9\nLemma 3.4. The Markov chain (ξn)n∈N with the transition kernel K has a unique in-\nvariant distribution π given by\n(3.24)\nπ(dx) = χ(x)2ν(dx).\nProof. We first show that π is invariant for K. Comparing (3.6) and (3.22) yields that\nK(x, A) = L[1Aχ](x)/χ(x) for every x ≥h∗. Therefore, writing the action of K on π\nas inner product on L2(ν), using that L is self-adjoint and χ is its eigenfunction with\neigenvalue 1 (see Proposition 3.1),\n(3.25)\n(πK)(A) = ⟨K(·, A), χ2⟩ν = ⟨L[1Aχ], χ⟩ν = ⟨1Aχ, L[χ]⟩ν = ⟨1Aχ, χ⟩ν = π(A),\nwhich shows that π is an invariant measure. The uniqueness follows from the irreducibility\nof (ξn)n∈N.\n□\nNext, we state several moment formulas that are frequently used throughout the paper.\nSuch formulas are well understood in the theory of branching processes, see, e.g., [HR17]\nand references therein. The proof of the following proposition is based on Lemma 8 of\nthat paper and can be found in Appendix B.\nProposition 3.5. For all functions f, g : [h∗, ∞) →R for which the expectations below\nare well defined,\nEx\nh X\nv∈N+\nn\nf(φv)\ni\n= Qx\nh\nf(ξn) χ(x)\nχ(ξn)\ni\n,\n(3.26)\nEx\nh X\nv,w∈N+\nn\nf(φv)g(φw)\ni\n= χ(x)d −1\nd\nn−1\nX\nk=0\nQx\n\u0014\nχ(ξk)Qξk\nhf(ξn−k)\nχ(ξn−k)\ni\nQξk\nh g(ξn−k)\nχ(ξn−k)\ni\u0015\n+ χ(x)Qx\nhf(ξn)g(ξn)\nχ(ξn)\ni\n.\n(3.27)\n3.3. Asymptotic behaviour of the moments. The main result of this section is Propo-\nsition 3.8 giving precise asymptotic estimates on the quantities appearing in (3.26), (3.27).\nTo prove them, we could, in principle, use this proposition together with the known results\non the convergence of Markov chains. However, it is easier, and for our purposes slightly\nmore practical, to use formula (3.8) together with L2-estimates on the operator L.\nSince L is a self-adjoint Hilbert-Schmidt operator (see Proposition 3.1), L2(ν) has an\northonormal basis consisting of the eigenfunctions {ek}k≥1 of L corresponding to the\neigenvalues {λk}k≥1. By Proposition 3.1 we may assume that 1 = λ1 > |λ2| ≥|λ3| ≥. . . ,\nand e1 = χ. We set γ = |λ2| ∈(0, 1). By decomposing f ∈L2(ν) as\n(3.28)\nf =\nX\nk≥1\n⟨ek, f⟩ek = ⟨χ, f⟩χ +\nX\nk≥2\n⟨ek, f⟩ek =: ⟨χ, f⟩χ + β[f],\nfor every n ∈N, it holds\n(3.29)\nLn[f] =\nX\nk≥1\nλn\nk⟨ek, f⟩ek = ⟨χ, f⟩χ +\nX\nk≥2\nλn\nk⟨ek, f⟩ek = ⟨χ, f⟩χ + Ln[β[f]],\nwith\n(3.30)\n∥Ln[β[f]]∥≤γn∥β[f]∥≤γn∥f∥.\nThe following simple lemma will later be used to deduce the pointwise convergence\nfrom the L2(ν)-one.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n10\nLemma 3.6. There is a function q : [h∗, ∞) →[0, ∞) such that for every x ≥h∗and\nf ∈L2(ν)\n(3.31)\n|L[f](x)| ≤∥f∥q(x).\nProof. Using the definition (3.6) of L and the Cauchy–Schwarz inequality,\n|L[f](x)| ≤d\nZ\nR\n|f(y)|ρY\n\u0010\ny −x\nd\n\u0011\ndy = d\nZ\nR\n|f(y)|ρY (y −x/d)\nρν(y)\nν(dy)\n≤d∥f∥L2(ν)\n\r\r\rρY (· −x/d)\nρν(·)\n\r\r\r\nL2(ν) =: ∥f∥L2(ν)q(x).\n(3.32)\nBy (3.3), σν > σY . Using this one can easily check by a direct computation that q(x) < ∞\nfor every x ≥h∗.\n□\nRemark 3.7. The same computation also shows that q(x) ∈L2(ν), but we will not use\nthis fact.\nWe can now give the asymptotic estimates on moments appearing in Proposition 3.5.\nProposition 3.8. (a) For every f ∈L2(ν), x ≥h∗, and n ∈N,\n(3.33)\nEx\nh X\nv∈N+\nn\nf(φv)\ni\n= χ(x)⟨χ, f⟩+ εf\nn(x),\nwhere the error term εf\nn satisfies (with q as in Lemma 3.6)\n(3.34)\n∥εf\nn∥≤γn∥f∥\nand\n|εf\nn(x)| ≤γn−1∥f∥q(x).\n(b) For every f, g ∈L2(ν), x ≥h∗, and n ∈N\nEx\nh X\nv,w∈N+\nn\nf(φv)g(φw)\ni\n−Ex\nh X\nv∈N+\nn\nf(φv)g(φv)\ni\n= χ(x)d −1\nd\n⟨χ, f⟩⟨χ, g⟩⟨χ2, χ⟩n + εf,g\nn (x),\n(3.35)\nwhere the error term εf,g\nn\nsatisfies\n(3.36)\n∥εf,g\nn ∥≤C∥f∥∥g∥\nand\nεf,g\nn (x) ≤∥f∥∥g∥˜q(x)\nwith ˜q(x) ≤C(q(x) + χ(x)) < ∞.\nProof. (a) By (3.8), the left-hand side of (3.33) equals Ln[f]. Therefore, by (3.29), εf\nn =\nLn[β[f]]. (3.30) then implies the first claim in (3.34). The second one follows from the\nfirst one and Lemma 3.6.\n(b) Combining (3.27) with (3.26) and (3.8), we obtain that\n(3.37)\nEx\nh X\nv,w∈N+\nn\nf(φv)g(φw)\ni\n= d −1\nd\nn−1\nX\nk=0\nLkh\nLn−k[f]Ln−k[g]\ni\n(x) + Ln[fg](x).\nBy (3.33), Ln−k[f]Ln−k[g] = (⟨χ, f⟩χ + εf\nn−k)(⟨χ, g⟩χ + εg\nn−k) and thus, again by (3.33),\nLkh\nLn−k[f]Ln−k[g]\ni\n= ⟨χ, Ln−k[f]Ln−k[g]⟩χ + εLn−k[f]Ln−k[g]\nk\n=\n\u0010\n⟨χ, f⟩⟨χ, g⟩⟨χ, χ2⟩+ ⟨χ, f⟩⟨χ, χεg\nn−k⟩+ ⟨χ, g⟩⟨χ, χεf\nn−k⟩\n+ ⟨χ, εf\nn−kεg\nn−k⟩\n\u0011\nχ + εLn−k[f]Ln−k[g]\nk\n.\n(3.38)\n\nSCALING LIMITS FOR THE CRITICAL GFF\n11\nNote also that the Ln[fg](x) summand in (3.37) equals Ex[P\nv∈N+\nn f(φv)g(φv)]. Therefore,\nthe error term in (3.35) satisfies\nεf,g\nn (x) = d −1\nd\nn−1\nX\nk=0\n\u0010\u0000⟨χ, f⟩⟨χ, χεg\nn−k⟩+ ⟨χ, g⟩⟨χ, χεf\nn−k⟩+ ⟨χ, εf\nn−kεg\nn−k⟩\n\u0001\nχ(x)\n+ εLn−k[f]Ln−k[g]\nk\n(x)\n\u0011\n.\n(3.39)\nTo bound this expression, note that |⟨χ, f⟩| ≤∥f∥and |⟨χ, χεg\nn−k⟩| ≤∥χ2∥∥εg\nn−k∥≤\nCγn−k∥g∥, by part (a). Therefore, the first summand in (3.39) satisfies |⟨χ, f⟩⟨χ, χεg\nn−k⟩| ≤\nCγn−k∥f∥∥g∥. Analogously, |⟨χ, g⟩⟨χ, χεf\nn−k⟩| ≤Cγn−k∥f∥∥g∥.\nTo estimate the third\nsummand, we observe that by (3.33) εf\nn−k = Ln−k[β[f]].\nTherefore, using Proposi-\ntion 3.2(b), ∥εf\nn−kεg\nn−k∥≤d2∥εf\nn−k−1∥∥εg\nn−k−1∥and thus\n(3.40)\n⟨χ, εf\nn−kεg\nn−k⟩≤d2∥εf\nn−k−1∥∥εg\nn−k−1∥≤d2γ2(n−k−1)∥f∥∥g∥.\nFinally, for the fourth summand, using again Proposition 3.2(b) and part (a), since the\noperator norm of L equals 1,\n∥εLn−k[f]Ln−k[g]\nk\n∥≤γk∥Ln−k[f]Ln−k[g]∥\n≤d2γk∥Ln−k−1[f]∥∥Ln−k−1[g]∥≤Cγk∥f∥∥g∥.\n(3.41)\nTaking the norm in (3.39) and making use of the above estimates yields\n(3.42)\n∥εf,g\nn ∥≤C∥f∥∥g∥\nn−1\nX\nk=0\n(γn−k + γ2(n−k−1) + γk) < C∥f∥∥g∥,\nsince γ < 1. This establishes the bound on ∥εf,g\nn ∥.\nTo establish the claimed pointwise bound for |εf,g\nn (x)|, observe that by part (a), we\nhave\n\f\fεLn−k[f]Ln−k[g]\nk\n(x)\n\f\f ≤γk∥Ln−k[f]Ln−k[g]∥q(x) ≤Cγk∥f∥∥g∥q(x).\nCombining this\nwith (3.39) and the previously derived bounds on the inner products appearing there, we\neasily conclude.\n□\n4. Proof of Theorem 2.1\nIn this section we prove Theorem 2.1 which describes the tail behaviour of the diameter\nof the critical cluster. We also show several estimates that will later be used in the proof\nof Theorem 2.2.\nLet F be the set of all non-increasing functions f : R →[0, 1). For every f ∈F and\nn ∈N, we introduce\n(4.1)\nuf\nn(x) :=\n(\nEx\n\u0002\n1{N+\nn ̸=∅}\n\u00001 −Q\nu∈N+\nn f(φu)\n\u0001\u0003\n,\nfor x ≥h∗,\n0,\nfor x < h∗.\nNote that uf\nn(x) ∈[0, 1] and by (3.5) it is increasing in x. Taking f ≡0,\n(4.2)\nu0\nn(x) = Px[N +\nn ̸= ∅],\nwhich explains the relevance of this definition for the proof of Theorem 2.1.\nWe will use the following inequality, which often allows us to consider the special case\nf ≡0 only: For every f ∈F\n(4.3)\nu0\nn(x)(1 −f(h∗)) ≤uf\nn(x) ≤u0\nn(x).\n\nSCALING LIMITS FOR THE CRITICAL GFF\n12\nIndeed, the second inequality follows directly from the definition of uf\nn, since f ≥0. To\nsee the first one, note that since f is non-increasing, Q\nv∈N+\nn f(φv) ≤f(h∗) when N +\nn ̸= ∅,\nand thus Ex[1{N+\nn ̸=∅}(1 −Q\nv∈N+\nn f(φv))] ≥Ex[1{N+\nn ̸=∅}(1 −f(h∗))] = (1 −f(h∗))u0\nn(x).\nSimilarly to (3.28) we decompose uf\nn as\n(4.4)\nuf\nn = ⟨χ, uf\nn⟩χ + β[uf\nn]\nand define af\nn := ⟨χ, uf\nn⟩and bf\nn := ∥β[uf\nn]∥, so that\n(4.5)\n∥uf\nn∥2 = (af\nn)2 + (bf\nn)2.\nDue to (4.2), in order to show Theorem 2.1, we need precise asymptotic estimates on af\nn\nand bf\nn. These will be proved step by step in the following several lemmas. The theorem\nis then shown at the end of the section.\nBy Proposition 3.3, Px[|Co ∩T+| = ∞] = 0. Therefore, Px[N +\nn ̸= ∅] →0 as n →∞. As\na consequence, using also (4.3), for every f ∈F,\n(4.6)\nuf\nn(x) →0 as n →∞, pointwise and thus in L2(ν).\nAs a consequence, for every f ∈F,\n(4.7)\nlim\nn→∞af\nn = lim\nn→∞bf\nn = 0.\nThe following lemma provides a basic recursive relation for uf\nn, based on the branching\nprocess representation. This relation will be important to obtain a more precise descrip-\ntion of the asymptotic behaviour of af\nn and bf\nn as n →∞.\nLemma 4.1. For every non-increasing f : R →[0, 1) and n ∈N,\n(4.8)\nuf\nn(x) = 1 −\n\u0010\n1 −1\ndL[uf\nn−1](x)\n\u0011d\nfor x ∈R.\nAs a consequence,\n(4.9)\nuf\nn(x) = L[uf\nn−1](x) −g(L[uf\nn−1](x)),\nwith g(x) = 1 −x −(1 −d−1x)d. This function satisfies, for c1, c2 ∈(0, ∞),\n(4.10)\nc1x2 ≤g(x) ≤c2x2\nfor all x ∈[0, 1].\nProof. For x < h∗, both sides of (4.8) are trivially zero by definition of uf\nn and L. For\nx ≥h∗, by the branching process construction of φ, recalling Fn from (3.20), for n0 < n,\nEx\nh Y\nv∈N+\nn\nf(φv)\n\f\f\fFn0\ni\n=\nY\nv∈N+\nn0\nEφv\nh\nY\nw∈N+\nn−n0\nf(φw)\ni\n=\nY\nv∈N+\nn0\n\u0012\n1 −Eφv\nh\n1 −\nY\nw∈N+\nn−n0\nf(φw)\ni\u0013\n=\nY\nv∈N+\nn0\n\u00001 −uf\nn−n0(φv)\n\u0001\n.\n(4.11)\nTherefore, setting n0 = 1,\n(4.12)\nuf\nn(x) = Ex\nh\n1 −Ex\nh Y\nv∈N+\nn\nf(φv)\n\f\f\fF1\nii\n= 1 −Ex\nh Y\nv∈N+\n1\n\u00001 −uf\nn−1(φv)\n\u0001i\n.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n13\nUsing the conditional independence of the (φv : v ∈S+\n1 ) given φo = x, and the fact that\nuf\nn(x) = 0 for x < h∗,\n(4.13)\nEx\nh Y\nv∈N+\n1\n\u00001 −uf\nn−1(φv)\n\u0001i\n= Ex\nh Y\nv∈S+\n1\n\u00001 −uf\nn−1(φv)\n\u0001i\n=\nY\nv∈S+\n1\nEx[1 −uf\nn−1(φv)].\nUsing (3.7), Ex[uf\nn−1(φv)] = d−1L[uf\nn−1](x) for every v ∈S+\n1 . Together with (4.12) and\n(4.13), this proves (4.8) and (4.9). Inequality (4.10) is proved in [ˇCL25, Lemma 5.3].\n□\nThe following lemma provides a rough lower bound for u0\nn and a0\nn.\nLemma 4.2. There exists a constant c > 0 such that for all n ≥1\n(4.14)\n∥u0\nn∥≥a0\nn ≥cn−1.\nProof. The statement will follow if we show\n(4.15)\n0 ≤g(x)n−1 ≤u0\nn(x)\nfor all n ≥1\nfor some non-trivial function g : R →[0, 1].\nTo show (4.15) we use the estimate on\nthe volume of Co from Proposition 3.3. By this proposition and (3.5) there is a positive\nconstant c1 such that, for all m ≥1 and x ≥h∗,\n(4.16)\nPx[|Co ∩T+| > m] ≥c1m−1/2.\nSince |Co ∩T+| = P\nk≥0|N +\nk |, this implies for all m, n ≥1\nc1m−1/2 ≤Px\nh X\nk≥0\n|N +\nk | > m, N +\nn = ∅\ni\n+ Px\nh X\nk≥0\n|N +\nk | > m, N +\nn ̸= ∅\ni\n(4.17)\n≤Px\nh n−1\nX\nk=0\n|N +\nk | > m\ni\n+ Px[N +\nn ̸= ∅].\n(4.18)\nBy the Markov inequality, Px[Pn−1\nk=0|N +\nk | > m] ≤\n1\nm\nPn−1\nk=0 Ex[|N +\nk |], where, by Proposi-\ntion 3.8, Ex[|N +\nk |] ≤c2(x) for some x-dependent constant c2(x) > 0. Thus,\n(4.19)\nc1m−1/2 ≤c2(x) n\nm + Px[N +\nn ̸= ∅].\nRecalling (4.2) and choosing m = ⌊δ2n2⌋with δ = 2c1c2(x)−1, it follows that\n(4.20)\nu0\nn(x) ≥\nc2\n1\n2c2(x)n−1\nfor all n ≥1,\nshowing (4.15) and thus the lemma.\n□\nThe next lemma gives upper bounds on af\nn and bf\nn.\nLemma 4.3. There is c < ∞such that for every f ∈F and n ≥1\n(4.21)\naf\nn ≤cn−1\nand\nbf\nn ≤cn−2.\nProof. To prove the first statement in (4.21) we recall the recursive relation (4.9) and\nproject it on span{χ}.\nUsing ⟨χ, L[uf\nn]⟩= ⟨χ, uf\nn⟩= af\nn, and the lower bound on g\nfrom (4.10), this yields\n(4.22)\naf\nn+1 = ⟨χ, uf\nn+1⟩= ⟨χ, L[uf\nn]⟩−⟨χ, g(L[uf\nn])⟩≤af\nn −c1⟨χ, L[uf\nn]2⟩.\nBy Proposition 3.1 and (3.17), χ(x) ≥c > 0 for all x ∈[h∗, ∞). Hence, ⟨χ, L[uf\nn]2⟩≥\nc⟨1, L[uf\nn]2⟩= c∥L[uf\nn]∥2 ≥c(af\nn)2. Applied to (4.22), this gives\n(4.23)\naf\nn+1 ≤af\nn −c(af\nn)2.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n14\nThis implies that af\nn is decreasing in n and, after rearranging, also\n(4.24)\nc ≤af\nn −af\nn+1\n(af\nn)2\n≤af\nn −af\nn+1\naf\nn+1af\nn\n=\n1\naf\nn+1\n−1\naf\nn\n.\nSumming this over n running from 0 to n −1 yields\n(4.25)\ncn ≤\nn−1\nX\nk=0\n\u0010 1\naf\nk+1\n−1\naf\nk\n\u0011\n= 1\naf\nn\n−1\naf\n0\n≤1\naf\nn\n,\nproving the first part of (4.21).\nTo prove the second part we project (4.9) onto the orthogonal complement of span {χ}\nand take norms. With the triangle inequality, this gives\n(4.26)\nbf\nn+1 ≤∥β[L[uf\nn]]∥+ ∥β[g(L[uf\nn])]∥.\nBy (3.30), ∥β[L[uf\nn]]∥= ∥L[β[uf\nn]]∥≤γ∥β[uf\nn]∥= γbf\nn.\nBy the upper bound of g\nfrom (4.10) and Proposition 3.2, since β is a projection, ∥β[g(L[uf\nn])]∥≤∥g(L[uf\nn])∥≤\nc2∥L[uf\nn]2∥≤c∥uf\nn∥2. Applied to (4.26), this gives\n(4.27)\nbf\nn+1 ≤γbf\nn + c∥uf\nn∥2 = γbf\nn + c(af\nn)2 + c(bf\nn)2.\nTaking now f = 0, since b0\nn →0 as n →∞, there is γ′ ∈(γ, 1) and n′ < ∞such that\nγb0\nn + c(b0\nn)2 ≤γ′b0\nn for all n ≥n′. Together with the first part of (4.21), this implies\n(4.28)\nb0\nn+1 ≤γ′b0\nn + cn−2\nfor n ≥n′.\nApplying this recursively yields\n(4.29) b0\nn+1 ≤γ′\u0000γ′b0\nn−1 + c(n −1)−2\u0001\n+ cn−2 ≤· · · ≤(γ′)n+1−n′b0\nn′ + c\nn−n′\nX\nk=0\n(γ′)k(n −k)−2.\nFor n ≥2n′, by splitting the sum at k = n/2, using also that b0\nn ≤1 for all n, this can be\nbounded by c(γ′)n/2 + c(n/2)−2 ≤cn−2, proving the second half of (4.21) for f = 0.\nFor a general f ∈F, by (4.3) and the already proven statements, ∥uf\nn∥2 ≤∥u0\nn∥2 =\n(a0\nn)2 + (b0\nn)2 ≤cn−2. Inserting this into (4.27) gives\n(4.30)\nbf\nn+1 ≤γbf\nn + cn−2,\nwhich looks like (4.28). The second part of (4.21) for general f ∈F then follows by the\nsame arguments as for f = 0.\n□\nWhen f ∈F is fixed, Lemmas 4.2 and 4.3 show that limn→∞uf\nn/∥uf\nn∥= χ in L2(ν) and\nc < n∥uf\nn∥< c′. In the special case f = 0, this is almost sufficient to prove Theorem 2.1,\nwe only need to improve the estimate on ∥uf\nn∥. In contrast, in the proof of Theorem 2.2 we\nwill need to consider functions f varying with n. There our estimates are not sufficient,\nsince the lower bound in Lemma 4.2 holds only for f = 0 and cannot be true uniformly\nover f ∈F. We now provide tools allowing us to deal with this case as well.\nAs it turns out, it is enough to prove the uniformity over a certain family of non-\nincreasing functions. Specifically, let\n(4.31)\nˆF = {fλ : λ ∈[0, ∞)} ⊂F,\nwhere\n(4.32)\nf0 ≡0\nand\nfλ(x) := exp\n\u0010\n−χ(x)\nλ\n\u0011\nfor λ > 0.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n15\nTo simplify the notation, we define (recall (4.1), (4.4))\n(4.33)\nuλ\nn = ufλ\nn ,\naλ\nn = afλ\nn = ⟨χ, uλ\nn⟩,\nbλ\nn = bfλ\nn = ∥β[uλ\nn]∥.\nThe first preliminary step in proving the uniformity over ˆF is the following lemma. In\nthe special case λ = 0, this already follows from Lemmas 4.2 and 4.3.\nLemma 4.4. There is a constant c < ∞so that\n(4.34)\nbλ\nn ≤caλ\nn\nand\n∥uλ\nn∥≤caλ\nn\nfor all n ∈N0 and λ ≥0.\nProof. As the case λ = 0 already follows from Lemmas 4.2 and 4.3, it is enough to show\n(4.34) with λ ≥0 replaced by λ > 0.\nBy (4.27) from the proof of Lemma 4.3,\n(4.35)\nbλ\nn+1 ≤γbλ\nn + c∥uλ\nn∥2,\nwith c that is uniform over f ∈F. An iterative application of this inequality yields\n(4.36)\nbλ\nn+1 ≤γnbλ\n0 + c\nn\nX\nl=0\nγn−l∥uλ\nl ∥2.\nTo continue, we argue that\n(4.37)\n∥uλ\nl ∥≤λ−1\nfor every l ∈N0, λ > 0.\nIndeed, by Lemma 4.1, uλ\nn+1(x) ≤L[uλ\nn](x). Applying this recursively, taking norms, and\nusing that the operator norm of L is one, we obtain\n(4.38)\n∥uλ\nl ∥≤∥Ll[uλ\n0]∥≤∥uλ\n0∥.\nMoreover, since by definition 0 ≤uλ\n0 = 1 −exp(−λ−1χ) ≤λ−1χ, we know that ∥uλ\n0∥≤\nλ−1∥χ∥= λ−1. Together with (4.38), this shows (4.37).\nSince also bλ\n0 ≤∥uλ\n0∥≤λ−1 and γ < 1, (4.36) and (4.37) imply\n(4.39)\nbλ\nn+1 ≤γnλ−1 + cλ−2.\nOn the other hand, by Lemma 4.3, bλ\nn ≤cn−2, and thus\n(4.40)\nbλ\nn ≤c min\n\u0010\nγnλ−1 + λ−2, 1\nn2\n\u0011\n.\nSince 1 −fλ(h∗) ≥cλ−1, (4.3) implies that\n(4.41)\naλ\nn = ⟨uλ\nn, χ⟩≥(1 −fλ(h∗))⟨u0\nn, χ⟩≥cλ−1a0\nn ≥cλ−1n−1,\nwhere the last inequality follows from Lemma 4.2. Therefore,\n(4.42)\nbλ\nn\naλ\nn\n≤c min\n\u0000γnλ−1 + λ−2, n−2\u0001\nλ−1n−1\n≤c min\n\u0010\ncγnn + n\nλ, λ\nn\n\u0011\n≤cγnn + c,\nwhich is bounded from above by some constant c independent of λ or n. This proves the\nfirst statement. The second follows from the first one and ∥uλ\nn∥2 = (aλ\nn)2 + (bλ\nn)2.\n□\nWe now improve the result of Lemma 4.4 and show that bλ\nn = o(aλ\nn), uniformly in λ.\nProposition 4.5. Uniformly over λ ≥0,\n(4.43)\nlim\nn→∞\nuλ\nn\n⟨χ, uλ\nn⟩= χ\nin L2(ν).\n\nSCALING LIMITS FOR THE CRITICAL GFF\n16\nProof. The proof consists of two steps. First we show that, for a suitable choice of n0(n),\nwhich slowly diverges with n,\n(4.44)\nuλ\nn\n⟨χ, uλ\nn−n0(n)⟩→χ\nin L2(ν) as n →∞, uniformly in λ ≥0.\nIn a second step, we then show that ⟨χ, uλ\nn−n0(n)⟩/⟨χ, uλ\nn⟩→1 as n →∞uniformly in\nλ ≥0, implying the claim of the proposition.\nBy (4.11) from the proof of Lemma 4.1,\nuλ\nn(x) = Ex\n\u0014\nEx\nh\n1 −\nY\nv∈N+\nn\nfλ(φv)\n\f\f\fFn0\ni\u0015\n= Ex\nh\n1 −\nY\nv∈N+\nn0\n\u00001 −uλ\nn−n0(φv)\n\u0001i\n= Ex\nh X\nv∈N+\nn0\nuλ\nn−n0(φv)\ni\n−Ex\nh Y\nv∈N+\nn0\n\u00001 −uλ\nn−n0(φv)\n\u0001\n−1 +\nX\nv∈N+\nn0\nuλ\nn−n0(φv)\ni\n.\n(4.45)\nBy Proposition 3.8(a), the first expectation on the right-hand side satisfies\n(4.46)\nEx\nh X\nu∈N+\nn0\nuλ\nn−n0(φu)\ni\n= χ(x)⟨χ, uλ\nn−n0⟩+ ελ,n−n0\nn0\n(x)\nwith ∥ελ,n−n0\nn0\n∥≤γn0∥uλ\nn−n0∥.\nTo estimate the second expectation we need the following inequality: For any finite\nindex set I and ai ∈[0, 1], i ∈I,\n(4.47)\n0 ≤\nY\ni∈I\n(1 −ai) −1 +\nX\ni∈I\nai ≤1\n2\nX\ni̸=j∈I\naiaj.\nThis inequality follows easily from Bonferroni inequalities for independent events Ai with\nP(Ai) = ai. Indeed,\n(4.48)\n1 −\nY\ni∈I\n(1 −ai) = P(∪i∈IAi) ≤\nX\ni∈I\nP(Ai) =\nX\ni∈I\nai\nimplies the lower bound in (4.47), and\n(4.49) 1−\nY\ni∈I\n(1−ai) = P(∪i∈IAi) ≥\nX\ni∈I\nP(Ai)−1\n2\nX\ni̸=j∈I\nP(Ai∩Aj) =\nX\ni∈I\nai−1\n2\nX\ni̸=j∈I\naiaj\nimplies the upper bound. Inequality (4.47) and Proposition 3.8(b) imply\n0 ≤Ex\nh Y\nu∈N+\nn0\n\u00001 −uλ\nn−n0(φu)\n\u0001\n−1 +\nX\nu∈N+\nn0\nuλ\nn−n0(φu)\ni\n≤cχ(x)⟨χ, uλ\nn−n0⟩2⟨χ, χ2⟩n0 + ¯ελ,n−n0\nn0\n(x),\n(4.50)\nwith ∥¯ελ,n−n0\nn0\n∥≤C∥uλ\nn−n0∥2.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n17\nDividing (4.45) by ⟨χ, uλ\nn−n0⟩and combining it with (4.46) and (4.50), we obtain\n\r\r\r\r\nuλ\nn\n⟨χ, uλ\nn−n0⟩−χ\n\r\r\r\r\n≤∥ελ,n−n0\nn0\n∥\n⟨χ, uλ\nn−n0⟩+ c∥χ⟨χ, uλ\nn−n0⟩2⟨χ, χ2⟩n0∥\n⟨χ, uλ\nn−n0⟩\n+ ∥¯ελ,n−n0\nn0\n(x)∥\n⟨χ, uλ\nn−n0⟩\n≤γn0 ∥uλ\nn−n0∥\n⟨χ, uλ\nn−n0⟩+ c⟨χ, χ2⟩⟨χ, uλ\nn−n0⟩n0 + c ∥uλ\nn−n0∥\n⟨χ, uλ\nn−n0⟩∥uλ\nn−n0∥\n≤cγn0 + c⟨χ, χ2⟩n0\nn −n0\n+\nc\nn −n0\n,\n(4.51)\nwhere in the last inequality we used that ∥uλ\nn∥/⟨χ, uλ\nn⟩is uniformly bounded by Lemma 4.4,\nand both ⟨χ, uλ\nn⟩and ∥uλ\nn∥are uniformly bounded above by cn−1 by Lemma 4.3. This\nestablishes (4.44) with, for example, n0(n) = ⌊log(n)⌋.\nNext we show that (4.51) also implies\n(4.52)\n\f\f\f\f\n⟨χ, uλ\nn⟩\n⟨χ, uλ\nn−n0(n)⟩−1\n\f\f\f\f →0\nas n →∞, uniformly in λ ≥0.\nTo see this, we use that 1 = ⟨χ, χ⟩and |⟨χ, f⟩| ≤∥f∥to get\n\f\f\f\f\n⟨χ, uλ\nn⟩\n⟨χ, uλ\nn−n0(n)⟩−1\n\f\f\f\f =\n\f\f\f\f\n⟨χ, uλ\nn⟩\n⟨χ, uλ\nn−n0(n)⟩−⟨χ, χ⟩\n\f\f\f\f =\n\f\f\f\f\n\u001c\nχ,\nuλ\nn\n⟨χ, uλ\nn−n0(n)⟩−χ\n\u001d\f\f\f\f\n≤\n\r\r\r\r\nuλ\nn\n⟨χ, uλ\nn−n0(n)⟩−χ\n\r\r\r\r,\n(4.53)\nwhich by (4.51) converges to zero uniformly in λ. This shows (4.52), which together with\n(4.44) finishes the proof.\n□\nUsing the hypercontractivity of L from Proposition 3.2, it is easy to slightly improve\nProposition 4.5. This will be useful to ensure that certain products still lie in L2(ν).\nLemma 4.6. Uniformly over λ ≥0,\n(4.54)\nlim\nn→∞\nuλ\nn\n⟨χ, uλ\nn⟩= χ\nin L5/2(ν) and pointwise over [h∗, ∞).\nProof. We will first show (4.54).\nWe set p = 5/2.\nBy Lemma 4.1, uλ\nn = L[uλ\nn−1] +\nPd\nl=2 clL[uλ\nn−1]l. Therefore, also using that L[χ] = χ,\n(4.55)\n\r\r\r\nuλ\nn\n⟨χ, uλ\nn⟩−χ\n\r\r\r\nLp(ν) ≤\n\r\r\r\rL\nh uλ\nn−1\n⟨χ, uλ\nn⟩−χ\ni\r\r\r\r\nLp(ν)\n+\nc\n⟨χ, uλ\nn⟩\nd\nX\nl=2\n∥L[uλ\nn−1]l∥Lp(ν).\nSince 0 ≤uλ\nn ≤1, it holds that 0 ≤L[uλ\nn] ≤d. Therefore,\n(4.56)\n∥L[uλ\nn−1]l∥Lp(ν) ≤dl−2∥L[uλ\nn−1]2∥Lp(ν) = dl−2∥L[uλ\nn−1]∥2\nL2p(ν) ≤c∥uλ\nn−1∥2\nL2(ν),\nwhere in the last inequality we used Proposition 3.2(a) with q = 2p = 5 and p = 2.\nTherefore, using also (4.52) (with n0 = 1), Lemma 4.4, and the fact that ∥uλ\nn∥≤∥u0\nn∥→0\nas n →∞, we get\n(4.57)\n∥L[uλ\nn−1]l∥Lp(ν)\n⟨χ, uλ\nn⟩\n≤\n∥uλ\nn−1∥2\nL2(ν)\nc⟨χ, uλ\nn−1⟩→0\nas n →∞uniformly in λ ≥0.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n18\nUsing Proposition 3.2(a) again for the remaining term on the right-hand side of (4.55),\n(4.58)\n\r\r\r\rL\nh uλ\nn−1\n⟨χ, uλ\nn⟩−χ\ni\r\r\r\r\nLp(ν)\n≤d\n\r\r\r\r\nuλ\nn−1\n⟨χ, uλ\nn⟩−χ\n\r\r\r\r\nL2(ν)\n→0\nas n →∞uniformly in λ ≥0.\nHere, the convergence follows from Proposition 4.5. Combining (4.55) with (4.57) and\n(4.58) shows (4.54).\nWe now show the pointwise convergence. By similar steps as in (4.55), using Lemma 3.6,\nfor x ≥h∗,\n\f\f\f uλ\nn(x)\n⟨χ, uλ\nn⟩−χ(x)\n\f\f\f ≤\n\f\f\f\fL\nh uλ\nn−1\n⟨χ, uλ\nn⟩−χ\ni\n(x)\n\f\f\f\f +\n1\n⟨χ, uλ\nn⟩\nd\nX\nl=2\n|L[uλ\nn−1](x)|l\n≤\n\r\r\r\r\nuλ\nn−1\n⟨χ, uλ\nn⟩−χ\n\r\r\r\rq(x) +\n1\n⟨χ, uλ\nn⟩\nd\nX\nl=2\n∥uλ\nn−1∥lq(x)l.\n(4.59)\nUsing (4.57) and (4.58), it is immediate that the right-hand side of (4.59) converges to 0\nas n →∞, uniformly in λ ≥0. This shows the pointwise convergence and finishes the\nproof.\n□\nThe following lemma provides the final ingredient for the proof of Theorem 2.1. Its\nresult allows us to determine the exact asymptotic behaviour of aλ\nn as n →∞.\nLemma 4.7. Let C1 be as in (2.11). Then,\n(4.60)\nlim\nn→∞\n1\nn\n\u0010 1\naλ\nn\n−1\naλ\n0\n\u0011\n= C−1\n1\nuniformly in λ ≥0.\nProof. We show that\n(4.61)\nlim\nn→∞\n\u0010 1\naλ\nn\n−\n1\naλ\nn−1\n\u0011\n= lim\nn→∞\naλ\nn−1 −aλ\nn\naλ\nn−1aλ\nn\n= C−1\n1\nuniformly in λ ≥0. The claim of the lemma then follows from (4.61) by telescoping the\ndifference therein.\nBy Lemma 4.1 again, this time writing the prefactor of the quadratic term explicitly,\naλ\nn = ⟨χ, uλ\nn⟩= ⟨χ, L[uλ\nn−1]⟩−\n\u0012d\n2\n\u0013 1\nd2⟨χ, L[uλ\nn−1]2⟩−\nd\nX\nl=3\ncl⟨χ, L[uλ\nn−1]l⟩.\n(4.62)\nUsing that L is self-adjoint and L[χ] = χ, we get ⟨χ, L[uλ\nn−1]⟩= ⟨χ, uλ\nn−1⟩= aλ\nn−1. Hence,\nafter dividing by (aλ\nn−1)2, (4.62) implies\naλ\nn−1 −aλ\nn\n(aλ\nn−1)2\n=\n\u0012d\n2\n\u0013\nd−2D\nχ, L[uλ\nn−1]2\n(aλ\nn−1)2\nE\n+\nd\nX\nl=3\ncl\nD\nχ, L[uλ\nn−1]l\n(aλ\nn−1)2\nE\n=\n\u0012d\n2\n\u0013\nd−2D\nχ, L\nhuλ\nn−1\naλ\nn−1\ni2E\n+\nd\nX\nl=3\ncl\nD\nχ, L\nhuλ\nn−1\naλ\nn−1\ni2\nL[uλ\nn−1]l−2E\n.\n(4.63)\nBy Propositions 4.5 and 3.2, L[uλ\nn−1/aλ\nn−1]2 →χ2 in L2(ν), uniformly in λ. By (4.6) and\n(4.3), 0 ≤uλ\nn ≤u0\nn →0 in L2(ν) as n →∞. Therefore, using that L[uλ\nn−1] ≤d and the\n\nSCALING LIMITS FOR THE CRITICAL GFF\n19\nboundedness of L, L[uλ\nn−1]l−2 ≤dl−3L[uλ\nn−1] →0 in L2(ν) uniformly in λ. Using this in\n(4.63) then yields\n(4.64)\nlim\nn→∞\naλ\nn−1 −aλ\nn\n(aλ\nn−1)2\n=\n\u0012d\n2\n\u0013\nd−2⟨χ, χ2⟩= C−1\n1\nuniformly in λ ≥0.\nFrom this (4.61) follows, if we show\n(4.65)\nlim\nn→∞\naλ\nn\naλ\nn−1\n= 1\nuniformly in λ ≥0.\nTo see this, note that by multiplying (4.64) by aλ\nn−1,\n(4.66)\n1 −aλ\nn/aλ\nn−1 = aλ\nn−1C−1\n1\n+ o(aλ\nn−1) = o(1),\nwhere the o(1) term is independent of λ since 0 ≤aλ\nn ≤a0\nn →0 for n →∞. This shows\n(4.65), and together with (4.64) completes the proof of (4.61).\n□\nWe are now ready to prove Theorem 2.1.\nProof of Theorem 2.1. Recall from (4.2) that u0\nn(x) = Px[N +\nn ̸= ∅]. Therefore, by Propo-\nsition 4.5 and Lemma 4.6,\n(4.67)\nlim\nn→∞\nP·[N +\nn ̸= ∅]\n⟨χ, u0\nn⟩\n= χ\nin L2(ν) and pointwise.\nMoreover, by Lemma 4.7,\n(4.68)\n⟨χ, u0\nn⟩= a0\nn = C1n−1(1 + o(1)).\nCombining these two facts one obtains (2.9), and by integrating over φ(o) which is ν-\ndistributed also (2.10).\nTo show the last statement of the theorem, let x1, . . . , xd+1 = ¯o be the neighbours of o\nin T, and let An,i be the event “the subtree of xi intersects Nn”. By the branching process\nconstruction, for every n ≥1, the events An,1, . . . , An,d+1 are independent and have the\nsame probability p = p(n, x). Further, Px[Nn ̸= ∅] = Px[∪d+1\ni=1 An,i] = 1 −(1 −p)d+1 and\nsimilarly Px[N +\nn ̸= ∅] = 1 −(1 −p)d and thus\n(4.69)\nPx[Nn ̸= ∅] = 1 −(1 −Px[N +\nn ̸= ∅])\nd+1\nd\n= d + 1\nd\nPx[N +\nn ̸= ∅] + O(Px[N +\nn ̸= ∅]2).\nThis directly implies the statement for Px[Nn ̸= ∅]. The statement for P[Nn ̸= ∅] is again\nobtained by integration over φo.\n□\n5. Proof of Theorem 2.2\nThe goal of this section is twofold: We show Theorem 2.2, and then, in Section 5.1,\nprovide further results concerning the behaviour of Co conditioned on {N +\nn ̸= ∅}. These\nresults are, more or less, direct consequences of Theorems 2.1 and 2.2, and will later be\nused to show Theorem 2.3.\nWe start with a preparatory lemma which is a special case of Theorem 2.2 for functions\nf orthogonal to χ.\nLemma 5.1. For every f ∈L2(ν) such that ⟨χ, f⟩= 0 there is a sequence εf\nn ∈L2(ν)\nconverging to zero in L2(ν) and pointwise, such that for every δ > 0, x ≥h∗, and n ≥1,\n(5.1)\nPx\nh\f\f\f1\nn\nX\nv∈N+\nn\nf(φv)\n\f\f\f > δ\n\f\f\f N +\nn ̸= ∅\ni\n≤cδ−2εf\nn(x).\n\nSCALING LIMITS FOR THE CRITICAL GFF\n20\nProof. By the conditional Markov inequality\n(5.2)\nPx\nh\f\f\f1\nn\nX\nv∈N+\nn\nf(φv)\n\f\f\f > δ\n\f\f\fN +\nn ̸= ∅\ni\n≤δ−2Ex\nh\u00101\nn\nX\nv∈N+\nn\nf(φv)\n\u00112\f\f\fN +\nn ̸= ∅\ni\n.\nThe conditional expectation on the right-hand side satisfies\n(5.3)\nEx\nh\u00101\nn\nX\nv∈N+\nn\nf(φv)\n\u00112\f\f\fN +\nn ̸= ∅\ni\n=\n\u0010\n1\nnPx[N +\nn ̸= ∅]\n\u0011\u00101\nnEx\nh\u0010 X\nv∈N+\nn\nf(φv)\n\u00112i\u0011\n.\nBy Proposition 3.8(a,b), since ⟨χ, f⟩= 0,\n(5.4)\n1\nnEx\nh\u0010 X\nv∈N+\nn\nf(φv)\n\u00112i\n= εf\nn(x),\nwith εf\nn →0 in L2 and pointwise. Further, by the stochastic domination (3.5), using\nTheorem 2.1, (nPx[N +\nn ̸= ∅])−1 ≤(nPh∗[N +\nn ̸= ∅])−1 ≤c and the statement of the lemma\nfollows.\n□\nProof of Theorem 2.2. We first consider the special case f = χ; the general case will then\neasily follow using Lemma 5.1.\nWe start by showing that for every α > 0,\n(5.5)\nE\n\u0002\ne−αZχ,x\nn \u0003\n=\nC1\nC1 + α + εα\nn(x),\nwith εα\nn →0 pointwise and in L2(ν). Using the definitions (4.1), (4.32) of uf\nn and fλ,\nsetting as before uλ\nn = ufλ\nn , and recalling (4.2), we obtain\nE\n\u0002\ne−αZχ,x\nn \u0003\n= Ex\nh\nexp\n\u0010\n−α\nn\nX\nv∈N+\nn\nχ(φv)\n\u0011\f\f\fN +\nn ̸= ∅\ni\n= 1 −un/α\nn\n(x)\nu0\nn(x) .\n(5.6)\nBy Lemma 4.6 and (4.68),\n(5.7)\nu0\nn(x) = Cn−1χ(x)(1 + ε(x)),\nwhere ε →0 pointwise. Since Lemma 4.6 holds uniformly in λ, we can apply it with\nλ = n/α to obtain\n(5.8)\nun/α\nn\n(x) = an/α\nn\nχ(x)(1 + ˜ε(x)),\nagain with ˜ε →0 pointwise. Moreover, by Lemma 4.7, again using the uniformity in λ,\n(5.9)\n(nan/α\nn\n)−1 = (nan/α\n0\n)−1 + C−1\n1\n+ o(1)\nas n →∞.\nBy the definitions of aλ\nn, uλ\nn and fλ, and by the monotone convergence theorem,\nnan/α\n0\n= n⟨χ, un/α\n0\n⟩= n\n\nχ, E·\n\u0002\n1 −fn/α(φo)\n\u0003\u000b\n= n⟨χ, 1 −fn/α⟩=\n\nχ, n\n\u00001 −e−αχ(·)/n\u0001\u000b n→∞\n−−−→α⟨χ, χ⟩= α.\n(5.10)\nInserting this into (5.9) yields\n(5.11)\nan/α\nn\n= αC1(1 + o(1))\nn(α + C1)\nas n →∞.\nCombining (5.6)–(5.8), (5.11) gives\n(5.12)\n1 −E\n\u0002\ne−αZχ,x\nn \u0003\n=\nα\nα + C1\n(1 + o(1)),\n\nSCALING LIMITS FOR THE CRITICAL GFF\n21\nwhich shows the pointwise convergence in (5.5). The L2(ν)-convergence follows by the\ndominated convergence theorem, since the left-hand side of (5.5) is bounded by 1.\nBy L´evy’s continuity theorem for the Laplace transform, (5.5) implies the statement\n(2.13) of the theorem in the case Zf,x\nn\nwith f = χ.\nFor general f ∈L2(ν) we write\nf = ⟨χ, f⟩χ+β[f] as usual. Then Zf,x\nn\n= ⟨χ, f⟩Zχ,x\nn\n+Zβ[f],x\nn\n. The statement for Zf,x\nn\nthen\ndirectly follows, since the first summand converges to an exponential random variable\nwith mean C−1\n1 ⟨χ, f⟩, by the first step of the proof, and the second summand converges\nto 0 in probability, by Lemma 5.1, since ⟨χ, β[f]⟩= 0.\nWe now show the statement of the theorem for Zf\nn. Let νn be the law of φo under\nP[ · |N +\nn ̸= ∅]. By integrating (5.5) over νn(dx)\n(5.13)\nE\n\u0002\ne−Zχ\nn\u0003\n=\nC1\nC1 + α +\nZ\nεα\nn(x)νn(dx).\nWe need to show that the last integral is o(1). Observe that νn(dx) = P[φo ∈dx|N +\nn ̸=\n∅] = P[φo ∈dx, N +\nn ̸= ∅]P[N +\nn ̸= ∅]−1 = Px[N +\nn ̸= ∅]P[N +\nn ̸= ∅]−1ν(dx). Therefore,\nusing the Cauchy–Schwarz inequality and that εα\nn →0 in L2(ν), the integral in (5.13) is\nbounded above by\n(5.14)\no(1)\n\u0010 Z Px[N +\nn ̸= ∅]2\nP[N +\nn ̸= ∅]2 ν(dx)\n\u00111/2\n.\nBy (4.67), Px[N +\nn ̸= ∅]P[N +\nn ̸= ∅]−1 = χ(x)⟨1, χ⟩−1 + εn(x), with εn(x) →0 in L2(ν)\nas n →∞. This implies that the integral in (5.14) is O(1) and together with (5.13)\nand again L´evy’s continuity theorem shows statement (2.13) for Zf\nn with f = χ. For Zf\nn\nand general f ∈L2(ν) the statement then follows by integrating (5.1) over νn(dx) and\napplying similar arguments as above.\n□\n5.1. Further results for the conditioned model. The following few results which all\nconcern the model conditioned to {N +\nn ̸= ∅} will later be used in the proof of Theorem 2.3.\nThe first one explains the role of the measure Qx introduced using the spine construction\nin Section 3.2. Similar results are well established in branching processes literature, see\nfor instance [HH07, Theorem 5] or [CR90, Theorem 4]. Proposition 1.5 in [Pow19] can be\nseen as the analogous result for branching diffusion on bounded domains.\nProposition 5.2. For all K ∈N, x ≥h∗and B ∈FK (see (3.20))\n(5.15)\nlim\nn→∞Px[B|N +\nn ̸= ∅] = Qx[B].\nProof. We follow similar steps as in the proof of [Pow19, Proposition 1.5].\nFor every\nn ≥K,\n(5.16)\nPx[B|N +\nn ̸= ∅] = Ex\nh1BPx[N +\nn ̸= ∅|FK]\nPx[N +\nn ̸= ∅]\ni\n.\nWriting N +\nK = {v1, . . . , v|N+\nK|} and defining the events Ai = {vi is root of a subtree in Co\nwith height of at least n −K},\n(5.17)\n{N +\nn ̸= ∅} =\n|N+\nK|\n[\nk=1\nAk =\n|N+\nK|\n[\nk=1\n\u0010\nAk ∩\n\\\nj<k\nAc\nj\n\u0011\n,\n\nSCALING LIMITS FOR THE CRITICAL GFF\n22\nwhere the last union is disjoint. Since the Ai are independent conditionally on FK,\n(5.18)\nPx[N +\nn ̸= ∅|FK] =\n|N+\nK|\nX\ni=1\nPφvi[N +\nn−K ̸= ∅]\nY\nj<i\nPφvj [N +\nn−K = ∅].\nBy Theorem 2.1, Pφvi[N +\nn−K ̸= ∅] ∼C1(n−K)−1χ(φvi) →0 and Px[N +\nn ̸= ∅] ∼C1n−1χ(x)\nas n →∞. Therefore,\n(5.19)\nlim\nn→∞\n1BPx[N +\nn ̸= ∅|FK]\nPx[N +\nn ̸= ∅]\n= 1B\nP\nv∈NK χ(φv)\nχ(x)\n,\nPx-a.s.\nIn order to insert this into (5.16), we use the generalised dominated convergence theorem.\nTo this end we bound the fraction on the left-hand side of (5.19) by a function gn satisfying\ngn →g in L1(Px) and Px-a.s.: Using (5.17) and Theorem 2.1 (or (4.67), (4.68) from its\nproof),\n(5.20)\nPx[N +\nn ̸= ∅|FK] ≤\nX\nv∈N+\nK\nPφv[N +\nn−K ̸= ∅] =\nX\nv∈N+\nK\nC1(n −K)−1(χ(φv) + ¯εn−K(φv))\nwith ¯εn →0 pointwise and in L2(ν) as n →∞. For the denominator, again by Theo-\nrem 2.1, Px[N +\nn ̸= ∅] ≥Ph∗[N +\nn ̸= ∅] ≥cn−1 for some constant c. Thus, for n > K,\n1BPx[N +\nn ̸= ∅|FK]\nPx[N +\nn ̸= ∅]\n≤\ncn\nn −K\nX\nv∈N+\nK\n(χ(φv) + ¯εn−K(φv))\n≤c′ X\nv∈N+\nK\n(χ(φv) + ¯εn−K(φv)) =: gn,\n(5.21)\nwhich converges a.s. to g := c′ P\nv∈N+\nK χ(φv). Moreover, by Proposition 3.8,\n(5.22)\nEx[|gn −g|] ≤Ex\n\u0002\nc′ X\nv∈N+\nK\n|¯εn−K(φv)|\n\u0003\n= ⟨χ, |¯εn−K|⟩χ(x) + ε|¯εn−K|\nK\n(x).\nSince ∥¯εn∥→0, the bounds established in (3.34) imply that the right-hand side of (5.22)\nconverges to 0, that is gn →g in L1(P x).\nTherefore, by the generalised dominated\nconvergence theorem, using (5.16), (5.19),\n(5.23)\nlim\nn→∞Px[B|N +\nn ̸= ∅] = Ex\nh\n1B\nP\nv∈NK χ(φv)\nχ(x)\ni\n= Qx[B],\nwhere the last equality follows from Lemma B.2 with k = 1 and Y (v1) = 1Bχ(φv1).\n□\nThe following lemma and its corollary follow almost directly from the results established\nin the proof of Theorem 2.2.\nLemma 5.3. For every f ∈L2(ν), x ≥h∗and δ > 0,\n(5.24)\nlim\nn→∞Px\n\u0014\f\f\f\f\nP\nv∈N+\nn f(φv)\nP\nv∈N+\nn χ(φv) −⟨χ, f⟩\n\f\f\f\f > δ\n\f\f\f\f N +\nn ̸= ∅\n\u0015\n= 0.\nProof. Observe that\nP\nv∈N+\nn f(φv)\nP\nv∈N+\nn χ(φv) −⟨χ, f⟩=\nP\nv∈N+\nn (f(φv) −⟨χ, f⟩χ(φv))\nP\nv∈N+\nn χ(φv)\n= n−1 P\nv∈N+\nn β[f](φv)\nn−1 P\nv∈N+\nn χ(φv) .\n(5.25)\n\nSCALING LIMITS FOR THE CRITICAL GFF\n23\nFor any A, B ∈R and ε, δ > 0, {|A/B| > ε} ⊂{|A| > δ} ∪{|B| < δ/ε}, which together\nwith (5.25) allows us to bound the left-hand side of (5.24) by\n(5.26)\nPx\nh\f\f\fn−1 X\nv∈N+\nn\nβ[f](φv)\n\f\f\f > ¯δn\n\f\f\f N +\nn ̸= ∅\ni\n+ Px\nh\nn−1 X\nv∈N+\nn\nχ(φv) < ¯δn/δ\n\f\f\f N +\nn ̸= ∅\ni\n.\nWe now show that there is a sequence ¯δn →0 such that both summands converge to\nzero. By Lemma 5.1, the first summand is bounded by ¯δ−2\nn ¯εf\nn(x), where the ¯εf\nn-term is\nindependent of ¯δn and converges to zero. Therefore, if ¯δn →0 sufficiently slowly, then\nalso ¯δ−2\nn ¯εf\nn(x) →0. For the second summand, fix ε > 0. Then for n large enough so that\n¯δn/δ ≤ε,\n(5.27)\nPx\nh\nn−1 X\nv∈N+\nn\nχ(φv) < ¯δn/δ\n\f\f\f N +\nn ̸= ∅\ni\n≤Px\nh\nn−1 X\nv∈N+\nn\nχ(φv) < ε\n\f\f\f N +\nn ̸= ∅\ni\n.\nBy Theorem 2.2, the right-hand side converges to P(Z < C1ε) which can be made ar-\nbitrarily small by choosing ε small.\nThis implies that the second summand in (5.26)\nconverges to zero and completes the proof.\n□\nCorollary 5.4. For every x ≥h∗, δ > 0 and f, g ∈L2(ν) with g > 0,\n(5.28)\nlim\nn→∞Px\n\u0014\f\f\f\f\nP\nv∈N+\nn f(φv)\nP\nv∈N+\nn g(φv) −⟨χ, f⟩\n⟨χ, g⟩\n\f\f\f\f > δ\n\f\f\f\f N +\nn ̸= ∅\n\u0015\n= 0.\nIn particular, setting g ≡1,\n(5.29)\nlim\nn→∞Px\n\u0014\f\f\f\f\n1\n|N +\nn |\nX\nv∈N+\nn\nf(φv) −⟨χ, f⟩\n⟨χ, 1⟩\n\f\f\f\f > δ\n\f\f\f\f N +\nn ̸= ∅\n\u0015\n= 0.\nProof. It is easy to see that there is δ′ = δ′(δ, f, g) > 0 such that the event on the left-\nhand side of (5.28) is contained in the union of {|P f(φv)/ P χ(φv) −⟨χ, f⟩| ≥δ′} and\n{|P g(φv)/ P χ(φv) −⟨χ, g⟩| ≥δ′}. The statement then follows by Lemma 5.3.\n□\nThe final result of this section is the following technical lemma which can be seen as a\nsomewhat stronger version of (5.29) in Corollary 5.4. It is tailored to be used in the proof\nof Proposition 6.7.\nLemma 5.5. Write N +\nn = {v1, v2, . . . , v|N+\nn |} and set N +\nn,M = {v1, v2, . . . , vM}. For δ, ρ >\n0 and f : R →R bounded, define the event\n(5.30)\nBn(f, δ, ρ) := {ρn ≤|N +\nn |} ∩\n\u0012\n[\nρn≤M≤|N+\nn |\n\u001a\f\f\f\f\nP\nv∈N+\nn,M f(φv)\nM\n−⟨χ, f⟩\n⟨χ, 1⟩\n\f\f\f\f > δ\n\u001b\u0013\n.\nThen for every x ≥h∗,\n(5.31)\nlim\nn→∞Px\n\u0002\nBn(f, δ, ρ)\n\f\fN +\nn ̸= ∅\n\u0003\n= 0.\nProof. We start by outlining the strategy of the proof. We will define events An = An(n0)\n(see (5.34)) and A′\nn = A′\nn(n0, q) (see (5.40)), so that when n0 = n0(n) and q = q(n) are\nchosen correctly,\n(5.32)\nlim\nn→∞Px\n\u0002\nAn(n0(n))\n\f\fN +\nn ̸= ∅\n\u0003\n= 0,\nlim\nn→∞Px\n\u0002\nA′\nn(n0(n), q(n))\n\f\fN +\nn ̸= ∅\n\u0003\n= 0,\nBn(f, δ, ρ) ⊆An(n0(n)) ∪A′\nn(n0(n), q(n))\nfor n large enough.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n24\nThe statement of the lemma follows directly from these claims by a union bound.\nTo define An = An(n0), we fix 0 ≤n0 ≤n, write N +\nn0 = {w1, . . . , w|N+\nn0|}, and set\nN i\nn = {v ∈N +\nn : wi is ancestor of v}. For 1 ≤i ≤|N +\nn0|, we set\n(5.33)\nmi =\n(\n|N i\nn|−1 P\nv∈Nin f(φv),\nif |N i\nn| > 0,\n⟨χ, f⟩/⟨χ, 1⟩,\notherwise,\nand define the events\n(5.34)\nAi\nn =\nn\f\f\fmi −⟨χ, f⟩\n⟨χ, 1⟩\n\f\f\f > δ\n2\no\nand\nAn = An(n0) =\n|N+\nn0|\n[\ni=1\nAi\nn.\nWe now prove an upper bound on P[An|N +\nn ̸= ∅].\nBy a union bound, using that\nAi\nn ⊂{N i\nn ̸= ∅} ⊂{N +\nn ̸= ∅},\nPx[An|N +\nn ̸= ∅] = Px[N +\nn ̸= ∅]−1Px[An ∩{N +\nn ̸= ∅}]\n≤Px[N +\nn ̸= ∅]−1Ex\n\u0014 |N+\nn0|\nX\ni=1\nPx[Ai\nn ∩{N i\nn ̸= ∅}|Fn0]\n\u0015\n.\n(5.35)\nUsing the branching process properties of the GFF,\nPx[Ai\nn ∩{N i\nn ̸= ∅}|Fn0]\n= Pφwi\n\u0014\f\f\f\f\n1\n|N +\nn−n0|\nX\nv∈N+\nn−n0\nf(φv) −⟨χ, f⟩\n⟨χ, 1⟩\n\f\f\f\f > δ\n2\n\f\f\f\f N +\nn−n0 ̸= ∅\n\u0015\nPφwi[N +\nn−n0 ̸= ∅].\n(5.36)\nBy Corollary 5.4(b), the first probability on the right-hand side is bounded by some\nεn−n0(φwi) satisfying 1 ≥εn →0 pointwise, and thus in any Lq(ν) by the bounded\nconvergence theorem. By Proposition 4.5 and Lemmas 4.6, 4.7, the second probability\nequals\n(5.37)\nPφwi[N +\nn−n0 ̸= ∅] = C1(n −n0)−1\u0000χ(φwi) + ¯εn−n0(φwi)\n\u0001\n,\nwhere ¯ε →0 pointwise and in L5/2(ν).\nCombining these statements with (5.35) and\n(5.36), and using then Proposition 3.8 in the numerator, we obtain\nPx[An|{N +\nn ̸= ∅}] ≤\nEx\nh P\nw∈N+\nn0 εn−n0(φw)C1(n −n0)−1(χ(φw) + ¯εn−n0(φw))\ni\nC1n−1(χ(x) + ¯εn(x))\n= c(x)n\nn −n0\n⟨χ, εn−n0(χ + ¯εn−n0)⟩χ(x) + ˜εn0(x),\n(5.38)\nwhere ˜εn →0 pointwise. Finally, using H¨older’s inequality on the inner product, since\nχ, χ2 ∈L2(ν) (by (3.18)), εn →0 in L5(ν) and ¯ε →0 in L5/2(ν), it follows that for every\nx ≥h∗,\n(5.39)\nif n0 →∞and n −n0 →∞, then Px[An(n0)|{N +\nn ̸= ∅}] →0.\nWe now turn to the events A′\nn. For given n0 ≤n and q, let\n(5.40)\nA′\nn = A′\nn(n0, q) =\n|N+\nn0|\n[\ni=1\n{|N i\nn| > q}.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n25\nTo bound the probability P[A′\nn|N +\nn ̸= ∅], we write\n(5.41)\nPx[A′\nn ∩{N +\nn ̸= ∅}] ≤Px[A′\nn] ≤Ex\nh |N+\nn0|\nX\ni=1\nPφwi[|N +\nn−n0| > q]\ni\n.\nBy the Markov inequality and Proposition 3.8(b) (with f = g = 1),\n(5.42)\nPφwi[|N +\nn−n0| > q] ≤q−2Eφwi[|N +\nn−n0|2] = Cq−2(χ(φwi)(n −n0) + εn−n0(φwi)),\nwhere ∥εn−n0∥≤c. Thus, by Proposition 3.8(a),\nPx[A′\nn ∩{N +\nn ̸= ∅}] ≤q−2Ex\nh X\nv∈N+\nn0\n(cχ(φv)(n −n0) + εn−n0(φv))\ni\n≤cq−2\u0000(n −n0 + 1)χ(x) + ε′\nn0(x)\n\u0001\n.\n(5.43)\nwith ε′\nn →0 pointwise. Together with Px[N +\nn ̸= ∅] ≥cn−1, by Theorem 2.1, this implies\n(5.44)\nPx[A′\nn|N +\nn ̸= ∅] ≤n\nq2\n\u0010\n(n −n0 + C)χ(x) + ε′\nn0(x)\n\u0011\n.\nWe will now give sufficient conditions on the functions n0(n) and q(n) so that for large\nenough n,\n(5.45)\nBn(f, δ, ρ) ⊂An ∪A′\nn, or equivalently Bn(f, δ, ρ)c ⊃(An)c ∩(A′\nn)c.\nBy definition, {ρn ≤|N +\nn |}c ⊆Bn(f, δ, ρ)c. Therefore, (5.45) is implied by\n(5.46)\n(An)c ∩(A′\nn)c ∩{ρn ≤|N +\nn |} ⊆Bn(f, δ, ρ)c.\nTo prove this, we assume that (An)c ∩(A′\nn)c ∩{ρn ≤|N +\nn |} holds. For M ≤|N +\nn |, let\n(5.47)\nk(M) := inf\nn\nℓ\nX\ni=1\n|N i\nn| : ℓ∈{0, . . . , |N +\nn0|} such that\nℓ\nX\ni=1\n|N i\nn| ≥M\no\n≥M.\nOn (An)c,\n\f\f|N i\nn|−1 P\nv∈Nin f(φv) −⟨χ, f⟩/⟨χ, 1⟩\n\f\f ≤δ/2 for all i = 1, . . . , |N +\nn0|. Therefore,\nfor all M with ρn ≤M ≤|N +\nn |\n(5.48)\n\f\f\f\f\nP\nv∈N+\nn,k(M) f(φv)\nk(M)\n−⟨χ, f⟩\n⟨χ, 1⟩\n\f\f\f\f ≤δ\n2.\nWriting P\nv∈N+\nn,M f(φv) = P\nv∈N+\nn,k(M) f(φv) −P\nv∈N+\nn,k(M)\\N+\nn,M f(φv), and using that the\nabsolute value of the second sum is bounded by (k(M) −M) sup|f|, we obtain that for\nevery M ≥ρn\n\f\f\f\f\nP\nv∈N+\nn,M f(φv)\nM\n−\nP\nv∈N+\nn,k(M) f(φv)\nk(M)\n\f\f\f\f\n=\n\f\f\f\f\nP\nv∈N+\nn,k(M) f(φv)\nM\n−\nP\nv∈N+\nn,k(M)\\N+\nn,M f(φv)\nM\n−\nP\nv∈N+\nn,k(M) f(φv)\nk(M)\n\f\f\f\f\n≤\n\f\f\f\f\nX\nv∈N+\nn,k(M)\nf(φv)\n\u0010 1\nM −\n1\nk(M)\n\u0011\f\f\f\f + k(M) −M\nM\nsup|f|\n= k(M) −M\nM\n\u0012\f\f\f\f\nP\nv∈N+\nn,k(M) f(φv)\nk(M)\n\f\f\f\f + sup|f|\n\u0013\n≤c(f, δ) q\nρn,\n(5.49)\n\nSCALING LIMITS FOR THE CRITICAL GFF\n26\nwhere in the last inequality we used (5.48) and the fact that k(M) −M ≤q on (A′\nn)c. If\nthe right-hand side of (5.49) is smaller than δ/2, then together with (5.48), this implies\nthat Bn(f, δ, ρ)c holds, implying (5.46) and thus (5.45).\nTo finish the proof of (5.32), we must choose n0 = n0(n) and q = q(n) so that (5.39)\napplies and the left-hand sides of (5.43), (5.49) tend to zero. This is easily done by setting,\ne.g., q(n) = n3/4 and n0 = n −n1/4.\n□\n6. The Sn martingale\nThe remaining three sections of this paper are dedicated to proving Theorem 2.3. In\nthis section, we will introduce a martingale based on the depth-first traversal of a sequence\nof copies of Co ∩T+, and prove that it satisfies an invariance principle, see Proposition 6.1.\nThis martingale can be seen as an analogue to the Lukasiewicz path used to study critical\nGalton-Watson trees.\nThe second part of the section then demonstrates another two\nscaling limit results, Proposition 6.8 and Proposition 6.10 which are both consequences\nof Proposition 6.1.\nTo define the martingale, we consider an i.i.d. sequence T = ((T 1, φ1), (T 2, φ2), . . . )\nwhere every (T i, φi) is distributed as (Co ∩T+, φ|Co∩T+) under Px. To keep the notation\nsimple, we keep using Px for the probability measure associated with the whole sequence\nT, and for v ∈T i we write φv instead of φi\nv. We further use oi to denote the root of T i,\nand set N +,i\nn\n= {v ∈T i : d(oi, v) = n}. Note that |N +,i\nn | has the same distribution as\n|N +\nn | which was studied in detail in the previous sections. In particular, we know that all\nT i are a.s. finite.\nThroughout this section, we use the notation for the parent p(v), direct descendants\ndesc(v), and siblings sib(v) of v ∈T i relative to the rooted tree T i (not T), w ⪯v means\nthat w is an ancestor of v, cf. below (3.1).\nWe now describe the depth-first traversal v = (v1, v2, . . . ) of T. It starts at the root of\nT 1, that is v1 = o1, and then explores the tree T 1 in a depth-first manner. After visiting\nall vertices of T 1, it proceeds to o2, explores T 2 in a depth-first manner, and so forth. vi\ndenotes the i-th vertex visited during this traversal. The notation v <v w indicates that\nv precedes w in v (with ≤v representing the reflexive version). For any v ∈∪iT i we write\nΛ(v) to denote the index of the tree which v belongs to (that is v ∈T Λ(v)). We define\n(6.1)\nΛn = Λ(vn)\nand\nHn = |vn| = d(vn, oΛn),\nthat is Λn is the index of the tree which is explored at step n, and Hn is the “height” of\nthe n-th explored vertex. For every v ∈∪iT i, we define the set Y (v) as the union of all\nsiblings of some ancestor of v that appear later in v than this ancestor itself, that is,\n(6.2)\nY (v) =\n[\nw⪯v\n{u ∈sib(w) : w <v u}.\nWith this notation we can introduce the key object of this section, the process\n(6.3)\nSn = χ(φvn) −\nX\ni≤Λn\nχ(φoi) +\nX\nw∈Y (vn)\nχ(φw),\nn ∈N,\nwhich is adapted to the filtration\n(6.4)\nHn = σ\n\u0010\n(v, φv) : v ∈\nn[\ni=1\nsib(vi)\n\u0011\n,\nn ∈N.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n27\nThe next proposition shows that Sn converges to a Brownian motion with variance\n(6.5)\nσ2 = ⟨χ, V⟩/⟨χ, 1⟩,\nwhere\n(6.6)\nV(x) := Px\nh\u0010 X\nw∈N+\n1\nχ(φw)\n\u00112i\n−Px\nh X\nw∈N+\n1\nχ(φw)\ni2\n= Varx\n\u0010 X\nw∈N+\n1\nχ(φw)\n\u0011\n.\nHere and below (Bt)t≥0 denotes the standard Brownian motion.\nProposition 6.1. For every x ≥h∗, in Px-distribution w.r.t. the Skorokhod topology,\n(6.7)\nlim\nn→∞\n\u0010 1\n√nS⌊nt⌋\n\u0011\nt≥0 = (σBt)t≥0.\nWe need a few preparatory steps to show this proposition. We start by proving that S\nis a martingale.\nLemma 6.2. The process S is a H-martingale under Px.\nProof. We first show that\n(6.8)\nSk+1 −Sk = −χ(φvk) +\nX\nw∈desc(vk)\nχ(φw).\nTo this end we need to distinguish between several possible scenarios:\n(1) If vk is not a leaf, that is desc(vk) ̸= ∅, then vk+1 is the first child of vk, and thus\nY (vk+1) = (desc(vk)\\{vk+1})∪Y (vk) where the union is disjoint. From this (6.8) follows.\n(2) If vk is a leaf, that is desc(vk) = ∅, then there are three possible cases for vk+1:\nEither it is the next sibling (with respect to <v) of vk, or the next sibling of some\nancestor of vk, or it is the root of the next tree in the tree sequence. In the first two cases,\nY (vk) = Y (vk+1) ∪{vk+1} where the union is disjoint, and thus Sk+1 −Sk = −χ(φvk) and\nthus (6.8) holds. In the last case, when vk+1 is the root of the next tree, both sets Y (vk)\nand Y (vk+1) must be empty and Λk+1 = Λk + 1, which implies (6.8) also in this case.\nFrom (6.8) and the branching process properties of φ it follows that\nEx[Sk+1 −Sk|Hk] = Ex\nh\n−χ(φvk) +\nX\nw∈desc(vk)\nχ(φw)\n\f\f\fHk\ni\n= −χ(φvk) + Eφvk\nh X\nw∈N+\n1\nχ(φw)\ni\n= −χ(φvk) + Lχ(φvk) = 0,\n(6.9)\nsince χ is the eigenfunction of L (see (3.12)). This finishes the proof.\n□\nThe next four simple lemmas will be used to control the quadratic variation of S.\nLemma 6.3. For every k ≥1, with V as in (6.6),\n(6.10)\nEx[(Sk+1 −Sk)2|Hk] = V(φvk).\nProof. By (6.8) the conditional expectation Ex[(Sk+1 −Sk)2|Hk] equals\nχ(φvk)2 −2χ(φvk)Ex\nh\nX\nw∈desc(vk)\nχ(φw)\n\f\f\fHk\ni\n+ Ex\nh\u0010\nX\nw∈desc(vk)\nχ(φw)\n\u00112\f\f\fHk\ni\n= χ(φvk)2 −2χ(φvk)Eφvk\nh X\nw∈N+\n1\nχ(φw)\ni\n+ Eφvk\nh\u0010 X\nw∈N+\n1\nχ(φw)\n\u00112i\n.\n(6.11)\nFrom this the statement follows using (3.12) again.\n□\n\nSCALING LIMITS FOR THE CRITICAL GFF\n28\nLemma 6.4. There is c < ∞such that V(x) < c for all x ≥h∗.\nProof. Let {w1, . . . , wd} be the children of the root in T+. Since (φwi)d\ni=1 are indepen-\ndent under Px and χ(x) = 0 for x < h∗, Varx(P\nv∈N+\n1 χ(φv)) = Varx(Pd\ni=1 χ(φwi)) =\nPd\ni=1 Varx(χ(φwi)) = d VarY (χ(Y + x/d)), where Y ∼N(0, σ2\nY ), see (3.4). Since χ is\nLipschitz on [h∗, ∞) by (3.19), the statement follows.\n□\nTo state the next result, let Un be a random variable distributed uniformly on {1, . . . , n},\ndefined on the same probability space as the sequence T, independent of T. We write\n(6.12)\nH∗\nn = HUn\nand\nΛ∗\nn = ΛUn,\nfor the height and the tree index of a vertex chosen uniformly amongst the first n explored\nvertices.\nLemma 6.5. For all x ≥h∗,\n(6.13)\nlim\nu→∞sup\nn≥1\nPx[H∗\nn ≥u√n] = 0.\nProof. By Theorem 2.1 and Proposition 3.3, there is c(x) < ∞such that for all n ≥1,\n(6.14)\nPx[N +\nn ̸= ∅] ≤c(x)\nn\nand\nEx[Λn] ≤c(x)√n.\nTo see the second inequality in (6.14), we note that by the independence of the trees T i\nand Proposition 3.3, Px[Λn ≥k] ≤Px[|T i| < n, i = 1, . . . , k −1] = Px[|T 1| < n]k−1 ≤\n(1 −c(x)n−1/2)k−1 and thus Ex[Λn] = P∞\nk=1 Px[Λn ≥k] ≤c(x)−1n1/2 as claimed.\nSince {H∗\nn ≥u√n} ⊆{∃i ≤Λn : N +,i\n⌈u√n⌉̸= ∅}, by a union bound,\n(6.15)\nPx[H∗\nn ≥u√n] ≤\nX\ni≥1\nPx\n\u0002\ni ≤Λn, N +,i\n⌈u√n⌉̸= ∅\n\u0003\n.\nThe events {i ≤Λn} and {|N +,i\n⌈u√n⌉| > 0} are independent. Hence, using the first half\nof (6.14), this is bounded by c(x)u−1n−1/2 P\ni≥1 Px[i ≤Λn]. Since P\ni≥1 Px[i ≤Λn] =\nEx[Λn] ≤c(x)√n, by the second half of (6.14), the claim follows.\n□\nLemma 6.6. For every δ > 0 and x ≥h∗there exists a sequence R(n) such that\nlimn→∞R(n) = ∞and\n(6.16)\nsup\nn≥1\nPx[H∗\nn ≤R(n)] ≤δ.\nProof. We will show that for every K < ∞\n(6.17)\nlim\nn→∞Px[H∗\nn ≤K] = 0,\nwhich implies the statement of the lemma.\nFor fixed K > 0, by definition of H∗\nn,\n(6.18)\nPx[H∗\nn ≤K] = n−1\nn\nX\nl=1\nPx[Hl ≤K] = n−1Ex\nh\nK\nX\nh=0\nn\nX\nl=1\n1Hl=h\ni\n.\nSince Pn\nl=1 1Hl=h ≤PΛn\ni=1|N +,i\nh | and thus PK\nh=0\nPn\nl=1 1Hl=h ≤PΛn\ni=1\nPK\nh=0|N +,i\nh |. More-\nover, (PK\nh=0|N +,i\nh |)Λn\ni=1 are i.i.d. and with finite mean. Noting that Λn is a stopping time\n\nSCALING LIMITS FOR THE CRITICAL GFF\n29\nwith respect to Gi = σ(T j : j ≤i) and using Wald’s equation (see e.g. [Dur19, Theorem\n2.6.2]),\n(6.19)\nPx[H∗\nn ≤K] ≤1\nnEx\nh Λn\nX\ni=1\nK\nX\nh=0\n|N +,i\nh |\ni\n= 1\nnEx[Λn]Ex\nh\nK\nX\nh=0\n|N +,1\nh\n|\ni\n,\nwhich by (6.14) converges to 0 as n →∞.\n□\nThe following law of large numbers will be important for the proof of Proposition 6.1.\nProposition 6.7. Let f be a bounded function and mf\nn := n−1 Pn\ni=1 f(φvi). Then\n(6.20)\nlim\nn→∞mf\nn = mf\n∞:= ⟨χ, f⟩\n⟨χ, 1⟩,\nin Px-probability.\nProof. Let\n(6.21)\nN +,i\nk,n = N +,i\nk\n∩{v1, . . . , vn}\nbe the part of N +,i\nk\ntraversed in the first n steps, and let N ∗\nn = N +,Λ∗\nn\nH∗n,n . We set\n(6.22)\nm∗\nn = |N ∗\nn|−1 X\nv∈N∗n\nf(φv).\nDenoting by FT the σ-algebra generated by the sequence T = (Ti, φi)i≥1, it holds that\n(6.23)\nmf\nn = Ex[m∗\nn|FT]\n(effectively, the expectation here is over Un only). To show the proposition it will thus be\nsufficient to show that\n(6.24)\nlim\nn→∞m∗\nn = mf\n∞\nin Px-probability.\nTo see that this is indeed enough, note that since f is bounded m∗\nn is dominated by a\nconstant. Hence, by the dominated convergence theorem, m∗\nn →mf\n∞also in L2(Px). By\n(6.23), since the conditional expectation is a contraction on L2(Px), mf\nn →mf\n∞in L2(Px),\nand thus in Px-probability as claimed.\nTo prove (6.24), we fix ε, δ > 0 and set\n(6.25)\nAn = {|m∗\nn −mf\n∞| > ε}.\nWe then fix C < ∞such that Px[H∗\nn ≥Cn1/2] ≤δ/3 for all n ≥1, which is possible\nby Lemma 6.5, and take R(n) as in Lemma 6.6, so that Px[H∗\nn ≤R(n)] ≤δ/3. Setting\nBn = {R(n) ≤H∗\nn ≤Cn1/2}, it follows that for all n ≥1,\n(6.26)\nPx[An] ≤2δ/3 + Px[An ∩Bn]\n\nSCALING LIMITS FOR THE CRITICAL GFF\n30\nWe further set mi\nh,n = |N +,i\nh,n|−1 P\nv∈N+,i\nh,n f(φv) and Ai\nh,n = {|mi\nh,n −mf\n∞| > ε}. Using the\ndefinitions of Un and m∗\nn, and then decomposing by possible values of Hk, Λk, we obtain\nPx[An ∩Bn] =\nn\nX\nk=1\nPx[An ∩Bn|Un = k]Px[Un = k]\n=\nn\nX\nk=1\n1\nnPx[AΛk\nHk,n, R(n) ≤Hk ≤Cn1/2]\n= 1\nn\nCn1/2\nX\nl=R(n)\nn\nX\ni=1\nEx\nh\n1Ai\nl,n\nn\nX\nk=1\n1{Hk=l,Λk=i}\ni\n.\n= 1\nn\nCn1/2\nX\nl=R(n)\nn\nX\ni=1\nEx\n\u0002\n1Ai\nl,n\n\f\fN +,i\nl,n\n\f\f\u0003\n,\n(6.27)\nwhere in the last step we used Pn\nk=1 1{Hk=l,Λk=i} = |N +,i\nl,n |. To analyse the expectation on\nthe right-hand side, we define σ-algebras Gi = σ(T 1, T 2, . . . , T i) and set τi = |T 1| + · · · +\n|T i−1| (which is Gi−1-measurable). Then, using that (T i, φi)i≥1 is an i.i.d. sequence,\nEx\n\u0002\n1Ai\nl,n|N +,i\nl,n |\n\f\fGi−1\n\u0003\n= 1{τi≤n}Ex\n\u0002\n1A1\nl,n−τi\n\f\fN +,1\nl,n−τi\n\f\f\u0003\n= 1{τi≤n}Ex\n\u0002\n1Al,n−τi|N +\nl,n−τi|\n\f\fN +\nl ̸= ∅\n\u0003\nPx[N +\nl ̸= ∅],\n(6.28)\nwhere on the last line we omitted the superscript ‘1’ since N +,1\nl,k has the same distribution\nas N +\nl,k. By Theorem 2.1, Proposition 3.8 and (6.14) there is a c = c(x) such that\n(6.29)\nPx\n\u0002\nN +\nl ̸= ∅\n\u0003\n≤cl−1,\nEx\n\u0002\f\fN +\nl\n\f\f2 \f\f N +\nl ̸= ∅\n\u00031/2 ≤cl,\nEx\n\u0002\nΛn\n\u0003\n≤cn1/2.\nFurther, for Bn(f, ε, ρ) as in Lemma 5.5, it holds that Al,k ∩{|N +\nl,k| ≥lρ} ⊆Bl(f, ε, ρ).\nTherefore, choosing ρ = δ/(6Cc2) and decomposing on whether |N +\nl,n−τi| is larger than\nlδ/(6Cc2), we obtain\nEx\n\u0002\n1A1\nl,n−τi|N +,1\nl,n−τi|\n\f\fN +,1\nl\n̸= ∅\n\u0003\n≤\nlδ\n6Cc2 + Ex\n\u0002\n1Bn(f,ε,ρ)|N +\nl,n−τi|\n\f\fN +\nl ̸= ∅\n\u0003\n≤\nlδ\n6Cc2 + Ex\n\u0002\n|N +\nl |2\f\fN +\nl ̸= ∅\n\u00031/2Px\n\u0002\nBl(f, ε, ρ)\n\f\fN +\nl ̸= ∅\n\u00031/2,\n(6.30)\nwhere in the last step we used the Cauchy–Schwarz inequality and |N +\nl,n−τi| ≤|N +\nl |.\nCombining (6.28) and (6.30) with the first two claims in (6.29) we obtain\n(6.31)\nEx\n\u0002\n1Ai\nl,n|N +,i\nl,n |\n\u0003\n≤\n\u0010 δ\n6Cc + c2Px\n\u0002\nBl(f, ε, ρ)\n\f\f N +\nl ̸= ∅\n\u00031/2\u0011\nPx[τi ≤n].\nInserting this into (6.26), (6.27), we obtain that Px[An] is bounded by\n(6.32)\n2δ\n3 + 1\nn\n\u0010 δ\n6Cc + c2 sup\nl≥R(n)\nPx\n\u0002\nBl(f, ε, ρ)\n\f\f N +\nl ̸= ∅\n\u00031/2\u0011 ⌊C√n⌋\nX\nl=R(n)\nn\nX\ni=1\nPx[Λn ≥i],\nSince R(n) diverges, Lemma 5.5 implies that the supremum in this formula tends to zero\nas n →∞. By the last property in (6.29), P⌊C√n⌋\nl=R(n)\nPn\ni=1 Px[Λn ≥i] ≤C√nEx[Λn] ≤Ccn.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n31\nTherefore\n(6.33)\nPx[An] ≤2δ\n3 +\n\u0010δ\n6 + Cc3o(1)\n\u0011\n≤δ\nfor n large enough. Since ε and δ are arbitrary, this shows (6.24) and with the initial\ncomment completes the proof.\n□\nWe are now ready to prove Proposition 6.1.\nProof of Proposition 6.1. We apply a martingale functional central limit theorem, see\ne.g. [Dur19, Theorem 8.2.8]. To check its assumptions we need to show that\n(a) limn→∞n−1 Pn\nk=1 Ex[(Sk −Sk−1)2|Hk−1] = ⟨χ,V⟩\n⟨χ,1⟩in Px-probability,\n(b) limn→∞n−1 Pn\nk=1 Ex[(Sk −Sk−1)21{|Sk−Sk−1|>ε√n}] = 0.\nCondition (a) follows from Lemma 6.3 and Proposition 6.7, using that V is bounded by\nLemma 6.4.\nWe now show (b). By the Cauchy–Schwarz inequality, Ex[(Sk−Sk−1)21{|Sk−Sk−1|>ε√n}] ≤\nEx[(Sk −Sk−1)4]1/2Px[(Sk −Sk−1)2 > ε2n]1/2. By Lemmas 6.3, 6.4, Ex[(Sk −Sk−1)2] ≤c\nuniformly in k and thus Px[(Sk −Sk−1)2 > ε2n] ≤cε−2n−1. Thus, to prove (b) it is enough\nto show\n(6.34)\nEx[(Sk −Sk−1)4] ≤c\nuniformly in k. By (6.8), Sk −Sk−1 = P\nv∈desc(vk−1) χ(φv) −χ(φvk−1), and thus\nEx[(Sk −Sk−1)4|Hk−1] = Eφvk−1\nh\u0010 X\nv∈N+\n1\nχ(φv) −χ(φvk−1)\n\u00114i\n= Eφvk−1\nh\u0010 X\nv∈N+\n1\nχ(φv) −Eφvk−1\n\u0002 X\nv∈N+\n1\nχ(φv)\n\u0003\u00114i\n,\n(6.35)\nwhere we used that χ(φvk−1) = Eφvk−1[P\nv∈N+\n1 χ(φv)] in the last line. We write N +\n1 =\n{w1, . . . , wd} and follow similar arguments as in the proof of Lemma 6.4, using that\nχ(x) = 0 for x < h∗and exploiting the independence of the φwi, to obtain that (6.35) is\nbounded above by cdEφvk−1[(χ(φw1) −Eφvk−1[χ(φw1)])4]. Since χ is Lipschitz on [h∗, ∞),\nχ(φw1)−Ex[χ(φw1)] is sub-Gaussian, and thus Ex[(χ(φw1)−Ex[χ(φw1)])4] ≤c, where the\nconstant c only depends on d and the Lipschitz constant of χ but not on x. Statement\n(6.34) then follows by taking the expectation of (6.35).\n□\n6.1. Further invariance principles. We now prove several further convergence results\nthat are a consequence of Proposition 6.1 and which will be useful later.\nProposition 6.8. Let Bt be a standard Brownian motion, let L0\nt be its local time at 0.\nThen, as n →∞, in Px-distribution with respect to the Skorokhod topology,\n(6.36)\n\u0012P\nw∈Y (v⌊nt⌋) χ(φw)\n√n\n, Λ(v⌊nt⌋)\n√n\n\u0013\nt≥0\n→\n\u0010\nσ|Bt|,\nσ\nχ(x)L0\nt\n\u0011\nt≥0.\nTo show Proposition 6.8 from Proposition 6.1, we need the following lemma. It states\nthat (under scaling) the first summand in the definition (6.3) of Sn is negligible.\nLemma 6.9. In Px-distribution with respect to the Skorokhod topology,\n(6.37)\nlim\nn→∞\n\u0010χ(φv⌊nt⌋)\n√n\n\u0011\nt≥0 = 0.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n32\nProof. As χ grows at most linearly (see (3.18)), it is enough to show that for any fixed\nt > 0 and ε > 0,\n(6.38)\nlim\nn→∞Px\nh\nmax\ni≤nt φvi/√n > ε\ni\n= 0.\nLet H(T i) = maxv∈T i H(v) denote the height of tree T i. For every j, k ∈N, the probability\nin (6.38) is bounded from above by\nPx\nh\nmax\ni≤nt φvi > ε√n, Λ⌊nt⌋≤j, H(Ti) ≤k for i = 1, . . . , j\ni\n+ Px[∃i ∈{1, . . . , j} : H(T i) > k, Λ⌊nt⌋≤j] + Px[Λ⌊nt⌋> j].\n(6.39)\nTo show (6.38), we thus need to choose jn →∞and kn →∞so that all three summands\nconverge to zero.\nWe start with the first summand. Restricted to {Λ⌊nt⌋≤j} and {H(T i) ≤k for i =\n1, . . . , j}, maxk≤nt φvk is dominated by the maximum of all φv with v such that Λ(v) ≤j\nand H(v) ≤k. Considering not only the maximum over the connected components of the\nlevel set, but over the first k generations in j copies of the whole forward tree T+, this is\ndominated by the maximum of jdk non-negatively correlated Gaussian random variables\nwith mean at most x and bounded variance. By Gaussian comparison techniques, the\nmean of this maximum is of order x + c\np\nlog jdk ≤c√k log j for j, k large enough. Thus,\nby the Markov inequality,\n(6.40)\nPx\nh\nmax\ni≤nt φvi > ε√n, Λ⌊nt⌋≤j, H(Ti) ≤k for i = 1, . . . , j\ni\n≤\nC\nεn1/2\np\nk log(j).\nBy a union bound, using Theorem 2.1, the second summand in (6.39) satisfies\n(6.41)\nPx[∃i ∈{1, . . . , j} : H(T i) > k, Λ⌊nt⌋≤j] ≤c(x)j/k.\nFinally, by the Markov inequality and (6.14), the third summand can be bounded by\n(6.42)\nPx[Λ⌊nt⌋> j] ≤c(x)\n√\ntn/j.\nSetting now, e.g., jn = n2/3 and kn = n5/6, all three summands in (6.39) converge to 0 as\nrequired.\n□\nProof of Proposition 6.8. We use the continuous mapping theorem together with the al-\nready established convergence statements. By Proposition 6.1, n−1/2S⌊n·⌋→σB·, and by\nLemma 6.9, n−1/2χ(φv⌊n·⌋) →0 as n →∞, in Px-distribution, in the Skorokhod topology.\nTherefore\n(6.43)\nn−1/2(S⌊n·⌋−χ(φv⌊n·⌋)) = n−1/2\u0010\nX\nw∈Y (v⌊n·⌋)\nχ(φw) −\nX\ni≤Λ⌊n·⌋\nχ(φoi)\n\u0011\nalso converges to σB· as n →∞. Next, note that\n(6.44)\ninf\nk≤n{Sk −χ(φvk)} = inf\nk≤n\nn X\nw∈Y (vk)\nχ(φw) −\nX\ni≤Λk\nχ(φoi)\no\n= −\nX\ni≤Λn\nχ(φoi).\nTherefore, setting Bt = infs≤t Bs and using the continuous mapping theorem with the\nmap g(Xt) = (Xt −infs≤t Xs, −infs≤t Xs),\n(6.45)\nn−1/2\u0010\nX\nw∈Y (v⌊n·⌋)\nχ(φw),\nX\ni≤Λ⌊n·⌋\nχ(φoi)\n\u0011\n→\n\u0000σ(B· −B·), −σB·\n\u0001\n\nSCALING LIMITS FOR THE CRITICAL GFF\n33\nin distribution as n →∞. By L´evy’s theorem (see, e.g., Theorem VI.2.3 in [RY99]), the\nright-hand side of (6.45) has the same distribution as (σ|B·|, σL0\n· (B)). Together with the\nfact that under Px, P\ni≤Λ⌊nt⌋χ(φoi) = χ(x)Λ⌊nt⌋, this implies the proposition.\n□\nAnother consequence of Proposition 6.1 is the following scaling limit result for Sn\nconditioned to reach a certain height on the first tree T1. To this end we define\n(6.46)\n¯Sn =\n(P\nw∈Y (vn) χ(φw)\nif n ≤|T 1|,\n0\nif n > |T 1|.\nNote that, up to an additive correction χ(φvn) −χ(x), ¯S· is equal to S· restricted to T1.\nProposition 6.10. For y > 0, let (e≥y/σ)t≥0 be a Brownian excursion conditioned to\nreach at least height y/σ. Then, in distribution under Px[ · | supk ¯Sk ≥√ny], with respect\nto the Skorokhod topology,\n(6.47)\nlim\nn→∞\n\u0000n−1/2 ¯S⌊nt⌋\n\u0001\nt≥0 = (σe≥y/σ)t≥0.\nWe omit the proof of this proposition as it would be identical to the proof of Proposi-\ntion 6.13 in [Pow19], which itself follows [DLG02, Proposition 2.5.2].\n7. Relation of Sn and Hn\nThe aim of this section will be to establish a connection between the martingale Sn and\nthe height process Hn (see (6.3) and (6.1) for definitions). This will be useful in the proof\nof Theorem 2.3 in Section 8. Throughout the whole section, we only consider the field on\nthe first tree (T, φ) = (T 1, φ1) in the infinite tree sequence T = ((T 1, φ1), (T 2, φ2), . . . )\nintroduced in Section 6. As a consequence, we only consider ¯S introduced in (6.46) instead\nof S. We will see that, approximately, ¯S(v) ≈H(v)/C1 with C1 as in (2.11).\nMotivated by this, for η > 0 we say that v ∈T is η-bad if\n(7.1)\n\f\f\f\n¯S(v)\nH(v) −C−1\n1\n\f\f\f > η.\nFixing in addition R > 0, we say that v ∈T is (η, R)-bad if there exists a w ≺v such\nthat H(w) ≥R and w is η-bad. This means that v is (η, R)-good (i.e. not (η, R)-bad) if\nall its ancestors in generations at least R are η-good. We set\n(7.2)\nN (η,R)\nn\n= {v ∈N +\nn : v is (η, R)-bad}.\nThe first main result of this section shows that this set is relatively small.\nProposition 7.1. For every ε > 0 and x ≥h∗,\n(7.3)\nlim\nR→∞sup\nn≥R\nPx\nh|N (η,R)\nn\n|\n|N +\nn |\n> ε\n\f\f\f N +\nn ̸= ∅\ni\n= 0.\nProof. The proof follows a strategy similar to the proof of Proposition 6.17 in [Pow19],\nwith adaptations that are necessary in order to handle the unbounded domain in our\nsetting. For ε > 0, R > 0 and n ≥0 we define the event\n(7.4)\nEε\nR,n =\nnP\nv∈N+\nn χ(φv)1{v is (η, R)-bad}\nP\nv∈N+\nn χ(φv)\n> ε\no\n.\nWe first claim that it suffices to show that\n(7.5)\nlim\nR→∞sup\nn≥R\nPx[Eε\nR,n|N +\nn ̸= ∅] = 0.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n34\nTo see that this indeed implies the lemma, we write\n(7.6)\n|N (η,R)\nn\n|\n|N +\nn |\n=\n|N (η,R)\nn\n|\nP\nv∈N+\nn χ(φv) ·\nP\nv∈N+\nn χ(φv)\n|N +\nn |\n.\nTherefore, for every δ > 0, the probability in (7.3) is bounded from above by\n(7.7)\nPx\nh\n|N (η,R)\nn\n|\nP\nv∈N+\nn χ(φv) > δ\n\f\f\f N +\nn ̸= ∅\ni\n+ Px\nh\n|N +\nn |\nP\nv∈N+\nn χ(φv) < δ/ε\n\f\f\f N +\nn ̸= ∅\ni\n.\nBy Lemma 5.3 (with f = 1), the second probability tends to 0 as n →∞if δ = δ(ε)\nis small enough. Since χ(x) ≥χ(h∗) > 0 for every x ≥h∗, it holds that |N (η,R)\nn\n| ≤\nc P\nv∈N+\nn χ(φv)1{v is (η, R)-bad}, and thus\n(7.8)\nPx\nh\n|N +,(η,R)\nn\n|\nP\nv∈N+\nn χ(φv) > δ\n\f\f\fN +\nn ̸= ∅\ni\n≤Px\n\u0002\nEδ/c\nR,n\n\f\fN +\nn ̸= ∅\n\u0003\n,\nwhich together with (7.5) implies the claim of the lemma.\nWe now prove (7.5). By first using the Markov inequality, and then Lemma B.2 with\nk = 1 and Y (v1) = χ(φv1)1{v1 is (η, R)-bad}1{N+\nn ̸=∅}/(P\nw∈N+\nn χ(φw)), it holds\nPx[Eε\nR,n|N +\nn ̸= ∅] ≤ε−1Ex\nhP\nv∈N+\nn χ(φv)1{v is (η, R)-bad}\nP\nv∈N+\nn χ(φv)\n\f\f\fN +\nn ̸= ∅\ni\n= ε−1Ex\nh1{N+\nn ̸=∅}\nP\nv∈N+\nn χ(φv)1{v is (η, R)-bad}\nP\nv∈N+\nn χ(φv)\ni\nPx[N +\nn ̸= ∅]−1\n= ε−1Qx\nhχ(x)Px[N +\nn ̸= ∅]−1\nP\nv∈N+\nn χ(φv)\n1{σn is (η, R)-bad}\ni\n,\n(7.9)\nwhere, as defined in Section 3, σn is the vertex on the spine in the n-th generation. To\ncontinue, we first show the following two claims:\n(i) supn≥R Qx[σn is (η, R)-bad] →0 as R →∞.\n(ii) Set Zn =\nχ(x)Px[N+\nn ̸=∅]−1\nP\nv∈N+\nn χ(φv) . Then for all δ > 0, there exist R′, K > 0 such that\nQx[Zn1{Zn>K}] ≤δ for all n ≥R′.\nClaim (i) will follow from the ergodic behaviour of the field along the spine (σn)n≥0\nunder Qx. Let sib<(v) = {w ∈sib(v) : v <v w}. Then ¯S(σn) = P\nk≤n\nP\nv∈sib<(σk) χ(φv).\nRecall that ξk := φ(σk). By conditioning on ξk−1, due to the fact that under Qx the spine\nmark at generation k is uniformly distributed on descendants of its position at generation\nk −1 and that non-spine vertices behave as under Px,\nQx\nh\nX\nv∈sib<(σk)\nχ(φv)\ni\n= Qx\nh\nQξk−1\nh\nX\nv∈sib<(σ1)\nχ(φv)\nii\n= Qx\nh1\n2Qξk−1\nh\nX\nv∈N+\n1 \\{σ1}\nχ(φv)\nii\n= Qx\nhd −1\n2d Pξk−1\nh X\nv∈N+\n1\nχ(φv)\nii\n= d −1\n2d Qx[χ(ξk−1)].\n(7.10)\n\nSCALING LIMITS FOR THE CRITICAL GFF\n35\nBy Lemma 3.4 the invariant measure of the Markov chain (ξk)k≥0 under Qx is given by\nχ(y)2ν(dy), so the last expression converges to C−1\n1\nas k →∞. Claim (i) then follows\nfrom an ergodicity argument.\nTo see (ii), note that again by Lemma B.2,\nQx[Zn1{Zn>K}] =\nPx[1{Zn>K}1{N+\nn ̸=∅}]\nP[N +\nn ̸= ∅]\n= Px[Zn > K|N +\nn ̸= ∅].\n≤Px\nhP\nv∈N+\nn χ(φv)\nn\n< 1\nKc\n\f\f\fN +\nn ̸= ∅\ni\n,\n(7.11)\nwhere in the last inequality we used the fact that χ(x)Px[N +\nn ̸= ∅]−1 ≥c, by Theorem 2.1.\nBy Theorem 2.2, this converges to the probability that an exponentially distributed ran-\ndom variable is smaller than (Kcx)−1. It is thus straightforward to find K and R′ such\nthat the last probability is smaller than δ for all n ≥R′, proving (ii).\nReturning back to (7.9), its right-hand side is bounded by\nQx[Zn1{σn is (η, R)-bad}]\n= Qx[Zn1{Zn>K}1{σn is (η, R)-bad}] + Q[Zn1{Zn≤K}1{σn is (η, R)-bad}]\n≤Qx[Zn1{Zn>K}] + KQ[σn is (η, R)-bad],\n(7.12)\nwhich together with (i) and (ii) implies (7.5) and concludes the proof.\n□\nTo state the second main result of this section, we introduce (enlarging the probability\nspace if necessary) random variables τ 1, . . . , τ k which under Px, conditionally on T, are\nindependent and uniformly distributed on {1, 2, . . . , |T|}. Using these random variables,\nwe define two k × k random matrices\n(D\n¯S\nn)i,j = n−1\u0000 ¯S(vτ i) + ¯S(vτ j) −2 ¯S(vτ i ∧vτ j)\n\u0001\n(DH\nn )i,j = n−1\u0000H(vτ i) + H(vτ j) −2H(vτ i ∧vτ j)\n\u0001\n,\n(7.13)\nwhere, as usual, (v1, v2, . . . ) denotes the depth-first traversal of T, and v ∧w denotes the\nmost recent common ancestor of v and w in T.\nProposition 7.2. For every k ≥1 and ε > 0,\n(7.14)\nlim\nn→∞Px\n\u0002\n∥C−1\n1 DH\nn −D\n¯S\nn∥> ε\n\f\fN +\nn ̸= ∅\n\u0003\n= 0,\nwhere ∥·∥denotes the Frobenius norm of k × k matrices.\nTo prove this proposition, we need two lemmas. The first one estimates from below the\nsize of T conditionally on {N +\nn ̸= ∅}.\nLemma 7.3. For every x ≥h∗,\n(7.15)\nlim\nq→0 sup\nn≥0\nPx\n\u0002\n|T| ≤qn2|N +\nn ̸= ∅\n\u0003\n= 0.\nProof. It will be sufficient to show that for every δ > 0 there exist q and n0 such that for\nevery n ≥n0,\n(7.16)\nPx[|T| ≤qn2|N +\nn ̸= ∅] ≤δ.\nIndeed, since there are only finitely many n < n0, by decreasing the value of q, the\ninequality (7.16) can be made true for all n ≥0. This then directly implies the claim of\nthe lemma.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n36\nTo show (7.16), observe that for every η > 0\nPx[|T| ≤qn2|N +\nn ̸= ∅]\n≤Px\n\u0002\n|T| ≤qn2, |N +\nn | ≥ηn|N +\nn ̸= ∅\n\u0003\n+ Px\n\u0002\n|N +\nn | < ηn|N +\nn ̸= ∅\n\u0003\n.\n(7.17)\nBy Theorem 2.2, there is η > 0 small such that the second probability on the right-hand\nside of (7.17) is bounded by δ/2 for all n large enough. It is thus sufficient to show that\nfor every δ > 0 and η > 0 there is n0 so that for all n ≥n0\n(7.18)\nPx\n\u0002\n|T| ≤qn2, |N +\nn | ≥ηn\n\f\fN +\nn ̸= ∅\n\u0003\n≤δ/2.\nNote first that\n(7.19)\nPx\n\u0002\n|T| ≤qn2, |N +\nn | ≥ηn\n\f\fN +\nn ̸= ∅\n\u0003\n≤Px\n\u0002\n|T| ≤qn2\f\f|N +\nn | ≥ηn\n\u0003\n.\nGiven {|N +\nn | ≥ηn}, let w1, . . . , w⌊ηn⌋be the first ⌊ηn⌋vertices in N +\nn , and let Tw be\nthe subtree of T rooted at w. Obviously, |T| ≤P⌊ηn⌋\ni=1 |Twi|. Under P[ · ||N +\nn | ≥ηn], the\nrandom variables |Twi| are independent. Moreover, since w ∈N +\nn implies φw ≥h∗, by\nstochastic domination (3.5), for any u > 0,\n(7.20)\nPx\n\u0002\n|Twi| ≥u\n\f\f|N +\nn | ≥ηn\n\u0003\n≥Ph∗\u0002\n|Co ∩T+| ≥u\n\u0003\n.\nDenoting T ′\ni, i ≥1, i.i.d. random variables distributed as |Co ∩T+| under Ph∗, it follows\nthat\n(7.21)\nPx\n\u0002\n|T| ≤qn2\f\f|N +\nn | ≥ηn\n\u0003\n≤P\nh ⌊ηn⌋\nX\ni=1\nT ′\ni ≤qn2i\n.\nBy Theorem 3.3, T ′\ni are in the domain of attraction of the 1/2-stable random distribution.\nTherefore, n−2 P\ni≤⌊ηn⌋T ′\ni converges in distribution to a non-negative 1/2-stable random\nvariable (see e.g. [Fel71, Theorem XIII.6.2]). As a consequence, since the distribution of\nthis random variable has no atom at 0, for any δ, η > 0 there exists q small so that for all\nn large enough the right-hand side of (7.21) is bounded by δ/2. This shows (7.18) and\ncompletes the proof.\n□\nThe second lemma needed to show Proposition 7.2 studies the probability that a ran-\ndomly chosen vertex vτ1 is (η, R)-bad.\nLemma 7.4. Let V := vτ1 be a uniformly chosen vertex of T. Then, for every x ≥h∗\nand η > 0,\n(7.22)\nlim\nR→∞sup\nn≥R\nPx\n\u0002\nV is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n= 0.\nProof. For arbitrary positive constants q, h1 < h2, it holds\nPx\n\u0002\nV is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n≤Px\n\u0002\n|T| ≤qn2\f\fN +\nn ̸= ∅\n\u0003\n+ Px\n\u0002\n|T| > qn2, H(V ) < h1n\n\f\fN +\nn ̸= ∅\n\u0003\n+ Px\n\u0002\n|T| > qn2, H(V ) > h2n\n\f\fN +\nn ̸= ∅\n\u0003\n+ Px\n\u0002\n|T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n.\n(7.23)\nWe will show that for every δ > 0 there are q, h1, h2 such that for every n ≥0 the first\nthree summands on the right-hand side are smaller than δ/3, and that for every fixed\nq, h1, h2 the fourth one satisfies\n(7.24)\nlim\nR→∞sup\nn≥R\nPx\n\u0002\n|T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n= 0.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n37\nThis will imply the statement of the lemma.\nConcerning the first summand, the fact that it is possible to choose q > 0 small so that\nPx[|T| ≤qn2\f\fN +\nn ̸= ∅] ≤δ/3 for all n ≥0 follows immediately from Lemma 7.3. From\nnow on we keep q fixed in this way.\nFor the second summand, it holds\nPx\n\u0002\n|T| > qn2, H(V ) < h1n\n\f\fN +\nn ̸= ∅\n\u0003\n≤\n1\nPx[N +\nn ̸= ∅]Px\n\u0002\n|T| > qn2, H(V ) < h1n\n\u0003\n≤\n1\nPx[N +\nn ̸= ∅]\n1\nqn2Ex\nh h1n\nX\nk=0\n|N +\nk |\ni\n,\n(7.25)\nwhere the last inequality follows from the fact that V is a uniformly chosen vertex of T,\nand thus on {|T| ≥qn2},\n(7.26)\nPx[HV ≤h1n|σ(T)] = |T|−1\nh1n\nX\nk=0\n|N +\nk | ≤\n1\nqn2\nh1n\nX\nk=0\n|N +\nk |.\nBy Theorem 2.1, Px[N +\nn ̸= ∅]−1 ≤c(x)n, and, by Proposition 3.8, Ex[|N +\nk |] ≤c′(x) for all\nk ≥0. Therefore, the right-hand side of (7.25) can be made smaller than δ/3 by choosing\nh1 small.\nFor the third summand in (7.23), note that {H(V ) ≥h2n} ⊂{N +\nh2n ̸= ∅}, and thus\n(7.27)\nPx\n\u0002\n|T| > qn2, H(V ) > h2n\n\f\fN +\nn ̸= ∅\n\u0003\n≤\n1\nPx[N +\nn ̸= ∅]Px[N +\nh2n ̸= ∅],\nwhich can be made arbitrarily small by using Theorem 2.1 and choosing h2 large.\nFinally, we show (7.24). Recall the notation from (7.2). Since V is a uniformly chosen\nvertex of T, for arbitrary q, h1, h2, n, R > 0 and ε ∈(0, 1), on the event {|T| ≥qn2} it\nholds that\nPx\n\u0002\nH(V ) ∈[h1n, h2n], V is (η, R)-bad\n\f\fσ(T, φ)\n\u0003\n= 1\n|T|\nh2n\nX\nk=h1n\n|N +,(η,R)\nk\n|\n= 1\n|T|\nh2n\nX\nk=h1n\n|N +,(η,R)\nk\n|\n|N +\nk |\n|N +\nk | ≤ε +\n1\nqn2\nh2n\nX\nk=h1n\n1{|N+,(η,R)\nk\n|/|N+\nk |≥ε}|N +\nk |.\n(7.28)\nAs a consequence, the probability in (7.24) satisfies\nPx\n\u0002\n|T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n≤ε +\n1\nqn2\nh2n\nX\nk=h1n\nEx\n\u0002\n1{|N+,(η,R)\nk\n|/|N+\nk |≥ε}|N +\nk |\n\f\fN +\nn ̸= ∅\n\u0003\n.\n(7.29)\nBy the Cauchy–Schwarz inequality, every summand in (7.29) is bounded by\n(7.30)\nEx\n\u0002\n|N +\nk |2\f\fN +\nn ̸= ∅\n\u00031/2Px\n\u0002\n|N +,(η,R)\nk\n|/|N +\nk | > ε, N +\nk ̸= ∅\n\f\fN +\nn ̸= ∅\n\u00031/2.\nBy Theorem 2.1 and Proposition 3.8, Ex[|N +\nk |2 | N +\nn ̸= ∅]1/2 ≤c(x)n. The second term\nsatisfies,\n(7.31)\nPx\nh|N +,(η,R)\nk\n|\n|N +\nk |\n> ε, N +\nk ̸= ∅\n\f\f\fN +\nn ̸= ∅\ni\n≤Px\nh|N +,(η,R)\nk\n|\n|N +\nk |\n> ε\n\f\f\fN +\nk ̸= ∅\niPx[N +\nk ̸= ∅]\nPx[N +\nn ̸= ∅].\n\nSCALING LIMITS FOR THE CRITICAL GFF\n38\nBy Theorem 2.1, for k ≥h1n, the fraction on the right-hand side can be bounded by a\nconstant c(x). Noting also that |N +,(η,R)\nk\n| = 0 for R > k, we obtain that (7.29) is bounded\nby\n(7.32)\nε +\n1\nqn2c(x)n2(h2 −h1) sup\nk≥R\nPx\nh|N +,(η,R)\nk\n|\n|N +\nk |\n> ε\n\f\f\fN +\nk ̸= ∅\ni1/2\n.\nBy Proposition 7.1, the supremum here converges to 0 as R →∞. Since ε > 0 is arbitrary,\nthis shows (7.24) and completes the proof.\n□\nWe can now prove Proposition 7.2. The proof resembles the one of Proposition 6.21 in\n[Pow19], with modifications required to account for the unbounded domain in our setting.\nProof of Proposition 7.2. It is enough to show the statement for k = 2. The general case\nwhere k > 2 is then obtained by a union bound. In the case k = 2, since the matrices\nare symmetric and the diagonal entries are zero, it is sufficient to control one off-diagonal\nentry. To this end, for ε > 0 let Gn be the event\n(7.33)\nGn :=\n\b\n|C−1\n1 (DH\nn )1,2 −(D\n¯S\nn)1,2| > ε\n\t\n.\nIn order to bound the probability of Gn, we will define events An(δ) (see (7.38)) satisfying\nP[An(δ)|N +\nn ̸= ∅] ≤δ and show that for n large enough and the right choice of δn →0,\nGn is a subset of An(δn). This will imply the statement of the lemma.\nIn order to define the events An(δ) for δ > 0, we first fix M = M(δ) ≥1 so that\n(7.34)\nlim\nn→∞Px[N +\nMn ̸= ∅|N +\nn ̸= ∅] ≤δ/3,\nwhich is possible by Theorem 2.1. We then set\n(7.35)\nη = η(δ) := ε/(4M),\nand, using Lemma 7.4 and the independence of τ 1 and τ 2, we fix R = R(δ) so that\n(7.36)\nlim\nn→∞Px\n\u0002\nvτ 1 or vτ 2 is (η, R)-bad\n\f\fN +\nn ̸= ∅\n\u0003\n≤δ/3.\nNext, we fix K = K(δ) such that\n(7.37)\nlim\nn→∞Px[sup{χ(φu) : u ∈∪R\nk=0N +\nk } ≥K|N +\nn ̸= ∅] ≤δ/3.\nThis is possible since the event in (7.37) is FR-measurable, and thus, by Proposition 5.2,\nthe left-hand side of (7.37) equals Qx[sup{χ(φu) : u ∈∪R\nk=0N +\nk } ≥K] (with Qx as in\nSection 3.2). It is then straightforward to choose K large enough so that (7.37) is satisfied.\nFinally, we define the events An by\nAn = An(δ) = {N +\nMn ̸= ∅} ∪{vτ 1 or vτ 2 is (η, R)-bad}\n∪{sup{χ(φu) : u ∈∪R\nk=0N +\nk } ≥K}.\n(7.38)\nBy the choice of M, η, R, and K, it holds that limn→∞Px[An(δ)|N +\nn ̸= ∅] ≤δ. Therefore,\nfor sequences δn converging to zero sufficiently slowly, it also holds that\n(7.39)\nlim\nn→∞Px[An(δn)|N +\nn ̸= ∅] = 0.\nNext, we show that Gn ⊆An(δn) for a well-chosen δn and large n. Let B(δ) = {H(vτ 1 ∧\nvτ 2) > R}. We will show that for large enough n and a suitable choice of δn,\n(7.40)\nGn ∩Ac\nn(δn) ∩B(δn) = ∅\nand\nGn ∩Ac\nn(δn) ∩Bc(δn) = ∅.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n39\nFrom this, it easily follows that Gn ∩Ac\nn(δn) = ∅, and thus, Gn ⊆An(δn). Then, if δn →0\nsufficiently slowly, by (7.39),\n(7.41)\nlim\nn→∞Px[Gn|N +\nn ̸= ∅] ≤lim\nn→∞Px[An(δn)|N +\nn ̸= ∅] = 0,\nwhich is sufficient to prove the statement of the proposition.\nIt remains to show the existence of δn such that (7.39) and (7.40) hold. By definition,\non Ac\nn it holds\n(7.42)\nH(vτ i) ≤Mn\nand\n\f\f\f\n¯S(vτ i)\nH(vτ i) −C−1\n1\n\f\f\f ≤η\nfor i = 1, 2.\nThus, on Ac\nn ∩B it holds that R < H(vτ 1 ∧vτ 2) ≤Mn. Since vτ 1 is not (η, R)-bad,\n| ¯S(vτ 1 ∧vτ 2)/H(vτ 1 ∧vτ 2)−C−1\n1 | ≤η. Together with (7.42), the definition of the matrices\nin (7.13), and the definition of η, this implies\n(7.43)\n|(D\n¯S\nn)1,2 −C−1\n1 (DH\nn )1,2| ≤1\nn4Mnη ≤ε,\nand thus Gn ∩B ∩Ac\nn = ∅, proving the first half of (7.40). Next, on event Bc ∩Ac\nn\nit holds H(vτ 1 ∧vτ 2) ≤R and sup{χ(φu) : u ∈∪k≤RN +\nk } < K. Therefore, ¯S(vτ 1 ∧\nvτ 2) ≤KdR, and as a consequence | ¯S(vτ 1 ∧vτ 2) −C−1\n1 H(vτ 1 ∧vτ 2)| ≤KdR + RC−1\n1 .\nWe choose a sequence δn converging to 0 sufficiently slowly so that (7.39) is satisfied and\nK(δn)dR(δn)+1 + R(δn)C−1\n1\n≤εn, and thus\n(7.44)\n|(D\n¯S\nn)1,2 −C−1\n1 (DH\nn )1,2| ≤ε\n2 + K(δn)dR(δn) + R(δn)C−1\n1\nn\n≤ε\nfor n large. Hence, Gn∩Bc(δn)∩Ac\nn(δn) = ∅for such n, completing the proof of (7.40).\n□\n8. Proof of Theorem 2.3\nIn this section, we prove Theorem 2.3, relying on Lemmas 8.1 and 8.2. The argument\nfollows the framework of random metric measure spaces, with key results originating\nfrom [GPW09] and [ADH13]. The proof closely mirrors that of Theorem 1.1 in [Pow19,\nSection 6.3.2], which itself builds on the aforementioned results.\nRecall from Section 2 that e denotes a Brownian excursion conditioned to reach height\n1. Also define ˆe to be (σC1) times a Brownian excursion conditioned to reach height\n(C1σ)−1, where the constants σ and C1 are given in (6.5) and (2.11), and write (Tˆe, dˆe)\nfor the real tree with contour function ˆe. By Brownian scaling, (Tˆe, dˆe) is isometrically\nequivalent to (Te, de).\nTherefore, to show Theorem 2.3, it is enough to prove that if\n(Tx,n, dx,n) is a random metric space whose law coincides with that of (Ch∗\no\n∩T+, d/n)\nunder Px[·|N +\nn ̸= ∅] then\n(8.1)\n(Tx,n, dx,n) →(Tˆe, dˆe)\nas n →∞\nin distribution with respect to the Gromov-Hausdorff topology.\nIt will prove useful to additionally equip (Tx,n, dx,n) and (Tˆe, dˆe) with measures and\nwork in the space of metric measure spaces which we will call X. For this purpose, we\nequip (Tx,n, dx,n) with the measure µx,n which is the uniform measure among the vertices\nin Ch∗\no ∩T+. We equip (Tˆe, dˆe) with the measure µˆe which is the uniform measure on [0, ˆτ)\nand ˆτ is the length of ˆe. We write P n\nx for the law of (Tx,n, dx,n, µx,n) and Pˆe for the law of\n(Tˆe, dˆe, µˆe), and use En\nx and Eˆe for the corresponding expectations.\nLemma 8.1. For fixed x ≥h∗, the family (Tx,n, dx,n, µx,n)n≥0 is tight with respect to the\nGromov-Hausdorff-Prokhorov metric.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n40\nProof. Fix x ≥h∗. Our aim is to show that for every ε > 0 there exists a relatively\ncompact subset K ⊂X (with respect to the Gromov-Hausdorff-Prokhorov metric) such\nthat\n(8.2)\ninf\nn≥0 P n\nx [(Tx,n, dx,n, µx,n) ∈K] ≥1 −ε.\nTo do this, we define the relatively compact set (where the relative compactness follows\nfrom Theorem 2.6 in [ADH13])\n(8.3)\nKR,M =\n\u001a\n(X, r, µ) ∈X\n\f\f\f\f\nµ(X) = 1, diam(X) ≤2R, for all k ≥1, X can be\ncovered by fewer than 24kM balls of radius 2−k\n\u001b\nand show that for ε > 0 there are R(ε) and M(ε) such that for K = KR,M, (8.2) is\nsatisfied. Define the events\nAx,n(R) = {diam(Tx,n) ≤2R}\nand\n(8.4)\nBδ\nx,n(M) = {Tx,n can be covered with < δ−4M balls of radius δ}.\n(8.5)\nThen, since µx,n(Tx,n) = 1, {(Tx,n, dx,n, µx,n) ∈KR,M} = Ax,n(R) ∩(∩k≥1B2−k\nx,n (M)), and\nthus P n\nx [(Tx,n, dx,n, µx,n) ̸∈KR,M] ≤P n\nx [Ax,n(R)c]+P\nk≥1 P n\nx [B2−k\nx,n (M)c ∩Ax,n(R)]. To see\n(8.2), it is therefore enough to show that for M = M(ε) and R = R(ε) large enough\n(8.6)\nP n\nx [Ax,n(R)c] ≤ε/2\nand\nP n\nx [Bδ\nx,n(M)c ∩Ax,n(R)] ≤δε/2,\nfor all δ > 0 and n ≥0.\nWe will first find a R(ε) such that the first equation of (8.6) is satisfied. Recall that the\nlaw of (Tx,n, dx,n) under P n\nx is that of (T, d/n) under Px[ · |N +\nn ̸= ∅]. Thus, P n\nx [diam(Tx,n) ≥\n2R] ≤Px[|N +\nRn| > 0|N +\nn ̸= ∅], which by Theorem 2.1, is smaller than ε/2 for R(ε) large\nenough. For such R(ε) the first part of (8.6) is valid.\nNext, we will find a M(ε) such that also the second equation of (8.6) is met. We will\ndistinguish thereby between the cases where nδ < 1 and nδ ≥1. Assume first that nδ < 1.\nThen, every δ-ball around a vertex v ∈Tx,n only contains the vertex v itself. This means\nthat in this case Tx,n can be covered with < δ−4M balls of radius δ if and only if the\ncardinality of Tx,n is < δ−4M. Therefore\n(8.7)\nP n\nx [Bδ\nx,n(M)c ∩Ax,n(R)] ≤Px[|T| ≥δ−4M|N +\nn ̸= ∅] ≤Px[|T| ≥δ−4M]\nPx[N +\nn ̸= ∅]\n.\nUsing Theorem 2.1 and Proposition 3.3 this can be further bounded from above by\n(8.8)\nC δ2n\n√\nM\n≤C\nδ\n√\nM\n,\nfor some constant C, where we also used that by assumption δn < 1. Choosing some\nM = M(ε) ≥(2C/ε)2, the second equation of (8.6) is true for all n, δ with nδ < 1.\nNext we consider the case nδ ≥1. Set bn,δ = ⌊nδ⌋and define the sets of vertices\nVj = {v ∈N +\njbn,δ : ∃w ∈N +\n(j+1)bn,δ with v ≺w} for j ≥0. We will show that on\n(8.9)\n{diam(Tx,n) ≤2R} ∩\n\u00002Rn/bn,δ−1\n\\\nj=0\n\b\n|Vj| ≤Mbn,δ\n2Rnδ4\n\t\u0001\n(Tx,n, dx,n) can be covered with fewer than δ−4M balls of radius δ.\nSpecifically, Tx,n\nis covered by {Bδ(v)}v∈V , where V = ∪\n2Rn/bn,δ−1\nj=0\nVj.\nTo see this, note that since on\n{diam(Tx,n) ≤2R} the height of Tx,n is trivially bounded by 2Rn and thus, for every\n\nSCALING LIMITS FOR THE CRITICAL GFF\n41\nv ∈Tx,n there is a w ∈Vj for some 0 ≤j ≤2Rn/bn,δ −1 so that dx,n(v, w) ≤δ. Further,\nit is clear that on (8.9) the cardinality of V is bounded by (2Rn/bn,δ)(Mbn,δ/2Rnδ4) =\nδ−4M. This shows the claimed covering property. To prove the second part of (8.6), it\nthus suffices to show that for M = M(R, ε) large enough,\n(8.10)\nPx\nh 2Rn/bn,δ−1\n[\nj=0\n\b\n|Vj| > Mbn,δ\n2Rnδ4\n\t\f\f\fN +\nn ̸= ∅\ni\n≤δε/2.\nBy conditioning on Fjbn,δ (recalling the definition of Fn from (3.20)) and using Theo-\nrem 2.1 and Proposition 3.8,\nEx\n\u0002\n|Vj|\n\u0003\n= Ex\n\u0002\nEx\n\u0002\n|Vj|\n\f\fFjbn,δ\n\u0003\u0003\n= Ex\nh\nX\nv∈N+\njbn,δ\nPφv[N +\nbn,δ ̸= ∅]\ni\n(8.11)\n= Ex\nh\nX\nv∈N+\njbn,δ\nCχ(φv)b−1\nn,δ(1 + εbn,δ(φv))\ni\n≤q(x)b−1\nn,δ\n(8.12)\nfor some function q(x) independent of n and δ. Therefore, by conditioning on {N +\nn ̸= ∅}\nand using the Markov inequality and Theorem 2.1,\n(8.13)\nPx\nh\n|Vj| > Mbn,δ\n2Rnδ4\n\f\f\fN +\nn ̸= ∅\ni\n≤2Rnδ4\nMbn,δ\nEx[|Vj|]\nPx[N +\nn ̸= ∅] ≤q′(x)2Rn2δ4\nMb2\nn,δ\n,\nwhere q′ is some other function independent of n and δ. This means that by a union\nbound, the left-hand side of (8.10) is bounded from above by\n(8.14)\nq′(x)2Rn2δ4\nMb2\nn,δ\n(2Rn/bn,δ) = q′(x)4R2n3δ4\nMb3\nn,δ\n= q′(x)4R2δ\nM\n\u0010 nδ\n⌊nδ⌋\n\u00113\n.\nUsing that for x ≥1, 1 ≤x/⌊x⌋≤2, we see that by choosing M ≥64R2q′(x)/ε, the\nright-hand side of (8.14) is bounded by δε/2. For such choices of M, (8.10) is satisfied.\nFinally, first choosing R as described above and then M as the maximum of the two\nM-values obtained for the cases nδ < 1 and nδ ≥1, gives (R, M) such that (8.6) is\nsatisfied. For such (R, M), (8.2) holds true for K = KR,M and the proof is finished.\n□\nLemma 8.2. For fixed x ≥h∗,\n(8.15)\n(Tx,n, dx,n, µx,n) →(Tˆe, dˆe, µˆe)\nas n →∞\nin distribution with respect to the Gromov-Prokhorov metric.\nProof. By Corollary 3.1 in [GPW09], the convergence (8.15) holds true if and only if\n(a) The family {P n\nx }n≥0 is relatively compact in the space of probability measures on\nX (with respect to the Gromov-weak topology).\n(b) For every function Ψ : X →R of the form,\n(8.16)\nΨ((X, r, µ)) =\nZ\nψ((r(xi, xj))1≤i<j≤k)µ⊗k(d(x1, . . . , xk)),\nwhere ψ : [0, ∞)(k\n2) →R is a continuous and bounded function, En\nx[Ψ] →Eˆe[Ψ]\nas n →∞.\nSince convergence in the Gromov-Hausdorff-Prokhorov sense implies convergence in the\nGromov-weak sense, part (a) is implied by Lemma 8.1 and only part (b) is left to be\nshown.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n42\nTo see (b), fix a function Ψ as described in (8.16) and recall the definition of DH\nn from\n(7.13). Note that dx,n(v, w) = n−1(H(v) + H(w) −2H(v ∧w)), and thus\n(8.17)\nEn\nx[Ψ] = Ex\n\u0002\nψ(DH\nn )\n\f\fN +\nn ̸= ∅\n\u0003\n.\nBy Proposition 7.2,\n(8.18)\nlim\nn→∞\n\u0000Ex\n\u0002\nψ(DH\nn )\n\f\fN +\nn ̸= ∅\n\u0003\n−Ex\n\u0002\nψ(D\n¯S\nn/C−1\n1 )\n\f\fN +\nn ̸= ∅\n\u0003\u0001\n= 0.\nAssume for now that also\n(8.19)\nlim\nn→∞\n\u0000Ex\n\u0002\nψ(D\n¯S\nn/C−1\n1 )\n\f\fN +\nn ̸= ∅\n\u0003\n−Ex\n\u0002\nψ(D\n¯S\nn/C−1\n1 )\n\f\f sup\nm\n¯Sm ≥C−1\n1 n\n\u0003\u0001\n= 0.\nBy Proposition 6.10, limn→∞Ex[ψ(D ¯S\nn/C−1\n1 )| supm ¯Sm ≥C−1\n1 n] = Eˆe[Ψ], and thus in\ncombination with (8.17), (8.18) and (8.19), it follows limn→∞En\nx[Ψ] = Eˆe[Ψ], concluding\nthe proof.\nIt remains to prove (8.19). Since ψ is bounded, it is enough to show that\n(8.20)\nlim\nn→∞Px\n\u0002\nN +\nn ̸= ∅\n\f\f sup\nm\n¯Sm ≥C−1\n1 n\n\u0003\n= 1\nand\nlim\nn→∞Px\n\u0002\nsup\nm\n¯Sm ≥C−1\n1 n\n\f\fN +\nn ̸= ∅\n\u0003\n= 1.\nUsing that P[A|B] = P[B|A]P[A]/P[B], it is enough to show that\n(8.21)\nlim\nn→∞Px[sup\nm\n¯Sm ≥C−1\n1 n|N +\nn ̸= ∅] = 1\nand\nlim\nn→∞\nPx[supm ¯Sm ≥C−1\n1 n]\nPx[N +\nn ̸= ∅]\n= 1.\nObserve that for every δ > 0,\nPx\n\u0002\nsup\nm\n¯Sm ≥C−1\n1 n\n\f\fN +\nn ̸= ∅\n\u0003\n≥Px\n\u0002\nsup\nm\n¯Sm ≥C−1\n1 n\n\f\f|N +\n⌊n(1+δ)⌋| > 0\n\u0003Px[|N +\n⌊n(1+δ)⌋| > 0]\nPx[N +\nn ̸= ∅]\n.\n(8.22)\nBy Proposition 7.1, the first term in the product on the right-hand side converges to 1 as\nn →∞, and the second term converges to (1 + δ)−1 by Theorem 2.1. This shows the first\nconvergence of (8.21). By Theorem 2.1, Px[N +\nn ̸= ∅] ∼χ(x)C1n−1 as n →∞. Therefore,\nwe are left to prove that\n(8.23)\nlim\nn→∞\nPx[supm ¯Sm ≥C−1\n1 n]\nχ(x)C1n−1\n= 1\nto conclude the second limit of (8.21). We will do this in a similar way as, e.g., in [LG05,\nSection 1.4, p. 263].\nConsider the depth-first traversal (v1, v2, . . . ) of the sequence of trees (T 1, T 2, . . . ). We\nset S′\nn = P\nw∈Y (vn) χ(φw) and Si = supn S′\nn1vn∈T i for i ≥1 (so that supn ¯Sn = S1). Recall\nthat by Proposition 6.8,\n(8.24)\nlim\nn→∞\n\u0010S′\n⌊nt⌋\n√n , Λ⌊nt⌋\n√n\n\u0011\nt≥0 =\n\u0010\nσ|Bt|,\nσ\nχ(x)L0\nt\n\u0011\nt≥0,\nwhere the limit is in Px-distribution with respect to the Skorokhod topology. Defining\nγr = inf{t ≥0 : L0\nt > r} and τn = inf{t ≥0 : n−1Λ⌊n2t⌋> 1} and using Brownian scaling,\nthis gives the joint convergence\n(8.25)\nlim\nn→∞\n\u0010\u00101\nnS′\n⌊n2t⌋\n\u0011\nt≥0, τn\n\u0011\n=\n\u0010\u0000σ|Bt|\n\u0001\nt≥0, γχ(x)/σ\n\u0011\n,\n\nSCALING LIMITS FOR THE CRITICAL GFF\n43\nand therefore also\n(8.26)\nlim\nn→∞\n\u00101\nnS′\n⌊n2(t∧τn)⌋\n\u0011\nt≥0 =\n\u0000σ|Bt∧γχ(x)/σ|\n\u0001\nt≥0.\nFrom this we deduce that for every y > 0, as n →∞,\n(8.27)\nlim\nn→∞Px\nh\nsup\n1≤i≤n\nSi > ny\ni\n= Px\nh\nsup\nt≤γχ(x)/σ\nσ|Bt| > y\ni\n= 1 −exp\n\u0010\n−χ(x)\ny\n\u0011\n,\nwhere the last equality is a consequence of excursion theory for Brownian motion (see e.g.\nChapter XII in [RY99]). By the independence of the trees T i (and thus the variables Si),\nwe also have that\n(8.28)\nPx\nh\nsup\n1≤i≤n\nSi > ny\ni\n= 1 −\n\u00001 −Px\n\u0002\nS1 > ny\n\u0003\u0001n = 1 −\n\u0010\n1 −Px\nh\nsup\nm\n¯Sm > ny\ni\u0011n\n.\nCombining (8.27) and (8.28) then gives\n(8.29)\nPx\nh\nsup\nm\n¯Sm > ny\ni\n∼χ(x)\ny\nn−1\nas n →∞,\nwhich by setting y = C−1\n1\nconcludes (8.23) and finishes the proof.\n□\nWe now present the proof of Theorem 2.3, which is identical to that of Theorem 1.1 in\n[Pow19].\nProof of Theorem 2.3. Since convergence in the Gromov-Hausdorff-Prokhorov metric im-\nplies convergence in the Gromov-Prokhorov metric, Lemma 8.2 characterizes subsequen-\ntial limits with respect to the Gromov-Hausdorff-Prokhorov topology. Hence, we obtain\nthe convergence in distribution\n(8.30)\n(Tx,n, dx,n, µx,n) →(Tˆe, dˆe, µˆe)\nas n →∞\nwith respect to the Gromov-Hausdorff-Prokhorov metric. Now, since Gromov-Hausdorff-\nProkhorov convergence further implies convergence in the Gromov-Hausdorff sense, claim\n(8.1) follows, completing the proof.\n□\nAppendix A. Properties of χ\nWe prove here that χ is Lipschitz continuous on [h∗, ∞), as stated in (3.19). The proof\nuses similar ideas as the proof of Proposition 3.1 of [AˇC20].\nProof of (3.19). We will show that the derivative χ′ is bounded on [h∗, ∞). Recall that\nfor x ≥h∗it holds χ(x) = d\nR\n[h∗,∞) χ(z)ρY (z −x/d) dz, where ρY is the density of the\nGaussian random variable Y . Differentiating this expression, using an integration by parts\nand the fact that χ′(z) = 0 for z < h∗, we obtain that for x ≥h∗\nχ′(x) = −\nZ ∞\nh∗χ(z)ρ′\nY\n\u0010\nz −x\nd\n\u0011\ndz\n=\nZ\n[h∗,∞)\nχ′(z)ρY\n\u0010\nz −x\nd\n\u0011\ndz + χ(h∗)ρY\n\u0010\nh∗−x\nd\n\u0011\n=: EY\nh\nχ′\u0010\nY + x\nd\n\u0011i\n+ e(x).\n(A.1)\nDefining e(x) = 0 for x < h∗, using that χ is increasing and thus χ′ ≥0, this equality can\nbe extended to inequality\n(A.2)\nχ′(x) ≤EY\nh\nχ′\u0010\nY + x\nd\n\u0011i\n+ e(x)\nfor all x ∈R.\n\nSCALING LIMITS FOR THE CRITICAL GFF\n44\nSimilarly as in the proof of Proposition 3.1 in [AˇC20], we obtain an upper bound on χ′ by\niterating (A.2) an appropriate amount of times. For this, let Y1, . . . , Yk be independent\nGaussian random variables having the same distribution as Y , and define Zi = Y1/di−1 +\nY2/di−2 + · · · + Yi. Note that Zi is a centred Gaussian random variable whose variance,\ndenoted σ2\ni , is bounded uniformly in i. Then, by applying (A.2) k-times,\nχ′(x) ≤EY1\nh\nEY2\nh\nχ′\u0010\nY2 + Y1 + x/d\nd\n\u0011\n+ e(Y1 + x/d) + e(x)\nii\n≤· · · ≤E\nh\nχ′\u0010 x\ndk + Y1\ndk−1 + Y2\ndk−2 + · · · + Yk\n\u0011i\n+ E\nh k−1\nX\ni=0\ne\n\u0010 x\ndi + Zi\n\u0011i\n≤E\nh\nχ′\u0010 x\ndk + Zk\n\u0011i\n+\nk−1\nX\ni=0\nE\nh\ne\n\u0010 x\ndi + Zi\n\u0011i\n.\n(A.3)\nWe now choose k = k(x) = ⌊logd(x)⌋. Then the boundedness of the first summand can\nbe proved exactly in the same way as the boundedness of E[χ(x/dk + Zk)] in the proof of\nProposition 3.1 in [AˇC20] (note that (d −1) in [AˇC20] corresponds to d in our setting),\none only needs to verify that χ′ ∈L2(ν). This can be easily proved using (A.1) and the\nfacts ρ′\nY (x) = −cxρY (x), L[χ] = χ and (3.18) which imply |χ′(x)| ≤cL[χ2](x) + cx2 for\nx ≥h∗. From this χ′ ∈L2(ν) follows from Proposition 3.2(b).\nTo bound the second summand on the right hand side of (A.3), we first observe that by\nthe choice of k(x) it holds that x0 := x/dk(x) ∈[1, d]. With this notation, by rearranging\nthe sum,\n(A.4)\nk(x)−1\nX\ni=0\nE[e(x/di + Zi)] =\nk(x)−1\nX\ni=0\nE[e(x0di + Zk(x)−i)].\nMoreover, it is easy to see that for some constants c, c′\n(A.5)\n0 ≤e(x) := 1x≥h∗χ(h∗)ρY\n\u0010\nh∗−x\nd\n\u0011\n≤c′1x≥h∗e−cx.\nTherefore,\n(A.6)\nk(x)−1\nX\ni=0\nE[e(x0di + Zk(x)−i)] ≤c′\n∞\nX\ni=0\nE\n\u0002\ne−c(x0di+Zk(x)−i)\u0003\n≤c′\n∞\nX\ni=0\ne−cx0diec2σ2\nk(x)−i/2,\nwhich, since σ2\ni are bounded and x0 ∈[1, d], is clearly bounded uniformly in x ≥h∗.\n□\nAppendix B. Proof of the many-to-few formulas\nWe give here a proof of Proposition 3.5. Throughout the proof we use the notation\nfrom Section 3.2. We start by introducing a useful standard martingale related to φ when\nviewed as branching process.\nLemma B.1. For x ∈R, n ∈N, let\n(B.1)\nζ(x, n) := dnχ(x)\nand\nζn := ζ(ξ1\nn, n).\nThen, for every k ∈N, (ζn)n∈N is a P k\nx -martingale with respect to Fk\nn.\nProof. By definition of ζn, for n ≥1,\n(B.2)\nd−(n−1)Ek\nx[ζn | Fk\nn−1] = dEk\nx[χ(φσ1n) | Fk\nn−1],\n\nSCALING LIMITS FOR THE CRITICAL GFF\n45\nwhere σ1\nn is the vertex carrying the first spine mark at level n. Conditionally on Fk\nn−1, σ1\nn\nis uniformly distributed on desc(σ1\nn−1) and independent of φ. Therefore, this equals\n(B.3)\nX\nv∈desc(σ1\nn−1)\nEk\nx[χ(φv) | Fk\nn−1] = L[χ](φσ1\nn−1) = χ(φσ1\nn−1) = χ(ξ1\nn−1).\nwhere in for the first equality we used (3.7) and the fact that χ(x) = 0 on (−∞, h∗), and\nwhere the second equality follows from (3.10). The martingale property of ζn then follows\ndirectly from this computation.\n□\nDue to Lemma B.1, Lemma 8 of [HR17] can directly be applied to our process. We\nrestate it here for reader’s convenience, with very minor adaptations coming from the fact\nthat in our process every node has always d descendants (some of them might be in the\ncemetery state).\nLemma B.2 (Lemma 8 in [HR17]). For any k ≥1, let Y be a Fk\nn-measurable random\nvariable which can be written as\n(B.4)\nY =\nX\nv1,...,vk∈S+\nn\nY (v1, . . . , vk)1{σ1n=v1,...,σkn=vk},\nwhere, for every v1, . . . , vk ∈S+\nn , Y (v1, . . . , vk) is a Fn-measurable random variable. Then\n(B.5)\nEx\nh\nX\nv1,...,vk∈N+\nn\nY (v1, . . . , vk)\ni\n= Qk\nx\nh\nY\nY\nv∈skel(n)\\{o}\nζ(φp(v), |v| −1)\nζ(φv, |v|)\ndlp(v)\ni\n.\nWe are now ready to give the proof of Proposition 3.5.\nProof of Proposition 3.5. Statement (3.26) follows directly from Lemma B.2 with k = 1\nand Y (v1) = f(φv1). Indeed, since k = 1, there is only one mark on every level and thus\nY = f(ξ1\nn), skel(n) \\ {o} = {σ1\n1, . . . , σ1\nn}, and lp(v) = 1 for all v ∈skel(n) \\ {o}. Therefore,\nby (B.5)\nEx\nh X\nv∈N+\nn\nf(φv)\ni\n= Qx\n\u0014\nf(ξn)\nY\nv∈{σ1,...,σn}\n\u0010ζ(φp(v), |v| −1)\nζ(φv, |v|)\nd\n\u0011\u0015\n= Qx\n\u0014\nf(ξn)\nn\nY\ni=1\n\u0010χ(ξi−1)di−1\nχ(ξi)di\nd\n\u0011\u0015\n= Qx\n\u0014\nf(ξ1\nn) χ(x)\nχ(ξn)\n\u0015\n,\n(B.6)\nas claimed in (3.26).\nSimilarly, statement (3.27) is a consequence of Lemma B.2 with k = 2.\nWe set\nY (v1, v2) = f(φv1)g(φv2). Then Y = f(ξ1\nn)g(ξ2\nn) and by (B.5),\n(B.7)\nEx\nh X\nv,w∈N+\nn\nf(φv)g(φw)\ni\n= Q2\nx\n\u0014\nf(ξ1\nn)g(ξ2\nn)\nY\nv∈skel(n)\\{o}\nζ(φp(v), |v| −1)\nζ(φv, |v|)\ndlp(v)\n\u0015\n.\nTo consider the possible structures of skel(n) \\ {o}, set s = max{k : σ1\nk = σ2\nk} to be the\nlast time where the two spines agree. Note that due to the dynamics of marks under Q2\nx,\n(B.8)\nQ2\nx[s = k] =\n(\n(d −1)d−(k+1),\nif k ∈{0, . . . , n −1},\nd−n,\nif k = n,\n\nSCALING LIMITS FOR THE CRITICAL GFF\n46\nand, by a simple computation,\nY\nv∈skel(n)\\{o}\nζ(φp(v), |v| −1)\nζ(φv, |v|)\ndlp(v) =\nY\nv∈skel(n)\\{o}\n\u0012χ(φp(v))\nχ(φv)\n1\nd\n\u0013\ndlp(v) = χ(ξ1\ns)χ(x)\nχ(ξ1\nn)χ(ξ2\nn)ds.\n(B.9)\nTherefore, (B.7) can be written as\nEx\n\u0014 X\nv∈N+\nn\nf(φv)\nX\nv∈N+\nn\ng(φv)\n\u0015\n=\nn\nX\nk=0\ndkQ2\nx[s = k]Q2\nx\n\u0014\nf(ξ1\nn)g(ξ2\nn) χ(ξ1\nk)χ(x)\nχ(ξ1\nn)χ(ξ2\nn)\n\f\f\f\fs = k\n\u0015\n= d −1\nd\nn−1\nX\nk=0\nQ2\nx\n\u0014f(ξ1\nn)\nχ(ξ1\nn)\ng(ξ2\nn)\nχ(ξ2\nn)χ(ξ1\ns)χ(x)\n\f\f\f\fs = k\n\u0015\n+ Q2\nx\n\u0014\nf(ξ1\nn)g(ξ1\nn) χ(x)\nχ(ξ1\nn)\n\f\f\f\fs = n\n\u0015\n.\n(B.10)\nBy construction, under Q2\nx, conditional on s = k, for times i = 1, . . . , k the processes ξ1\ni\nand ξ2\ni are Markov chains which follow the same trajectory and have the same dynamics\nas ξi under Q1\nx. Further, for later times i = k + 1, . . . , n, they are independent Markov\nchains distributed according to Qξ1\nk. Therefore,\n(B.11) Q2\nx\nhf(ξ1\nn)\nχ(ξ1\nn)\ng(ξ2\nn)\nχ(ξ2\nn)χ(ξ1\ns)χ(x)\n\f\f\fs = k\ni\n= χ(x)Qx\n\u0014\nχ(ξk)Q2\nξ1\nk\nhf(ξ1\nn−k)\nχ(ξ1\nn−k)\ni\nQ2\nξ1\nk\nh g(ξ2\nn−k)\nχ(ξ2\nn−k)\ni\u0015\n,\nand similarly\n(B.12)\nQ2\nx\n\u0014\nf(ξ1\nn)g(ξ1\nn) χ(x)\nχ(ξ1\nn)\n\f\f\f\fs = n\n\u0015\n= χ(x)Qx\nhf(ξn)g(ξn)\nχ(ξn)\ni\n.\nCombining (B.10)–(B.12) directly implies (3.27).\n□\nReferences\n[AˇC20]\nAngelo Ab¨acherli and Jiˇr´ı ˇCern´y, Level-set percolation of the Gaussian free field on regular\ngraphs I: regular trees, Electron. J. Probab. 25 (2020), Paper No. 65, 24. MR4115734\n[ADH13]\nRomain Abraham, Jean-Fran¸cois Delmas, and Patrick Hoscheit, A note on the Gromov-\nHausdorff-Prokhorov distance between (locally) compact metric measure spaces, Electron. J.\nProbab. 18 (2013), no. 14, 21. MR3035742\n[Ald91]\nDavid Aldous, The continuum random tree. I, Ann. 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We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in [ˇCL25]. First, we derive a sharp asymptotic estimate for the probability that the connected component of the critical level set containing the root of the tree reaches generation n. In particular, we show that the one-arm exponent satisfies ρ = 1. Next, we establish a Yaglom-type limit theorem for the values of the GFF at generation n within this component. Finally, we show that, after a correct rescaling, this component conditioned on reaching generation n converges, as n →∞, to Aldous' continuum random tree. 1. Introduction The Gaussian free field's level-set percolation, especially on Zd, is a significant model in percolation theory that is characterised by its long-range dependencies. Initial investigations into this model trace back to the 1980s, with pioneering studies [BLM87, MS83, LS86]. Over the last decade, renewed interest has been ignited by the findings in [RS13], demonstrating that on Zd, the model undergoes a distinctive percolative phase transition at a critical threshold h∗= h∗(d) for any dimension d ≥3. Follow-up research, including papers [DRS14, PR15, DPR18, Szn19, CN20, GRS22, PS22], has provided a comprehensive understanding of the model's behaviour in both subcritical and supercritical phases, often making use of additional natural critical points in order to work in strongly sub- /super-critical regime. Notably, [DCGRS23] confirmed the alignment of these critical points with h∗, indicating a precise phase transition. In this paper, we continue the study of level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, building on the work initiated in [ˇCL25]. Our main contributions are threefold: First, we derive a sharp asymptotic estimate for the one-arm probability. Second, we establish a Yaglom-type limit theorem for the field at vertices located at distance n from the root. Finally, we show that the connected component of the critical level set, conditioned to be large, converges to Aldous' Continuum Random Tree. These results are stated precisely in Theorems 2.1, 2.2, and 2.3, respectively. The considered model was initially studied in [Szn16] where the critical value h∗was identified in terms of the largest eigenvalue of a specific integral operator. Additionally, a comparison with random interlacements was employed to establish bounds on h∗, notably demonstrating that 0 6 in [CD25], and for the intermediate regime 3 ≤d ≤6 in [CD24]. Our remaining main results consider the critical component conditioned on being large, more precisely conditioned on the rare event {N + n ̸= ∅}. The first such result is a Yaglomtype limit theorem for the GFF restricted to N + n . SCALING LIMITS FOR THE CRITICAL GFF 4 Theorem 2.2. For f ∈L2(ν), n ≥1, and x ≥h∗, let Zf,x n , resp. Zf n, be a random variable distributed as n-1 P v∈N+ n f(φv) under the conditional measure Px[ · |N + n ̸= ∅], resp. P[ · |N + n ̸= ∅]. Let further Z be an exponential random variable with mean one. Then, with C1 as in (2.11), (2.13) lim n→∞Zf,x n = lim n→∞Zf n = C-1 1 ⟨χ, f⟩Z in distribution. In the case of the critical single-type Galton-Watson process, results analogous to Theorem 2.2 trace back to the classical work of Yaglom [Yag47]. For more general branching processes with a finite type space, similar theorems are also well established; see, for instance, [Mod71, Theorem 10.1] and other foundational texts in the branching process literature. In recent years, Yaglom-type limits have been proven in various extended settings. These include critical non-local branching Markov processes [HHKW22], branching Brownian motion with absorption [MS22], and branching diffusions in bounded domains [Pow19]. Additionally, [GLLP22] establishes a Yaglom-type result for critical branching processes in a random Markovian environment with finite state space. Our third result concerns a scaling limit for the critical component Co ∩T+, viewed as a metric space, under the conditional law Px[ · |N + n ̸= ∅]. To this end, let (Tn,x, dn,x) be a random compact metric space whose law coincides with that of (Co ∩T+, n-1d) under Px[ · |N + n ̸= ∅] (recall that d denotes the distance on T). We show that the sequence (Tn,x, dn,x) converges in distribution to a conditioned Brownian continuum random tree (Te, de), whose contour function e is a Brownian excursion conditioned to reach height at least 1. Theorem 2.3. For every x ≥h∗, as n →∞, (2.14) (Tn,x, dn,x) →(Te, de) in distribution, with respect to the Gromov-Hausdorff topology. General scaling limits of critical Galton-Watson processes, in the spirit of Theorem 2.3, were first introduced in [Ald91]. A corresponding result for critical multi-type processes with finitely many types was established in [Mie08], and later extended to processes with a countably infinite type space in [dR17]. More recently, [CKKM24] extended the classical continuous random tree convergence to Galton-Watson trees evolving in a random environment, where each generation has a random offspring distribution with mean one and finite expected variance. In the context of critical branching diffusions, [Pow19] proves an invariance principle under the assumptions of a bounded domain, finite second moment of the offspring distribution, and an elliptic diffusion generator. However, for branching diffusions in general (unbounded) domains, analogous results are not yet available (see [Pow19, Question 1.8]). Remark 2.4. Theorems 2.2 and 2.3 hold without any further change if the conditioning therein is changed from Px[ · |N + n ̸= ∅] to Px[ · |Nn ̸= ∅]. For the sake of brevity, we refrain from providing detailed proofs of these results. We briefly discuss the organisation of this article. In Section 3, we collect relevant background on the GFF on regular trees along with the framework of branching processes with spines. Section 4 presents the proof of Theorem 2.1. In Section 5, we prove Theorem 2.2 and, in a dedicated subsection, establish additional results for the model conditioned on the event {N + n ̸= ∅}. Section 6 introduces an auxiliary martingale Sn and establishes scaling limit results for this martingale and some related processes. Section 7 focuses on SCALING LIMITS FOR THE CRITICAL GFF 5 the \"height process\" Hn, defined via the distance to the origin in a depth-first traversal of Co ∩T+, and investigates its connection to the martingale Sn. Finally, Section 8 concludes the article with the proof of Theorem 2.3, which combines topological arguments with the results from Sections 6 and 7. 3. Notation and useful results In this section we introduce the notation used throughout the paper and recall some known facts about the level set percolation of the Gaussian free field on trees. We then briefly present the formalism of branching processes with spines and apply it to our model. As already stated in the introduction, we use T to denote the (d+1)-regular tree, d ≥2, that is an infinite tree whose every vertex has exactly d + 1 neighbours. For two vertices v, w ∈T we use d(v, w) to denote the usual graph distance. The tree is rooted at an arbitrary fixed vertex o ∈T, ̄o ∈T denotes a fixed neighbour of o, and T+ stands for the forward tree, see (2.2). We set |v| = d(o, v) and write (3.1) Sn = {v ∈T : |v| = n}, S+ n = Sn ∩T+ for the spheres with radius n centred at o. For every v ∈T \\ {o} we use p(v) to denote its parent in T, that is the only vertex on the geodesic path from v to o with |p(v)| = |v| -1. We write desc(v) for the set of direct descendants of v, and sib(v) = desc(p(v)) for the set of its siblings, including itself. Finally, if w is an ancestor of v, that is w lies on the geodesics from o to v, we write w ⪯v. Throughout the paper we use the usual notation for the asymptotic relation of two functions f and g: We will write f(s) ∼g(s) as s →∞if lims→∞f(s)/g(s) = 1, f(s) = o(g(s)) as s →∞if lims→∞|f(s)|/g(s) = 0, and f(s) = O(g(s)) as s →∞if lim sups→∞|f(s)|/g(s) n] = Cχ(x)n-1/2(1 + o(1)). In particular, for every x ∈R, (3.16) Px[|Co ∩T+| = ∞] = 0. We will also need the following three properties of χ, the first one is proved in Remark 2.5 of [ˇCL25], the second is a direct consequence of Proposition 3.1 in [AˇC20], and the last one is proved in Appendix A: x 7→χ(x) is non-decreasing, (3.17) c1x ≤χ(x) ≤c2x for all x ≥h∗and some c1, c2 ∈(0, ∞), (3.18) x 7→χ(x) is Lipschitz on [h∗, ∞). (3.19) 3.2. Branching processes with spines. We now recall the machinery of branching processes with spines which is frequently used in the theory of branching processes, and specialize it to our model, in order to study the critical component Co. We then state many-to-few formulas that express certain moments for the original branching process in terms of the dynamics along the spines, see Proposition 3.5 below. Later, in Section 5.1, we will see that the processes with spines can be used to describe the distribution of Co conditioned on being infinite. The content of this section is mostly based on [HR17]. The main idea of the machinery is to designate several lines of descent in the branching process, called spines. These spines are then used to introduce a certain change of measure, under which the vertices on the spine exhibit modified branching behaviour, while the non-spine vertices behave as in the original process. For the construction, we need some notation. For x ∈R and k ∈N, we first introduce measures P k x under which the behaviour of the field is the same as under Px, but in SCALING LIMITS FOR THE CRITICAL GFF 8 addition there are k distinguished spines. Formally, the measure P k x is the distribution of a (R ∪{†}) × {0, 1, . . . , k}-valued stochastic process (φv, lv)v∈T+. This process assigns to every v ∈T+ a field value φv ∈R ∪{†} (where † is a cemetery state) and a number lv ∈{0, . . . , k} that represents the number of spine marks on v. Under P k x , the law of (φv)v∈T+ is a straightforward modification of the original measure Px, the only difference is that we set φv = † for every v ̸∈Co. The random variables (lv)v∈T+ are independent of the field values (φv)v∈T+. The root node o has exactly k marks, lo = k. The remaining random variables (lv)v̸=o are constructed recursively as follows: If a node v ∈T+ carries j marks, then each of its j marks 'moves' to one of its d direct descendants independently uniformly at random. (Note that nodes in the cemetery state can carry marks.) As consequence, under P k x , in every generation n, there are exactly k marks present, that is P v∈S+ n lv = k. We use P k x also for the corresponding expectations. For i = 1, . . . , k, we denote by σi n the node that carries the i-th mark in generation n (that is |σi n| = n), and set ξi n = φσin to be its type. We also define skel(n) = {v ∈ T+ : |v| ≤n, lv ≥1} to be the set of nodes of generation at most n having at least one mark. We let Fn stand for the natural filtration of the branching process, and Fk n for the filtration containing in addition the information about the k spine marks, (3.20) Fn = σ(φv : v ∈T+, |v| ≤n) and Fk n = σ(φv, lv : v ∈T+, |v| ≤n). Any f : R →R is extended to R ∪{†} by setting f(†) = 0. Then, by definition of P k x , (3.21) Ex X v∈N+ n f(φv) = P k x X v∈S+ n f(φv) . We now define another measure Qk x, where the nodes without a spine mark behave as under P k x but the nodes with a mark have a modified branching behaviour: Under Qk x the movement of the marks, and thus the distribution of (lv)v∈T+, is exactly the same as under P k x : If a node v carries k marks, each of the marks is given to one of its d direct descendants independently uniformly at random. To describe the distribution of the field φ under Qk x, we first define a transition kernel (recall ρY from (3.6)) (3.22) K(x, dy) = dχ(y) χ(x)ρY y -x d dy, x ≥h∗, y ∈R. Note that, by (3.6) and Proposition 3.1, for every x ≥h∗, K(x, ·) is a probability measure with support [h∗, ∞). Conditionally on the marks (lv)v∈T+, the field (φv)v∈T+ under Qk x is recursively constructed by (3.23) (a) φo := x, (b) If v ̸= o and lv = 0, then φv = d-1φp(v) + Yv (as under P k x ). (c) If v ̸= o and lv ≥1, then φv is K(φp(v), ·)-distributed, independently of previous randomness. To simplify notation, we write Qx instead of Q1 x; in this case we also set σn = σ1 n and ξn = ξ1 n. Note that, unlike under P k x , under the measure Qk x the nodes in the cemetery state cannot carry any mark. Consequently, Qk x-a.s., there are nodes not in the cemetery state in every generation. By construction, under Qk x, the process (ξi n)n∈N recording the value of the field along the i-th spine is a Markov chain with transition kernel K. This chain never enters the cemetery state. The following lemma determines its invariant distribution. SCALING LIMITS FOR THE CRITICAL GFF 9 Lemma 3.4. The Markov chain (ξn)n∈N with the transition kernel K has a unique invariant distribution π given by (3.24) π(dx) = χ(x)2ν(dx). Proof. We first show that π is invariant for K. Comparing (3.6) and (3.22) yields that K(x, A) = L[1Aχ](x)/χ(x) for every x ≥h∗. Therefore, writing the action of K on π as inner product on L2(ν), using that L is self-adjoint and χ is its eigenfunction with eigenvalue 1 (see Proposition 3.1), (3.25) (πK)(A) = ⟨K(·, A), χ2⟩ν = ⟨L[1Aχ], χ⟩ν = ⟨1Aχ, L[χ]⟩ν = ⟨1Aχ, χ⟩ν = π(A), which shows that π is an invariant measure. The uniqueness follows from the irreducibility of (ξn)n∈N. □ Next, we state several moment formulas that are frequently used throughout the paper. Such formulas are well understood in the theory of branching processes, see, e.g., [HR17] and references therein. The proof of the following proposition is based on Lemma 8 of that paper and can be found in Appendix B. Proposition 3.5. For all functions f, g : [h∗, ∞) →R for which the expectations below are well defined, Ex h X v∈N+ n f(φv) i = Qx h f(ξn) χ(x) χ(ξn) i , (3.26) Ex h X v,w∈N+ n f(φv)g(φw) i = χ(x)d -1 d n-1 X k=0 Qx χ(ξk)Qξk hf(ξn-k) χ(ξn-k) i Qξk h g(ξn-k) χ(ξn-k) i + χ(x)Qx hf(ξn)g(ξn) χ(ξn) i . (3.27) 3.3. Asymptotic behaviour of the moments. The main result of this section is Proposition 3.8 giving precise asymptotic estimates on the quantities appearing in (3.26), (3.27). To prove them, we could, in principle, use this proposition together with the known results on the convergence of Markov chains. However, it is easier, and for our purposes slightly more practical, to use formula (3.8) together with L2-estimates on the operator L. Since L is a self-adjoint Hilbert-Schmidt operator (see Proposition 3.1), L2(ν) has an orthonormal basis consisting of the eigenfunctions {ek}k≥1 of L corresponding to the eigenvalues {λk}k≥1. By Proposition 3.1 we may assume that 1 = λ1 > |λ2| ≥|λ3| ≥. . . , and e1 = χ. We set γ = |λ2| ∈(0, 1). By decomposing f ∈L2(ν) as (3.28) f = X k≥1 ⟨ek, f⟩ek = ⟨χ, f⟩χ + X k≥2 ⟨ek, f⟩ek =: ⟨χ, f⟩χ + β[f], for every n ∈N, it holds (3.29) Ln[f] = X k≥1 λn k⟨ek, f⟩ek = ⟨χ, f⟩χ + X k≥2 λn k⟨ek, f⟩ek = ⟨χ, f⟩χ + Ln[β[f]], with (3.30) ∥Ln[β[f]]∥≤γn∥β[f]∥≤γn∥f∥. The following simple lemma will later be used to deduce the pointwise convergence from the L2(ν)-one. SCALING LIMITS FOR THE CRITICAL GFF 10 Lemma 3.6. There is a function q : [h∗, ∞) →[0, ∞) such that for every x ≥h∗and f ∈L2(ν) (3.31) |L[f](x)| ≤∥f∥q(x). Proof. Using the definition (3.6) of L and the Cauchy-Schwarz inequality, |L[f](x)| ≤d Z R |f(y)|ρY y -x d dy = d Z R |f(y)|ρY (y -x/d) ρν(y) ν(dy) ≤d∥f∥L2(ν) ρY (· -x/d) ρν(·) L2(ν) =: ∥f∥L2(ν)q(x). (3.32) By (3.3), σν > σY . Using this one can easily check by a direct computation that q(x) 0 such that for all n ≥1 (4.14) ∥u0 n∥≥a0 n ≥cn-1. Proof. The statement will follow if we show (4.15) 0 ≤g(x)n-1 ≤u0 n(x) for all n ≥1 for some non-trivial function g : R →[0, 1]. To show (4.15) we use the estimate on the volume of Co from Proposition 3.3. By this proposition and (3.5) there is a positive constant c1 such that, for all m ≥1 and x ≥h∗, (4.16) Px[|Co ∩T+| > m] ≥c1m-1/2. Since |Co ∩T+| = P k≥0|N + k |, this implies for all m, n ≥1 c1m-1/2 ≤Px h X k≥0 |N + k | > m, N + n = ∅ i + Px h X k≥0 |N + k | > m, N + n ̸= ∅ i (4.17) ≤Px h n-1 X k=0 |N + k | > m i + Px[N + n ̸= ∅]. (4.18) By the Markov inequality, Px[Pn-1 k=0|N + k | > m] ≤ 1 m Pn-1 k=0 Ex[|N + k |], where, by Proposition 3.8, Ex[|N + k |] ≤c2(x) for some x-dependent constant c2(x) > 0. Thus, (4.19) c1m-1/2 ≤c2(x) n m + Px[N + n ̸= ∅]. Recalling (4.2) and choosing m = ⌊δ2n2⌋with δ = 2c1c2(x)-1, it follows that (4.20) u0 n(x) ≥ c2 1 2c2(x)n-1 for all n ≥1, showing (4.15) and thus the lemma. □ The next lemma gives upper bounds on af n and bf n. Lemma 4.3. There is c 0 for all x ∈[h∗, ∞). Hence, ⟨χ, L[uf n]2⟩≥ c⟨1, L[uf n]2⟩= c∥L[uf n]∥2 ≥c(af n)2. Applied to (4.22), this gives (4.23) af n+1 ≤af n -c(af n)2. SCALING LIMITS FOR THE CRITICAL GFF 14 This implies that af n is decreasing in n and, after rearranging, also (4.24) c ≤af n -af n+1 (af n)2 ≤af n -af n+1 af n+1af n = 1 af n+1 -1 af n . Summing this over n running from 0 to n -1 yields (4.25) cn ≤ n-1 X k=0 1 af k+1 -1 af k = 1 af n -1 af 0 ≤1 af n , proving the first part of (4.21). To prove the second part we project (4.9) onto the orthogonal complement of span {χ} and take norms. With the triangle inequality, this gives (4.26) bf n+1 ≤∥β[L[uf n]]∥+ ∥β[g(L[uf n])]∥. By (3.30), ∥β[L[uf n]]∥= ∥L[β[uf n]]∥≤γ∥β[uf n]∥= γbf n. By the upper bound of g from (4.10) and Proposition 3.2, since β is a projection, ∥β[g(L[uf n])]∥≤∥g(L[uf n])∥≤ c2∥L[uf n]2∥≤c∥uf n∥2. Applied to (4.26), this gives (4.27) bf n+1 ≤γbf n + c∥uf n∥2 = γbf n + c(af n)2 + c(bf n)2. Taking now f = 0, since b0 n →0 as n →∞, there is γ′ ∈(γ, 1) and n′ 0. SCALING LIMITS FOR THE CRITICAL GFF 15 To simplify the notation, we define (recall (4.1), (4.4)) (4.33) uλ n = ufλ n , aλ n = afλ n = ⟨χ, uλ n⟩, bλ n = bfλ n = ∥β[uλ n]∥. The first preliminary step in proving the uniformity over ˆF is the following lemma. In the special case λ = 0, this already follows from Lemmas 4.2 and 4.3. Lemma 4.4. There is a constant c 0. By (4.27) from the proof of Lemma 4.3, (4.35) bλ n+1 ≤γbλ n + c∥uλ n∥2, with c that is uniform over f ∈F. An iterative application of this inequality yields (4.36) bλ n+1 ≤γnbλ 0 + c n X l=0 γn-l∥uλ l ∥2. To continue, we argue that (4.37) ∥uλ l ∥≤λ-1 for every l ∈N0, λ > 0. Indeed, by Lemma 4.1, uλ n+1(x) ≤L[uλ n](x). Applying this recursively, taking norms, and using that the operator norm of L is one, we obtain (4.38) ∥uλ l ∥≤∥Ll[uλ 0]∥≤∥uλ 0∥. Moreover, since by definition 0 ≤uλ 0 = 1 -exp(-λ-1χ) ≤λ-1χ, we know that ∥uλ 0∥≤ λ-1∥χ∥= λ-1. Together with (4.38), this shows (4.37). Since also bλ 0 ≤∥uλ 0∥≤λ-1 and γ 0, x ≥h∗, and n ≥1, (5.1) Px h 1 n X v∈N+ n f(φv) > δ N + n ̸= ∅ i ≤cδ-2εf n(x). SCALING LIMITS FOR THE CRITICAL GFF 20 Proof. By the conditional Markov inequality (5.2) Px h 1 n X v∈N+ n f(φv) > δ N + n ̸= ∅ i ≤δ-2Ex h 1 n X v∈N+ n f(φv) 2 N + n ̸= ∅ i . The conditional expectation on the right-hand side satisfies (5.3) Ex h 1 n X v∈N+ n f(φv) 2 N + n ̸= ∅ i = 1 nPx[N + n ̸= ∅] 1 nEx h X v∈N+ n f(φv) 2i . By Proposition 3.8(a,b), since ⟨χ, f⟩= 0, (5.4) 1 nEx h X v∈N+ n f(φv) 2i = εf n(x), with εf n →0 in L2 and pointwise. Further, by the stochastic domination (3.5), using Theorem 2.1, (nPx[N + n ̸= ∅])-1 ≤(nPh∗[N + n ̸= ∅])-1 ≤c and the statement of the lemma follows. □ Proof of Theorem 2.2. We first consider the special case f = χ; the general case will then easily follow using Lemma 5.1. We start by showing that for every α > 0, (5.5) E e-αZχ,x n = C1 C1 + α + εα n(x), with εα n →0 pointwise and in L2(ν). Using the definitions (4.1), (4.32) of uf n and fλ, setting as before uλ n = ufλ n , and recalling (4.2), we obtain E e-αZχ,x n = Ex h exp -α n X v∈N+ n χ(φv) N + n ̸= ∅ i = 1 -un/α n (x) u0 n(x) . (5.6) By Lemma 4.6 and (4.68), (5.7) u0 n(x) = Cn-1χ(x)(1 + ε(x)), where ε →0 pointwise. Since Lemma 4.6 holds uniformly in λ, we can apply it with λ = n/α to obtain (5.8) un/α n (x) = an/α n χ(x)(1 + ̃ε(x)), again with ̃ε →0 pointwise. Moreover, by Lemma 4.7, again using the uniformity in λ, (5.9) (nan/α n )-1 = (nan/α 0 )-1 + C-1 1 + o(1) as n →∞. By the definitions of aλ n, uλ n and fλ, and by the monotone convergence theorem, nan/α 0 = n⟨χ, un/α 0 ⟩= n χ, E· 1 -fn/α(φo) = n⟨χ, 1 -fn/α⟩= χ, n 1 -e-αχ(·)/n n→∞ ---→α⟨χ, χ⟩= α. (5.10) Inserting this into (5.9) yields (5.11) an/α n = αC1(1 + o(1)) n(α + C1) as n →∞. Combining (5.6)-(5.8), (5.11) gives (5.12) 1 -E e-αZχ,x n = α α + C1 (1 + o(1)), SCALING LIMITS FOR THE CRITICAL GFF 21 which shows the pointwise convergence in (5.5). The L2(ν)-convergence follows by the dominated convergence theorem, since the left-hand side of (5.5) is bounded by 1. By L ́evy's continuity theorem for the Laplace transform, (5.5) implies the statement (2.13) of the theorem in the case Zf,x n with f = χ. For general f ∈L2(ν) we write f = ⟨χ, f⟩χ+β[f] as usual. Then Zf,x n = ⟨χ, f⟩Zχ,x n +Zβ[f],x n . The statement for Zf,x n then directly follows, since the first summand converges to an exponential random variable with mean C-1 1 ⟨χ, f⟩, by the first step of the proof, and the second summand converges to 0 in probability, by Lemma 5.1, since ⟨χ, β[f]⟩= 0. We now show the statement of the theorem for Zf n. Let νn be the law of φo under P[ · |N + n ̸= ∅]. By integrating (5.5) over νn(dx) (5.13) E e-Zχ n = C1 C1 + α + Z εα n(x)νn(dx). We need to show that the last integral is o(1). Observe that νn(dx) = P[φo ∈dx|N + n ̸= ∅] = P[φo ∈dx, N + n ̸= ∅]P[N + n ̸= ∅]-1 = Px[N + n ̸= ∅]P[N + n ̸= ∅]-1ν(dx). Therefore, using the Cauchy-Schwarz inequality and that εα n →0 in L2(ν), the integral in (5.13) is bounded above by (5.14) o(1) Z Px[N + n ̸= ∅]2 P[N + n ̸= ∅]2 ν(dx) 1/2 . By (4.67), Px[N + n ̸= ∅]P[N + n ̸= ∅]-1 = χ(x)⟨1, χ⟩-1 + εn(x), with εn(x) →0 in L2(ν) as n →∞. This implies that the integral in (5.14) is O(1) and together with (5.13) and again L ́evy's continuity theorem shows statement (2.13) for Zf n with f = χ. For Zf n and general f ∈L2(ν) the statement then follows by integrating (5.1) over νn(dx) and applying similar arguments as above. □ 5.1. Further results for the conditioned model. The following few results which all concern the model conditioned to {N + n ̸= ∅} will later be used in the proof of Theorem 2.3. The first one explains the role of the measure Qx introduced using the spine construction in Section 3.2. Similar results are well established in branching processes literature, see for instance [HH07, Theorem 5] or [CR90, Theorem 4]. Proposition 1.5 in [Pow19] can be seen as the analogous result for branching diffusion on bounded domains. Proposition 5.2. For all K ∈N, x ≥h∗and B ∈FK (see (3.20)) (5.15) lim n→∞Px[B|N + n ̸= ∅] = Qx[B]. Proof. We follow similar steps as in the proof of [Pow19, Proposition 1.5]. For every n ≥K, (5.16) Px[B|N + n ̸= ∅] = Ex h1BPx[N + n ̸= ∅|FK] Px[N + n ̸= ∅] i . Writing N + K = {v1, . . . , v|N+ K|} and defining the events Ai = {vi is root of a subtree in Co with height of at least n -K}, (5.17) {N + n ̸= ∅} = |N+ K| [ k=1 Ak = |N+ K| [ k=1 Ak ∩ \\ j K, 1BPx[N + n ̸= ∅|FK] Px[N + n ̸= ∅] ≤ cn n -K X v∈N+ K (χ(φv) + ̄εn-K(φv)) ≤c′ X v∈N+ K (χ(φv) + ̄εn-K(φv)) =: gn, (5.21) which converges a.s. to g := c′ P v∈N+ K χ(φv). Moreover, by Proposition 3.8, (5.22) Ex[|gn -g|] ≤Ex c′ X v∈N+ K | ̄εn-K(φv)| = ⟨χ, | ̄εn-K|⟩χ(x) + ε| ̄εn-K| K (x). Since ∥ ̄εn∥→0, the bounds established in (3.34) imply that the right-hand side of (5.22) converges to 0, that is gn →g in L1(P x). Therefore, by the generalised dominated convergence theorem, using (5.16), (5.19), (5.23) lim n→∞Px[B|N + n ̸= ∅] = Ex h 1B P v∈NK χ(φv) χ(x) i = Qx[B], where the last equality follows from Lemma B.2 with k = 1 and Y (v1) = 1Bχ(φv1). □ The following lemma and its corollary follow almost directly from the results established in the proof of Theorem 2.2. Lemma 5.3. For every f ∈L2(ν), x ≥h∗and δ > 0, (5.24) lim n→∞Px P v∈N+ n f(φv) P v∈N+ n χ(φv) -⟨χ, f⟩ > δ N + n ̸= ∅ = 0. Proof. Observe that P v∈N+ n f(φv) P v∈N+ n χ(φv) -⟨χ, f⟩= P v∈N+ n (f(φv) -⟨χ, f⟩χ(φv)) P v∈N+ n χ(φv) = n-1 P v∈N+ n β[f](φv) n-1 P v∈N+ n χ(φv) . (5.25) SCALING LIMITS FOR THE CRITICAL GFF 23 For any A, B ∈R and ε, δ > 0, {|A/B| > ε} ⊂{|A| > δ} ∪{|B| ̄δn N + n ̸= ∅ i + Px h n-1 X v∈N+ n χ(φv) 0. Then for n large enough so that ̄δn/δ ≤ε, (5.27) Px h n-1 X v∈N+ n χ(φv) 0 and f, g ∈L2(ν) with g > 0, (5.28) lim n→∞Px P v∈N+ n f(φv) P v∈N+ n g(φv) -⟨χ, f⟩ ⟨χ, g⟩ > δ N + n ̸= ∅ = 0. In particular, setting g ≡1, (5.29) lim n→∞Px 1 |N + n | X v∈N+ n f(φv) -⟨χ, f⟩ ⟨χ, 1⟩ > δ N + n ̸= ∅ = 0. Proof. It is easy to see that there is δ′ = δ′(δ, f, g) > 0 such that the event on the lefthand side of (5.28) is contained in the union of {|P f(φv)/ P χ(φv) -⟨χ, f⟩| ≥δ′} and {|P g(φv)/ P χ(φv) -⟨χ, g⟩| ≥δ′}. The statement then follows by Lemma 5.3. □ The final result of this section is the following technical lemma which can be seen as a somewhat stronger version of (5.29) in Corollary 5.4. It is tailored to be used in the proof of Proposition 6.7. Lemma 5.5. Write N + n = {v1, v2, . . . , v|N+ n |} and set N + n,M = {v1, v2, . . . , vM}. For δ, ρ > 0 and f : R →R bounded, define the event (5.30) Bn(f, δ, ρ) := {ρn ≤|N + n |} ∩ [ ρn≤M≤|N+ n | P v∈N+ n,M f(φv) M -⟨χ, f⟩ ⟨χ, 1⟩ > δ . Then for every x ≥h∗, (5.31) lim n→∞Px Bn(f, δ, ρ) N + n ̸= ∅ = 0. Proof. We start by outlining the strategy of the proof. We will define events An = An(n0) (see (5.34)) and A′ n = A′ n(n0, q) (see (5.40)), so that when n0 = n0(n) and q = q(n) are chosen correctly, (5.32) lim n→∞Px An(n0(n)) N + n ̸= ∅ = 0, lim n→∞Px A′ n(n0(n), q(n)) N + n ̸= ∅ = 0, Bn(f, δ, ρ) ⊆An(n0(n)) ∪A′ n(n0(n), q(n)) for n large enough. SCALING LIMITS FOR THE CRITICAL GFF 24 The statement of the lemma follows directly from these claims by a union bound. To define An = An(n0), we fix 0 ≤n0 ≤n, write N + n0 = {w1, . . . , w|N+ n0|}, and set N i n = {v ∈N + n : wi is ancestor of v}. For 1 ≤i ≤|N + n0|, we set (5.33) mi = ( |N i n|-1 P v∈Nin f(φv), if |N i n| > 0, ⟨χ, f⟩/⟨χ, 1⟩, otherwise, and define the events (5.34) Ai n = n mi -⟨χ, f⟩ ⟨χ, 1⟩ > δ 2 o and An = An(n0) = |N+ n0| [ i=1 Ai n. We now prove an upper bound on P[An|N + n ̸= ∅]. By a union bound, using that Ai n ⊂{N i n ̸= ∅} ⊂{N + n ̸= ∅}, Px[An|N + n ̸= ∅] = Px[N + n ̸= ∅]-1Px[An ∩{N + n ̸= ∅}] ≤Px[N + n ̸= ∅]-1Ex |N+ n0| X i=1 Px[Ai n ∩{N i n ̸= ∅}|Fn0] . (5.35) Using the branching process properties of the GFF, Px[Ai n ∩{N i n ̸= ∅}|Fn0] = Pφwi 1 |N + n-n0| X v∈N+ n-n0 f(φv) -⟨χ, f⟩ ⟨χ, 1⟩ > δ 2 N + n-n0 ̸= ∅ Pφwi[N + n-n0 ̸= ∅]. (5.36) By Corollary 5.4(b), the first probability on the right-hand side is bounded by some εn-n0(φwi) satisfying 1 ≥εn →0 pointwise, and thus in any Lq(ν) by the bounded convergence theorem. By Proposition 4.5 and Lemmas 4.6, 4.7, the second probability equals (5.37) Pφwi[N + n-n0 ̸= ∅] = C1(n -n0)-1 χ(φwi) + ̄εn-n0(φwi) , where ̄ε →0 pointwise and in L5/2(ν). Combining these statements with (5.35) and (5.36), and using then Proposition 3.8 in the numerator, we obtain Px[An|{N + n ̸= ∅}] ≤ Ex h P w∈N+ n0 εn-n0(φw)C1(n -n0)-1(χ(φw) + ̄εn-n0(φw)) i C1n-1(χ(x) + ̄εn(x)) = c(x)n n -n0 ⟨χ, εn-n0(χ + ̄εn-n0)⟩χ(x) + ̃εn0(x), (5.38) where ̃εn →0 pointwise. Finally, using H ̈older's inequality on the inner product, since χ, χ2 ∈L2(ν) (by (3.18)), εn →0 in L5(ν) and ̄ε →0 in L5/2(ν), it follows that for every x ≥h∗, (5.39) if n0 →∞and n -n0 →∞, then Px[An(n0)|{N + n ̸= ∅}] →0. We now turn to the events A′ n. For given n0 ≤n and q, let (5.40) A′ n = A′ n(n0, q) = |N+ n0| [ i=1 {|N i n| > q}. SCALING LIMITS FOR THE CRITICAL GFF 25 To bound the probability P[A′ n|N + n ̸= ∅], we write (5.41) Px[A′ n ∩{N + n ̸= ∅}] ≤Px[A′ n] ≤Ex h |N+ n0| X i=1 Pφwi[|N + n-n0| > q] i . By the Markov inequality and Proposition 3.8(b) (with f = g = 1), (5.42) Pφwi[|N + n-n0| > q] ≤q-2Eφwi[|N + n-n0|2] = Cq-2(χ(φwi)(n -n0) + εn-n0(φwi)), where ∥εn-n0∥≤c. Thus, by Proposition 3.8(a), Px[A′ n ∩{N + n ̸= ∅}] ≤q-2Ex h X v∈N+ n0 (cχ(φv)(n -n0) + εn-n0(φv)) i ≤cq-2 (n -n0 + 1)χ(x) + ε′ n0(x) . (5.43) with ε′ n →0 pointwise. Together with Px[N + n ̸= ∅] ≥cn-1, by Theorem 2.1, this implies (5.44) Px[A′ n|N + n ̸= ∅] ≤n q2 (n -n0 + C)χ(x) + ε′ n0(x) . We will now give sufficient conditions on the functions n0(n) and q(n) so that for large enough n, (5.45) Bn(f, δ, ρ) ⊂An ∪A′ n, or equivalently Bn(f, δ, ρ)c ⊃(An)c ∩(A′ n)c. By definition, {ρn ≤|N + n |}c ⊆Bn(f, δ, ρ)c. Therefore, (5.45) is implied by (5.46) (An)c ∩(A′ n)c ∩{ρn ≤|N + n |} ⊆Bn(f, δ, ρ)c. To prove this, we assume that (An)c ∩(A′ n)c ∩{ρn ≤|N + n |} holds. For M ≤|N + n |, let (5.47) k(M) := inf n l X i=1 |N i n| : l∈{0, . . . , |N + n0|} such that l X i=1 |N i n| ≥M o ≥M. On (An)c, |N i n|-1 P v∈Nin f(φv) -⟨χ, f⟩/⟨χ, 1⟩ ≤δ/2 for all i = 1, . . . , |N + n0|. Therefore, for all M with ρn ≤M ≤|N + n | (5.48) P v∈N+ n,k(M) f(φv) k(M) -⟨χ, f⟩ ⟨χ, 1⟩ ≤δ 2. Writing P v∈N+ n,M f(φv) = P v∈N+ n,k(M) f(φv) -P v∈N+ n,k(M) + n,M f(φv), and using that the absolute value of the second sum is bounded by (k(M) -M) sup|f|, we obtain that for every M ≥ρn P v∈N+ n,M f(φv) M - P v∈N+ n,k(M) f(φv) k(M) = P v∈N+ n,k(M) f(φv) M - P v∈N+ n,k(M) + n,M f(φv) M - P v∈N+ n,k(M) f(φv) k(M) ≤ X v∈N+ n,k(M) f(φv) 1 M - 1 k(M) + k(M) -M M sup|f| = k(M) -M M P v∈N+ n,k(M) f(φv) k(M) + sup|f| ≤c(f, δ) q ρn, (5.49) SCALING LIMITS FOR THE CRITICAL GFF 26 where in the last inequality we used (5.48) and the fact that k(M) -M ≤q on (A′ n)c. If the right-hand side of (5.49) is smaller than δ/2, then together with (5.48), this implies that Bn(f, δ, ρ)c holds, implying (5.46) and thus (5.45). To finish the proof of (5.32), we must choose n0 = n0(n) and q = q(n) so that (5.39) applies and the left-hand sides of (5.43), (5.49) tend to zero. This is easily done by setting, e.g., q(n) = n3/4 and n0 = n -n1/4. □ 6. The Sn martingale The remaining three sections of this paper are dedicated to proving Theorem 2.3. In this section, we will introduce a martingale based on the depth-first traversal of a sequence of copies of Co ∩T+, and prove that it satisfies an invariance principle, see Proposition 6.1. This martingale can be seen as an analogue to the Lukasiewicz path used to study critical Galton-Watson trees. The second part of the section then demonstrates another two scaling limit results, Proposition 6.8 and Proposition 6.10 which are both consequences of Proposition 6.1. To define the martingale, we consider an i.i.d. sequence T = ((T 1, φ1), (T 2, φ2), . . . ) where every (T i, φi) is distributed as (Co ∩T+, φ|Co∩T+) under Px. To keep the notation simple, we keep using Px for the probability measure associated with the whole sequence T, and for v ∈T i we write φv instead of φi v. We further use oi to denote the root of T i, and set N +,i n = {v ∈T i : d(oi, v) = n}. Note that |N +,i n | has the same distribution as |N + n | which was studied in detail in the previous sections. In particular, we know that all T i are a.s. finite. Throughout this section, we use the notation for the parent p(v), direct descendants desc(v), and siblings sib(v) of v ∈T i relative to the rooted tree T i (not T), w ⪯v means that w is an ancestor of v, cf. below (3.1). We now describe the depth-first traversal v = (v1, v2, . . . ) of T. It starts at the root of T 1, that is v1 = o1, and then explores the tree T 1 in a depth-first manner. After visiting all vertices of T 1, it proceeds to o2, explores T 2 in a depth-first manner, and so forth. vi denotes the i-th vertex visited during this traversal. The notation v 0} are independent. Hence, using the first half of (6.14), this is bounded by c(x)u-1n-1/2 P i≥1 Px[i ≤Λn]. Since P i≥1 Px[i ≤Λn] = Ex[Λn] ≤c(x)√n, by the second half of (6.14), the claim follows. □ Lemma 6.6. For every δ > 0 and x ≥h∗there exists a sequence R(n) such that limn→∞R(n) = ∞and (6.16) sup n≥1 Px[H∗ n ≤R(n)] ≤δ. Proof. We will show that for every K 0, by definition of H∗ n, (6.18) Px[H∗ n ≤K] = n-1 n X l=1 Px[Hl ≤K] = n-1Ex h K X h=0 n X l=1 1Hl=h i . Since Pn l=1 1Hl=h ≤PΛn i=1|N +,i h | and thus PK h=0 Pn l=1 1Hl=h ≤PΛn i=1 PK h=0|N +,i h |. Moreover, (PK h=0|N +,i h |)Λn i=1 are i.i.d. and with finite mean. Noting that Λn is a stopping time SCALING LIMITS FOR THE CRITICAL GFF 29 with respect to Gi = σ(T j : j ≤i) and using Wald's equation (see e.g. [Dur19, Theorem 2.6.2]), (6.19) Px[H∗ n ≤K] ≤1 nEx h Λn X i=1 K X h=0 |N +,i h | i = 1 nEx[Λn]Ex h K X h=0 |N +,1 h | i , which by (6.14) converges to 0 as n →∞. □ The following law of large numbers will be important for the proof of Proposition 6.1. Proposition 6.7. Let f be a bounded function and mf n := n-1 Pn i=1 f(φvi). Then (6.20) lim n→∞mf n = mf ∞:= ⟨χ, f⟩ ⟨χ, 1⟩, in Px-probability. Proof. Let (6.21) N +,i k,n = N +,i k ∩{v1, . . . , vn} be the part of N +,i k traversed in the first n steps, and let N ∗ n = N +,Λ∗ n H∗n,n . We set (6.22) m∗ n = |N ∗ n|-1 X v∈N∗n f(φv). Denoting by FT the σ-algebra generated by the sequence T = (Ti, φi)i≥1, it holds that (6.23) mf n = Ex[m∗ n|FT] (effectively, the expectation here is over Un only). To show the proposition it will thus be sufficient to show that (6.24) lim n→∞m∗ n = mf ∞ in Px-probability. To see that this is indeed enough, note that since f is bounded m∗ n is dominated by a constant. Hence, by the dominated convergence theorem, m∗ n →mf ∞also in L2(Px). By (6.23), since the conditional expectation is a contraction on L2(Px), mf n →mf ∞in L2(Px), and thus in Px-probability as claimed. To prove (6.24), we fix ε, δ > 0 and set (6.25) An = {|m∗ n -mf ∞| > ε}. We then fix C ε}. Using the definitions of Un and m∗ n, and then decomposing by possible values of Hk, Λk, we obtain Px[An ∩Bn] = n X k=1 Px[An ∩Bn|Un = k]Px[Un = k] = n X k=1 1 nPx[AΛk Hk,n, R(n) ≤Hk ≤Cn1/2] = 1 n Cn1/2 X l=R(n) n X i=1 Ex h 1Ai l,n n X k=1 1{Hk=l,Λk=i} i . = 1 n Cn1/2 X l=R(n) n X i=1 Ex 1Ai l,n N +,i l,n , (6.27) where in the last step we used Pn k=1 1{Hk=l,Λk=i} = |N +,i l,n |. To analyse the expectation on the right-hand side, we define σ-algebras Gi = σ(T 1, T 2, . . . , T i) and set τi = |T 1| + · · · + |T i-1| (which is Gi-1-measurable). Then, using that (T i, φi)i≥1 is an i.i.d. sequence, Ex 1Ai l,n|N +,i l,n | Gi-1 = 1{τi≤n}Ex 1A1 l,n-τi N +,1 l,n-τi = 1{τi≤n}Ex 1Al,n-τi|N + l,n-τi| N + l ̸= ∅ Px[N + l ̸= ∅], (6.28) where on the last line we omitted the superscript '1' since N +,1 l,k has the same distribution as N + l,k. By Theorem 2.1, Proposition 3.8 and (6.14) there is a c = c(x) such that (6.29) Px N + l ̸= ∅ ≤cl-1, Ex N + l 2 N + l ̸= ∅ 1/2 ≤cl, Ex Λn ≤cn1/2. Further, for Bn(f, ε, ρ) as in Lemma 5.5, it holds that Al,k ∩{|N + l,k| ≥lρ} ⊆Bl(f, ε, ρ). Therefore, choosing ρ = δ/(6Cc2) and decomposing on whether |N + l,n-τi| is larger than lδ/(6Cc2), we obtain Ex 1A1 l,n-τi|N +,1 l,n-τi| N +,1 l ̸= ∅ ≤ lδ 6Cc2 + Ex 1Bn(f,ε,ρ)|N + l,n-τi| N + l ̸= ∅ ≤ lδ 6Cc2 + Ex |N + l |2 N + l ̸= ∅ 1/2Px Bl(f, ε, ρ) N + l ̸= ∅ 1/2, (6.30) where in the last step we used the Cauchy-Schwarz inequality and |N + l,n-τi| ≤|N + l |. Combining (6.28) and (6.30) with the first two claims in (6.29) we obtain (6.31) Ex 1Ai l,n|N +,i l,n | ≤ δ 6Cc + c2Px Bl(f, ε, ρ) N + l ̸= ∅ 1/2 Px[τi ≤n]. Inserting this into (6.26), (6.27), we obtain that Px[An] is bounded by (6.32) 2δ 3 + 1 n δ 6Cc + c2 sup l≥R(n) Px Bl(f, ε, ρ) N + l ̸= ∅ 1/2 ⌊C√n⌋ X l=R(n) n X i=1 Px[Λn ≥i], Since R(n) diverges, Lemma 5.5 implies that the supremum in this formula tends to zero as n →∞. By the last property in (6.29), P⌊C√n⌋ l=R(n) Pn i=1 Px[Λn ≥i] ≤C√nEx[Λn] ≤Ccn. SCALING LIMITS FOR THE CRITICAL GFF 31 Therefore (6.33) Px[An] ≤2δ 3 + δ 6 + Cc3o(1) ≤δ for n large enough. Since ε and δ are arbitrary, this shows (6.24) and with the initial comment completes the proof. □ We are now ready to prove Proposition 6.1. Proof of Proposition 6.1. We apply a martingale functional central limit theorem, see e.g. [Dur19, Theorem 8.2.8]. To check its assumptions we need to show that (a) limn→∞n-1 Pn k=1 Ex[(Sk -Sk-1)2|Hk-1] = ⟨χ,V⟩ ⟨χ,1⟩in Px-probability, (b) limn→∞n-1 Pn k=1 Ex[(Sk -Sk-1)21{|Sk-Sk-1|>ε√n}] = 0. Condition (a) follows from Lemma 6.3 and Proposition 6.7, using that V is bounded by Lemma 6.4. We now show (b). By the Cauchy-Schwarz inequality, Ex[(Sk-Sk-1)21{|Sk-Sk-1|>ε√n}] ≤ Ex[(Sk -Sk-1)4]1/2Px[(Sk -Sk-1)2 > ε2n]1/2. By Lemmas 6.3, 6.4, Ex[(Sk -Sk-1)2] ≤c uniformly in k and thus Px[(Sk -Sk-1)2 > ε2n] ≤cε-2n-1. Thus, to prove (b) it is enough to show (6.34) Ex[(Sk -Sk-1)4] ≤c uniformly in k. By (6.8), Sk -Sk-1 = P v∈desc(vk-1) χ(φv) -χ(φvk-1), and thus Ex[(Sk -Sk-1)4|Hk-1] = Eφvk-1 h X v∈N+ 1 χ(φv) -χ(φvk-1) 4i = Eφvk-1 h X v∈N+ 1 χ(φv) -Eφvk-1 X v∈N+ 1 χ(φv) 4i , (6.35) where we used that χ(φvk-1) = Eφvk-1[P v∈N+ 1 χ(φv)] in the last line. We write N + 1 = {w1, . . . , wd} and follow similar arguments as in the proof of Lemma 6.4, using that χ(x) = 0 for x 0 and ε > 0, (6.38) lim n→∞Px h max i≤nt φvi/√n > ε i = 0. Let H(T i) = maxv∈T i H(v) denote the height of tree T i. For every j, k ∈N, the probability in (6.38) is bounded from above by Px h max i≤nt φvi > ε√n, Λ⌊nt⌋≤j, H(Ti) ≤k for i = 1, . . . , j i + Px[∃i ∈{1, . . . , j} : H(T i) > k, Λ⌊nt⌋≤j] + Px[Λ⌊nt⌋> j]. (6.39) To show (6.38), we thus need to choose jn →∞and kn →∞so that all three summands converge to zero. We start with the first summand. Restricted to {Λ⌊nt⌋≤j} and {H(T i) ≤k for i = 1, . . . , j}, maxk≤nt φvk is dominated by the maximum of all φv with v such that Λ(v) ≤j and H(v) ≤k. Considering not only the maximum over the connected components of the level set, but over the first k generations in j copies of the whole forward tree T+, this is dominated by the maximum of jdk non-negatively correlated Gaussian random variables with mean at most x and bounded variance. By Gaussian comparison techniques, the mean of this maximum is of order x + c p log jdk ≤c√k log j for j, k large enough. Thus, by the Markov inequality, (6.40) Px h max i≤nt φvi > ε√n, Λ⌊nt⌋≤j, H(Ti) ≤k for i = 1, . . . , j i ≤ C εn1/2 p k log(j). By a union bound, using Theorem 2.1, the second summand in (6.39) satisfies (6.41) Px[∃i ∈{1, . . . , j} : H(T i) > k, Λ⌊nt⌋≤j] ≤c(x)j/k. Finally, by the Markov inequality and (6.14), the third summand can be bounded by (6.42) Px[Λ⌊nt⌋> j] ≤c(x) √ tn/j. Setting now, e.g., jn = n2/3 and kn = n5/6, all three summands in (6.39) converge to 0 as required. □ Proof of Proposition 6.8. We use the continuous mapping theorem together with the already established convergence statements. By Proposition 6.1, n-1/2S⌊n·⌋→σB·, and by Lemma 6.9, n-1/2χ(φv⌊n·⌋) →0 as n →∞, in Px-distribution, in the Skorokhod topology. Therefore (6.43) n-1/2(S⌊n·⌋-χ(φv⌊n·⌋)) = n-1/2 X w∈Y (v⌊n·⌋) χ(φw) - X i≤Λ⌊n·⌋ χ(φoi) also converges to σB· as n →∞. Next, note that (6.44) inf k≤n{Sk -χ(φvk)} = inf k≤n n X w∈Y (vk) χ(φw) - X i≤Λk χ(φoi) o = - X i≤Λn χ(φoi). Therefore, setting Bt = infs≤t Bs and using the continuous mapping theorem with the map g(Xt) = (Xt -infs≤t Xs, -infs≤t Xs), (6.45) n-1/2 X w∈Y (v⌊n·⌋) χ(φw), X i≤Λ⌊n·⌋ χ(φoi) → σ(B· -B·), -σB· SCALING LIMITS FOR THE CRITICAL GFF 33 in distribution as n →∞. By L ́evy's theorem (see, e.g., Theorem VI.2.3 in [RY99]), the right-hand side of (6.45) has the same distribution as (σ|B·|, σL0 · (B)). Together with the fact that under Px, P i≤Λ⌊nt⌋χ(φoi) = χ(x)Λ⌊nt⌋, this implies the proposition. □ Another consequence of Proposition 6.1 is the following scaling limit result for Sn conditioned to reach a certain height on the first tree T1. To this end we define (6.46) ̄Sn = (P w∈Y (vn) χ(φw) if n ≤|T 1|, 0 if n > |T 1|. Note that, up to an additive correction χ(φvn) -χ(x), ̄S· is equal to S· restricted to T1. Proposition 6.10. For y > 0, let (e≥y/σ)t≥0 be a Brownian excursion conditioned to reach at least height y/σ. Then, in distribution under Px[ · | supk ̄Sk ≥√ny], with respect to the Skorokhod topology, (6.47) lim n→∞ n-1/2 ̄S⌊nt⌋ t≥0 = (σe≥y/σ)t≥0. We omit the proof of this proposition as it would be identical to the proof of Proposition 6.13 in [Pow19], which itself follows [DLG02, Proposition 2.5.2]. 7. Relation of Sn and Hn The aim of this section will be to establish a connection between the martingale Sn and the height process Hn (see (6.3) and (6.1) for definitions). This will be useful in the proof of Theorem 2.3 in Section 8. Throughout the whole section, we only consider the field on the first tree (T, φ) = (T 1, φ1) in the infinite tree sequence T = ((T 1, φ1), (T 2, φ2), . . . ) introduced in Section 6. As a consequence, we only consider ̄S introduced in (6.46) instead of S. We will see that, approximately, ̄S(v) ≈H(v)/C1 with C1 as in (2.11). Motivated by this, for η > 0 we say that v ∈T is η-bad if (7.1) ̄S(v) H(v) -C-1 1 > η. Fixing in addition R > 0, we say that v ∈T is (η, R)-bad if there exists a w ≺v such that H(w) ≥R and w is η-bad. This means that v is (η, R)-good (i.e. not (η, R)-bad) if all its ancestors in generations at least R are η-good. We set (7.2) N (η,R) n = {v ∈N + n : v is (η, R)-bad}. The first main result of this section shows that this set is relatively small. Proposition 7.1. For every ε > 0 and x ≥h∗, (7.3) lim R→∞sup n≥R Px h|N (η,R) n | |N + n | > ε N + n ̸= ∅ i = 0. Proof. The proof follows a strategy similar to the proof of Proposition 6.17 in [Pow19], with adaptations that are necessary in order to handle the unbounded domain in our setting. For ε > 0, R > 0 and n ≥0 we define the event (7.4) Eε R,n = nP v∈N+ n χ(φv)1{v is (η, R)-bad} P v∈N+ n χ(φv) > ε o . We first claim that it suffices to show that (7.5) lim R→∞sup n≥R Px[Eε R,n|N + n ̸= ∅] = 0. SCALING LIMITS FOR THE CRITICAL GFF 34 To see that this indeed implies the lemma, we write (7.6) |N (η,R) n | |N + n | = |N (η,R) n | P v∈N+ n χ(φv) · P v∈N+ n χ(φv) |N + n | . Therefore, for every δ > 0, the probability in (7.3) is bounded from above by (7.7) Px h |N (η,R) n | P v∈N+ n χ(φv) > δ N + n ̸= ∅ i + Px h |N + n | P v∈N+ n χ(φv) 0 for every x ≥h∗, it holds that |N (η,R) n | ≤ c P v∈N+ n χ(φv)1{v is (η, R)-bad}, and thus (7.8) Px h |N +,(η,R) n | P v∈N+ n χ(φv) > δ N + n ̸= ∅ i ≤Px Eδ/c R,n N + n ̸= ∅ , which together with (7.5) implies the claim of the lemma. We now prove (7.5). By first using the Markov inequality, and then Lemma B.2 with k = 1 and Y (v1) = χ(φv1)1{v1 is (η, R)-bad}1{N+ n ̸=∅}/(P w∈N+ n χ(φw)), it holds Px[Eε R,n|N + n ̸= ∅] ≤ε-1Ex hP v∈N+ n χ(φv)1{v is (η, R)-bad} P v∈N+ n χ(φv) N + n ̸= ∅ i = ε-1Ex h1{N+ n ̸=∅} P v∈N+ n χ(φv)1{v is (η, R)-bad} P v∈N+ n χ(φv) i Px[N + n ̸= ∅]-1 = ε-1Qx hχ(x)Px[N + n ̸= ∅]-1 P v∈N+ n χ(φv) 1{σn is (η, R)-bad} i , (7.9) where, as defined in Section 3, σn is the vertex on the spine in the n-th generation. To continue, we first show the following two claims: (i) supn≥R Qx[σn is (η, R)-bad] →0 as R →∞. (ii) Set Zn = χ(x)Px[N+ n ̸=∅]-1 P v∈N+ n χ(φv) . Then for all δ > 0, there exist R′, K > 0 such that Qx[Zn1{Zn>K}] ≤δ for all n ≥R′. Claim (i) will follow from the ergodic behaviour of the field along the spine (σn)n≥0 under Qx. Let sib K}] = Px[1{Zn>K}1{N+ n ̸=∅}] P[N + n ̸= ∅] = Px[Zn > K|N + n ̸= ∅]. ≤Px hP v∈N+ n χ(φv) n K}1{σn is (η, R)-bad}] + Q[Zn1{Zn≤K}1{σn is (η, R)-bad}] ≤Qx[Zn1{Zn>K}] + KQ[σn is (η, R)-bad], (7.12) which together with (i) and (ii) implies (7.5) and concludes the proof. □ To state the second main result of this section, we introduce (enlarging the probability space if necessary) random variables τ 1, . . . , τ k which under Px, conditionally on T, are independent and uniformly distributed on {1, 2, . . . , |T|}. Using these random variables, we define two k × k random matrices (D ̄S n)i,j = n-1 ̄S(vτ i) + ̄S(vτ j) -2 ̄S(vτ i ∧vτ j) (DH n )i,j = n-1 H(vτ i) + H(vτ j) -2H(vτ i ∧vτ j) , (7.13) where, as usual, (v1, v2, . . . ) denotes the depth-first traversal of T, and v ∧w denotes the most recent common ancestor of v and w in T. Proposition 7.2. For every k ≥1 and ε > 0, (7.14) lim n→∞Px ∥C-1 1 DH n -D ̄S n∥> ε N + n ̸= ∅ = 0, where ∥·∥denotes the Frobenius norm of k × k matrices. To prove this proposition, we need two lemmas. The first one estimates from below the size of T conditionally on {N + n ̸= ∅}. Lemma 7.3. For every x ≥h∗, (7.15) lim q→0 sup n≥0 Px |T| ≤qn2|N + n ̸= ∅ = 0. Proof. It will be sufficient to show that for every δ > 0 there exist q and n0 such that for every n ≥n0, (7.16) Px[|T| ≤qn2|N + n ̸= ∅] ≤δ. Indeed, since there are only finitely many n 0 Px[|T| ≤qn2|N + n ̸= ∅] ≤Px |T| ≤qn2, |N + n | ≥ηn|N + n ̸= ∅ + Px |N + n | 0 small such that the second probability on the right-hand side of (7.17) is bounded by δ/2 for all n large enough. It is thus sufficient to show that for every δ > 0 and η > 0 there is n0 so that for all n ≥n0 (7.18) Px |T| ≤qn2, |N + n | ≥ηn N + n ̸= ∅ ≤δ/2. Note first that (7.19) Px |T| ≤qn2, |N + n | ≥ηn N + n ̸= ∅ ≤Px |T| ≤qn2 |N + n | ≥ηn . Given {|N + n | ≥ηn}, let w1, . . . , w⌊ηn⌋be the first ⌊ηn⌋vertices in N + n , and let Tw be the subtree of T rooted at w. Obviously, |T| ≤P⌊ηn⌋ i=1 |Twi|. Under P[ · ||N + n | ≥ηn], the random variables |Twi| are independent. Moreover, since w ∈N + n implies φw ≥h∗, by stochastic domination (3.5), for any u > 0, (7.20) Px |Twi| ≥u |N + n | ≥ηn ≥Ph∗ |Co ∩T+| ≥u . Denoting T ′ i, i ≥1, i.i.d. random variables distributed as |Co ∩T+| under Ph∗, it follows that (7.21) Px |T| ≤qn2 |N + n | ≥ηn ≤P h ⌊ηn⌋ X i=1 T ′ i ≤qn2i . By Theorem 3.3, T ′ i are in the domain of attraction of the 1/2-stable random distribution. Therefore, n-2 P i≤⌊ηn⌋T ′ i converges in distribution to a non-negative 1/2-stable random variable (see e.g. [Fel71, Theorem XIII.6.2]). As a consequence, since the distribution of this random variable has no atom at 0, for any δ, η > 0 there exists q small so that for all n large enough the right-hand side of (7.21) is bounded by δ/2. This shows (7.18) and completes the proof. □ The second lemma needed to show Proposition 7.2 studies the probability that a randomly chosen vertex vτ1 is (η, R)-bad. Lemma 7.4. Let V := vτ1 be a uniformly chosen vertex of T. Then, for every x ≥h∗ and η > 0, (7.22) lim R→∞sup n≥R Px V is (η, R)-bad N + n ̸= ∅ = 0. Proof. For arbitrary positive constants q, h1 qn2, H(V ) qn2, H(V ) > h2n N + n ̸= ∅ + Px |T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad N + n ̸= ∅ . (7.23) We will show that for every δ > 0 there are q, h1, h2 such that for every n ≥0 the first three summands on the right-hand side are smaller than δ/3, and that for every fixed q, h1, h2 the fourth one satisfies (7.24) lim R→∞sup n≥R Px |T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad N + n ̸= ∅ = 0. SCALING LIMITS FOR THE CRITICAL GFF 37 This will imply the statement of the lemma. Concerning the first summand, the fact that it is possible to choose q > 0 small so that Px[|T| ≤qn2 N + n ̸= ∅] ≤δ/3 for all n ≥0 follows immediately from Lemma 7.3. From now on we keep q fixed in this way. For the second summand, it holds Px |T| > qn2, H(V ) qn2, H(V ) qn2, H(V ) > h2n N + n ̸= ∅ ≤ 1 Px[N + n ̸= ∅]Px[N + h2n ̸= ∅], which can be made arbitrarily small by using Theorem 2.1 and choosing h2 large. Finally, we show (7.24). Recall the notation from (7.2). Since V is a uniformly chosen vertex of T, for arbitrary q, h1, h2, n, R > 0 and ε ∈(0, 1), on the event {|T| ≥qn2} it holds that Px H(V ) ∈[h1n, h2n], V is (η, R)-bad σ(T, φ) = 1 |T| h2n X k=h1n |N +,(η,R) k | = 1 |T| h2n X k=h1n |N +,(η,R) k | |N + k | |N + k | ≤ε + 1 qn2 h2n X k=h1n 1{|N+,(η,R) k |/|N+ k |≥ε}|N + k |. (7.28) As a consequence, the probability in (7.24) satisfies Px |T| > qn2, H(V ) ∈[h1n, h2n], V is (η, R)-bad N + n ̸= ∅ ≤ε + 1 qn2 h2n X k=h1n Ex 1{|N+,(η,R) k |/|N+ k |≥ε}|N + k | N + n ̸= ∅ . (7.29) By the Cauchy-Schwarz inequality, every summand in (7.29) is bounded by (7.30) Ex |N + k |2 N + n ̸= ∅ 1/2Px |N +,(η,R) k |/|N + k | > ε, N + k ̸= ∅ N + n ̸= ∅ 1/2. By Theorem 2.1 and Proposition 3.8, Ex[|N + k |2 | N + n ̸= ∅]1/2 ≤c(x)n. The second term satisfies, (7.31) Px h|N +,(η,R) k | |N + k | > ε, N + k ̸= ∅ N + n ̸= ∅ i ≤Px h|N +,(η,R) k | |N + k | > ε N + k ̸= ∅ iPx[N + k ̸= ∅] Px[N + n ̸= ∅]. SCALING LIMITS FOR THE CRITICAL GFF 38 By Theorem 2.1, for k ≥h1n, the fraction on the right-hand side can be bounded by a constant c(x). Noting also that |N +,(η,R) k | = 0 for R > k, we obtain that (7.29) is bounded by (7.32) ε + 1 qn2c(x)n2(h2 -h1) sup k≥R Px h|N +,(η,R) k | |N + k | > ε N + k ̸= ∅ i1/2 . By Proposition 7.1, the supremum here converges to 0 as R →∞. Since ε > 0 is arbitrary, this shows (7.24) and completes the proof. □ We can now prove Proposition 7.2. The proof resembles the one of Proposition 6.21 in [Pow19], with modifications required to account for the unbounded domain in our setting. Proof of Proposition 7.2. It is enough to show the statement for k = 2. The general case where k > 2 is then obtained by a union bound. In the case k = 2, since the matrices are symmetric and the diagonal entries are zero, it is sufficient to control one off-diagonal entry. To this end, for ε > 0 let Gn be the event (7.33) Gn := |C-1 1 (DH n )1,2 -(D ̄S n)1,2| > ε . In order to bound the probability of Gn, we will define events An(δ) (see (7.38)) satisfying P[An(δ)|N + n ̸= ∅] ≤δ and show that for n large enough and the right choice of δn →0, Gn is a subset of An(δn). This will imply the statement of the lemma. In order to define the events An(δ) for δ > 0, we first fix M = M(δ) ≥1 so that (7.34) lim n→∞Px[N + Mn ̸= ∅|N + n ̸= ∅] ≤δ/3, which is possible by Theorem 2.1. We then set (7.35) η = η(δ) := ε/(4M), and, using Lemma 7.4 and the independence of τ 1 and τ 2, we fix R = R(δ) so that (7.36) lim n→∞Px vτ 1 or vτ 2 is (η, R)-bad N + n ̸= ∅ ≤δ/3. Next, we fix K = K(δ) such that (7.37) lim n→∞Px[sup{χ(φu) : u ∈∪R k=0N + k } ≥K|N + n ̸= ∅] ≤δ/3. This is possible since the event in (7.37) is FR-measurable, and thus, by Proposition 5.2, the left-hand side of (7.37) equals Qx[sup{χ(φu) : u ∈∪R k=0N + k } ≥K] (with Qx as in Section 3.2). It is then straightforward to choose K large enough so that (7.37) is satisfied. Finally, we define the events An by An = An(δ) = {N + Mn ̸= ∅} ∪{vτ 1 or vτ 2 is (η, R)-bad} ∪{sup{χ(φu) : u ∈∪R k=0N + k } ≥K}. (7.38) By the choice of M, η, R, and K, it holds that limn→∞Px[An(δ)|N + n ̸= ∅] ≤δ. Therefore, for sequences δn converging to zero sufficiently slowly, it also holds that (7.39) lim n→∞Px[An(δn)|N + n ̸= ∅] = 0. Next, we show that Gn ⊆An(δn) for a well-chosen δn and large n. Let B(δ) = {H(vτ 1 ∧ vτ 2) > R}. We will show that for large enough n and a suitable choice of δn, (7.40) Gn ∩Ac n(δn) ∩B(δn) = ∅ and Gn ∩Ac n(δn) ∩Bc(δn) = ∅. SCALING LIMITS FOR THE CRITICAL GFF 39 From this, it easily follows that Gn ∩Ac n(δn) = ∅, and thus, Gn ⊆An(δn). Then, if δn →0 sufficiently slowly, by (7.39), (7.41) lim n→∞Px[Gn|N + n ̸= ∅] ≤lim n→∞Px[An(δn)|N + n ̸= ∅] = 0, which is sufficient to prove the statement of the proposition. It remains to show the existence of δn such that (7.39) and (7.40) hold. By definition, on Ac n it holds (7.42) H(vτ i) ≤Mn and ̄S(vτ i) H(vτ i) -C-1 1 ≤η for i = 1, 2. Thus, on Ac n ∩B it holds that R 0 there exists a relatively compact subset K ⊂X (with respect to the Gromov-Hausdorff-Prokhorov metric) such that (8.2) inf n≥0 P n x [(Tx,n, dx,n, μx,n) ∈K] ≥1 -ε. To do this, we define the relatively compact set (where the relative compactness follows from Theorem 2.6 in [ADH13]) (8.3) KR,M = (X, r, μ) ∈X μ(X) = 1, diam(X) ≤2R, for all k ≥1, X can be covered by fewer than 24kM balls of radius 2-k and show that for ε > 0 there are R(ε) and M(ε) such that for K = KR,M, (8.2) is satisfied. Define the events Ax,n(R) = {diam(Tx,n) ≤2R} and (8.4) Bδ x,n(M) = {Tx,n can be covered with 0 and n ≥0. We will first find a R(ε) such that the first equation of (8.6) is satisfied. Recall that the law of (Tx,n, dx,n) under P n x is that of (T, d/n) under Px[ · |N + n ̸= ∅]. Thus, P n x [diam(Tx,n) ≥ 2R] ≤Px[|N + Rn| > 0|N + n ̸= ∅], which by Theorem 2.1, is smaller than ε/2 for R(ε) large enough. For such R(ε) the first part of (8.6) is valid. Next, we will find a M(ε) such that also the second equation of (8.6) is met. We will distinguish thereby between the cases where nδ Mbn,δ 2Rnδ4 N + n ̸= ∅ i ≤δε/2. By conditioning on Fjbn,δ (recalling the definition of Fn from (3.20)) and using Theorem 2.1 and Proposition 3.8, Ex |Vj| = Ex Ex |Vj| Fjbn,δ = Ex h X v∈N+ jbn,δ Pφv[N + bn,δ ̸= ∅] i (8.11) = Ex h X v∈N+ jbn,δ Cχ(φv)b-1 n,δ(1 + εbn,δ(φv)) i ≤q(x)b-1 n,δ (8.12) for some function q(x) independent of n and δ. Therefore, by conditioning on {N + n ̸= ∅} and using the Markov inequality and Theorem 2.1, (8.13) Px h |Vj| > Mbn,δ 2Rnδ4 N + n ̸= ∅ i ≤2Rnδ4 Mbn,δ Ex[|Vj|] Px[N + n ̸= ∅] ≤q′(x)2Rn2δ4 Mb2 n,δ , where q′ is some other function independent of n and δ. This means that by a union bound, the left-hand side of (8.10) is bounded from above by (8.14) q′(x)2Rn2δ4 Mb2 n,δ (2Rn/bn,δ) = q′(x)4R2n3δ4 Mb3 n,δ = q′(x)4R2δ M nδ ⌊nδ⌋ 3 . Using that for x ≥1, 1 ≤x/⌊x⌋≤2, we see that by choosing M ≥64R2q′(x)/ε, the right-hand side of (8.14) is bounded by δε/2. For such choices of M, (8.10) is satisfied. Finally, first choosing R as described above and then M as the maximum of the two M-values obtained for the cases nδ 0, Px sup m ̄Sm ≥C-1 1 n N + n ̸= ∅ ≥Px sup m ̄Sm ≥C-1 1 n |N + ⌊n(1+δ)⌋| > 0 Px[|N + ⌊n(1+δ)⌋| > 0] Px[N + n ̸= ∅] . (8.22) By Proposition 7.1, the first term in the product on the right-hand side converges to 1 as n →∞, and the second term converges to (1 + δ)-1 by Theorem 2.1. This shows the first convergence of (8.21). By Theorem 2.1, Px[N + n ̸= ∅] ∼χ(x)C1n-1 as n →∞. Therefore, we are left to prove that (8.23) lim n→∞ Px[supm ̄Sm ≥C-1 1 n] χ(x)C1n-1 = 1 to conclude the second limit of (8.21). We will do this in a similar way as, e.g., in [LG05, Section 1.4, p. 263]. Consider the depth-first traversal (v1, v2, . . . ) of the sequence of trees (T 1, T 2, . . . ). We set S′ n = P w∈Y (vn) χ(φw) and Si = supn S′ n1vn∈T i for i ≥1 (so that supn ̄Sn = S1). Recall that by Proposition 6.8, (8.24) lim n→∞ S′ ⌊nt⌋ √n , Λ⌊nt⌋ √n t≥0 = σ|Bt|, σ χ(x)L0 t t≥0, where the limit is in Px-distribution with respect to the Skorokhod topology. Defining γr = inf{t ≥0 : L0 t > r} and τn = inf{t ≥0 : n-1Λ⌊n2t⌋> 1} and using Brownian scaling, this gives the joint convergence (8.25) lim n→∞ 1 nS′ ⌊n2t⌋ t≥0, τn = σ|Bt| t≥0, γχ(x)/σ , SCALING LIMITS FOR THE CRITICAL GFF 43 and therefore also (8.26) lim n→∞ 1 nS′ ⌊n2(t∧τn)⌋ t≥0 = σ|Bt∧γχ(x)/σ| t≥0. From this we deduce that for every y > 0, as n →∞, (8.27) lim n→∞Px h sup 1≤i≤n Si > ny i = Px h sup t≤γχ(x)/σ σ|Bt| > y i = 1 -exp -χ(x) y , where the last equality is a consequence of excursion theory for Brownian motion (see e.g. Chapter XII in [RY99]). By the independence of the trees T i (and thus the variables Si), we also have that (8.28) Px h sup 1≤i≤n Si > ny i = 1 - 1 -Px S1 > ny n = 1 - 1 -Px h sup m ̄Sm > ny i n . Combining (8.27) and (8.28) then gives (8.29) Px h sup m ̄Sm > ny i ∼χ(x) y n-1 as n →∞, which by setting y = C-1 1 concludes (8.23) and finishes the proof. □ We now present the proof of Theorem 2.3, which is identical to that of Theorem 1.1 in [Pow19]. Proof of Theorem 2.3. Since convergence in the Gromov-Hausdorff-Prokhorov metric implies convergence in the Gromov-Prokhorov metric, Lemma 8.2 characterizes subsequential limits with respect to the Gromov-Hausdorff-Prokhorov topology. Hence, we obtain the convergence in distribution (8.30) (Tx,n, dx,n, μx,n) →(Tˆe, dˆe, μˆe) as n →∞ with respect to the Gromov-Hausdorff-Prokhorov metric. Now, since Gromov-HausdorffProkhorov convergence further implies convergence in the Gromov-Hausdorff sense, claim (8.1) follows, completing the proof. □ Appendix A. Properties of χ We prove here that χ is Lipschitz continuous on [h∗, ∞), as stated in (3.19). The proof uses similar ideas as the proof of Proposition 3.1 of [AˇC20]. Proof of (3.19). We will show that the derivative χ′ is bounded on [h∗, ∞). Recall that for x ≥h∗it holds χ(x) = d R [h∗,∞) χ(z)ρY (z -x/d) dz, where ρY is the density of the Gaussian random variable Y . Differentiating this expression, using an integration by parts and the fact that χ′(z) = 0 for z < h∗, we obtain that for x ≥h∗ χ′(x) = - Z ∞ h∗χ(z)ρ′ Y z -x d dz = Z [h∗,∞) χ′(z)ρY z -x d dz + χ(h∗)ρY h∗-x d =: EY h χ′ Y + x d i + e(x). (A.1) Defining e(x) = 0 for x < h∗, using that χ is increasing and thus χ′ ≥0, this equality can be extended to inequality (A.2) χ′(x) ≤EY h χ′ Y + x d i + e(x) for all x ∈R. SCALING LIMITS FOR THE CRITICAL GFF 44 Similarly as in the proof of Proposition 3.1 in [AˇC20], we obtain an upper bound on χ′ by iterating (A.2) an appropriate amount of times. For this, let Y1, . . . , Yk be independent Gaussian random variables having the same distribution as Y , and define Zi = Y1/di-1 + Y2/di-2 + · · · + Yi. Note that Zi is a centred Gaussian random variable whose variance, denoted σ2 i , is bounded uniformly in i. Then, by applying (A.2) k-times, χ′(x) ≤EY1 h EY2 h χ′ Y2 + Y1 + x/d d + e(Y1 + x/d) + e(x) ii ≤· · · ≤E h χ′ x dk + Y1 dk-1 + Y2 dk-2 + · · · + Yk i + E h k-1 X i=0 e x di + Zi i ≤E h χ′ x dk + Zk i + k-1 X i=0 E h e x di + Zi i . (A.3) We now choose k = k(x) = ⌊logd(x)⌋. Then the boundedness of the first summand can be proved exactly in the same way as the boundedness of E[χ(x/dk + Zk)] in the proof of Proposition 3.1 in [AˇC20] (note that (d -1) in [AˇC20] corresponds to d in our setting), one only needs to verify that χ′ ∈L2(ν). This can be easily proved using (A.1) and the facts ρ′ Y (x) = -cxρY (x), L[χ] = χ and (3.18) which imply |χ′(x)| ≤cL[χ2](x) + cx2 for x ≥h∗. From this χ′ ∈L2(ν) follows from Proposition 3.2(b). To bound the second summand on the right hand side of (A.3), we first observe that by the choice of k(x) it holds that x0 := x/dk(x) ∈[1, d]. With this notation, by rearranging the sum, (A.4) k(x)-1 X i=0 E[e(x/di + Zi)] = k(x)-1 X i=0 E[e(x0di + Zk(x)-i)]. Moreover, it is easy to see that for some constants c, c′ (A.5) 0 ≤e(x) := 1x≥h∗χ(h∗)ρY h∗-x d ≤c′1x≥h∗e-cx. Therefore, (A.6) k(x)-1 X i=0 E[e(x0di + Zk(x)-i)] ≤c′ ∞ X i=0 E e-c(x0di+Zk(x)-i) ≤c′ ∞ X i=0 e-cx0diec2σ2 k(x)-i/2, which, since σ2 i are bounded and x0 ∈[1, d], is clearly bounded uniformly in x ≥h∗. □ Appendix B. Proof of the many-to-few formulas We give here a proof of Proposition 3.5. Throughout the proof we use the notation from Section 3.2. We start by introducing a useful standard martingale related to φ when viewed as branching process. Lemma B.1. For x ∈R, n ∈N, let (B.1) ζ(x, n) := dnχ(x) and ζn := ζ(ξ1 n, n). Then, for every k ∈N, (ζn)n∈N is a P k x -martingale with respect to Fk n. Proof. By definition of ζn, for n ≥1, (B.2) d-(n-1)Ek x[ζn | Fk n-1] = dEk x[χ(φσ1n) | Fk n-1], SCALING LIMITS FOR THE CRITICAL GFF 45 where σ1 n is the vertex carrying the first spine mark at level n. Conditionally on Fk n-1, σ1 n is uniformly distributed on desc(σ1 n-1) and independent of φ. Therefore, this equals (B.3) X v∈desc(σ1 n-1) Ek x[χ(φv) | Fk n-1] = L[χ](φσ1 n-1) = χ(φσ1 n-1) = χ(ξ1 n-1). where in for the first equality we used (3.7) and the fact that χ(x) = 0 on (-∞, h∗), and where the second equality follows from (3.10). The martingale property of ζn then follows directly from this computation. □ Due to Lemma B.1, Lemma 8 of [HR17] can directly be applied to our process. We restate it here for reader's convenience, with very minor adaptations coming from the fact that in our process every node has always d descendants (some of them might be in the cemetery state). Lemma B.2 (Lemma 8 in [HR17]). For any k ≥1, let Y be a Fk n-measurable random variable which can be written as (B.4) Y = X v1,...,vk∈S+ n Y (v1, . . . , vk)1{σ1n=v1,...,σkn=vk}, where, for every v1, . . . , vk ∈S+ n , Y (v1, . . . , vk) is a Fn-measurable random variable. Then (B.5) Ex h X v1,...,vk∈N+ n Y (v1, . . . , vk) i = Qk x h Y Y v∈skel(n)\\{o} ζ(φp(v), |v| -1) ζ(φv, |v|) dlp(v) i . We are now ready to give the proof of Proposition 3.5. Proof of Proposition 3.5. Statement (3.26) follows directly from Lemma B.2 with k = 1 and Y (v1) = f(φv1). Indeed, since k = 1, there is only one mark on every level and thus Y = f(ξ1 n), skel(n) \\ {o} = {σ1 1, . . . , σ1 n}, and lp(v) = 1 for all v ∈skel(n) \\ {o}. Therefore, by (B.5) Ex h X v∈N+ n f(φv) i = Qx f(ξn) Y v∈{σ1,...,σn} ζ(φp(v), |v| -1) ζ(φv, |v|) d = Qx f(ξn) n Y i=1 χ(ξi-1)di-1 χ(ξi)di d = Qx f(ξ1 n) χ(x) χ(ξn) , (B.6) as claimed in (3.26). Similarly, statement (3.27) is a consequence of Lemma B.2 with k = 2. We set Y (v1, v2) = f(φv1)g(φv2). Then Y = f(ξ1 n)g(ξ2 n) and by (B.5), (B.7) Ex h X v,w∈N+ n f(φv)g(φw) i = Q2 x f(ξ1 n)g(ξ2 n) Y v∈skel(n)\\{o} ζ(φp(v), |v| -1) ζ(φv, |v|) dlp(v) . To consider the possible structures of skel(n) \\ {o}, set s = max{k : σ1 k = σ2 k} to be the last time where the two spines agree. Note that due to the dynamics of marks under Q2 x, (B.8) Q2 x[s = k] = ( (d -1)d-(k+1), if k ∈{0, . . . , n -1}, d-n, if k = n, SCALING LIMITS FOR THE CRITICAL GFF 46 and, by a simple computation, Y v∈skel(n)\\{o} ζ(φp(v), |v| -1) ζ(φv, |v|) dlp(v) = Y v∈skel(n)\\{o} χ(φp(v)) χ(φv) 1 d dlp(v) = χ(ξ1 s)χ(x) χ(ξ1 n)χ(ξ2 n)ds. (B.9) Therefore, (B.7) can be written as Ex X v∈N+ n f(φv) X v∈N+ n g(φv) = n X k=0 dkQ2 x[s = k]Q2 x f(ξ1 n)g(ξ2 n) χ(ξ1 k)χ(x) χ(ξ1 n)χ(ξ2 n) s = k = d -1 d n-1 X k=0 Q2 x f(ξ1 n) χ(ξ1 n) g(ξ2 n) χ(ξ2 n)χ(ξ1 s)χ(x) s = k + Q2 x f(ξ1 n)g(ξ1 n) χ(x) χ(ξ1 n) s = n . (B.10) By construction, under Q2 x, conditional on s = k, for times i = 1, . . . , k the processes ξ1 i and ξ2 i are Markov chains which follow the same trajectory and have the same dynamics as ξi under Q1 x. Further, for later times i = k + 1, . . . , n, they are independent Markov chains distributed according to Qξ1 k. 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Elboulqe†2\n1Department of Mathematics and Computer, Faculty of Sciences A¨ın Chock, Hassan II\nUniversity, Casablanca, BP. 5366, Morocco\n2Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech,\nBP. 2390, Morocco\nOctober 17, 2025\nAbstract\nIn a recent work, we presented the reduced Jacobian method (RJM) as an extension\nof Wolfe’s reduced gradient method to multicriteria (multiobjective) optimization problems dealing\nwith linear constraints. This approach reveals that using a reduction technique of the Jacobian\nmatrix of the objective avoids scalarization. In the present work, we intend to generalize RJM\nto handle nonlinear constraints too. In fact, we propose a generalized reduced Jacobian (GRJ)\nmethod that extends Abadie-Carpentier’s approach for single-objective programs. To this end, we\nadopt a global reduction strategy based on the fundamental theorem of implicit functions. In this\nperspective, only a reduced descent direction common to all the criteria is computed by solving\na simple convex program. After establishing an Armijo-type line search condition that ensures\nfeasibility, the resulting algorithm is shown to be globally convergent, under mild assumptions, to\na Pareto critical (KKT-stationary) point. Finally, experimental results are presented, including\ncomparisons with other deterministic and evolutionary approaches.\nKeywords\nMulticriteria optimization. Pareto optima. Nonlinear programming. Pareto KKT-\nstationarity. Reduced Jacobian method. Nonlinear constraints.\nMathematics Subject Classification (2020)\n90C29 · 90C30 · 90C52\n1\nIntroduction\nIn many areas, like economics, medicine, design, transportation and so on, we are faced with\nmulticriteria (or multiobjective) optimization problems (MOP), where several functions must be\noptimized at the same time. Because of the conflicts which may arise between these functions,\nalmost never a single point will minimize or maximize them at once. So the Pareto optimality\nconcept have to be considered. A point is said to be Pareto optimum or efficient solution, if it\ncannot be improved with respect to a criterion without degrading at least one of the others. Taking\naccount the principle, Pareto solutions are incomparable and no single point can represent all the\nothers. The problem then consists of determining the set of all Pareto solutions or their objective\nvalues called the Pareto front or more generally the set of non-dominated solutions.\nIn recent years, it has been observed that this task has become more tractable with the advent of\na new generation of algorithms grounded in the principle of multiobjective search descent directions,\nwhich aim to identify a direction that decreases all the criteria simultaneously (e.g. [5, 13, 16, 18,\n20, 21, 23, 29, 31, 32, 33, 34]). In most cases, the resulting algorithms are merely direct extensions\nof well-known nonlinear descent methods, which tackle the given problem without resorting to\nintermediate transformations or introducing artificial parameters that may be sensitive to the\n∗Corresponding author. Email: elmaghri@yahoo.com\n†Email: elboulqeyoussef@gmail.com\n1\narXiv:2510.14785v1 [math.OC] 16 Oct 2025\n\n2\nm. el maghri, y. elboulqe\noriginal problem. From a theoretical perspective, convergence toward Pareto-KKT stationarity is\nwell established, and numerically, these approaches have also proven very promising, both in terms\nof convergence to the true Pareto front and the diversity of non-dominated approximate solutions.\nFor instance, in [16], we proposed a direct extension of the Wolfe reduced gradient method (RGM)\n[42] (see also [15] for a new RGM variant) to the so-called reduced Jacobian method (RJM) for\nsolving linearly constrained MOPs. We also refer the reader to [18], where we introduced an RJM\nvariant that possesses the full convergence property.\nIn the present study, we build upon our previous work by focusing on the so-called generalized\nreduced gradient (GRG) method, originally introduced by Abadie and Carpentier [1], which extends\nWolfe’s reduction strategy to problems with nonlinear constraints. Accordingly, we propose a novel\napproach, which we term the generalized reduced Jacobian (GRJ) method, to effectively handle\nnonlinear constraints in multicriteria optimization.\nThe GRG method is widely recognized as\none of the fundamental and practical approaches to nonlinear mathematical programming. Its\nconvergence properties were rigorously established in subsequent works by Smeers [37], Mokhtar-\nKharroubi [30], and Schittkowski [36]. As far as we are concerned, our investigations cover both the\ntheoretical and algorithmic aspects of this method. Abadie-Carpentier’s reduction strategy, based\non the well-known implicit function theorem, is adopted in this work, thereby enabling a complete\ncharacterization of Pareto-KKT stationarity through multiobjective reduced descent directions. It\nis subsequently proven that such directions guarantee simultaneous descent of all criteria and ensure\nthe existence of Armijo-like steplengths, while preserving feasibility throughout the process. From\na computational point of view, a simple direction-finding subproblem is introduced to iteratively\ncompute the reduced directions. Under mild assumptions, the resulting GRJ algorithm is proven\nto be globally convergent to a Pareto KKT-stationary point in the sense of accumulation points.\nFinally, numerical experiments are reported, illustrating the effectiveness of the GRJ method\ncompared with other methods, including a Zoutendijk-like method [31], an SQP-type approach\n[22], and the well-known evolutionary method NSGA-II [10].\n2\nEfficiency and multiobjective descent\nWe are dealing with the following nonlinearly constrained multicriteria optimization problem:\n(MOP)\nMin F(x)\nsubject to\nG(x) = 0, a ≤x ≤b,\nwhere\n\u0002\nF, G\n\u0003\n: x ∈Rn 7→\n\u0002\nF(x), G(x)\n\u0003\n=\n\u0002\u0000f1(x), . . . , fr(x)\n\u0001\n,\n\u0000g1(x), . . . , gm(x)\n\u0001\u0003\n∈Rr × Rm\nare two vector-valued functions continuously differentiable on the feasible set denoted by S, and,\na = (a1, . . . , an), b = (b1, . . . , bn) ∈Rn. For y = (y1, . . . , yp) ∈Rp and y′ = (y′\n1, . . . , y′\np) ∈Rp,\ny ≤y′ (resp. y < y′), iff yi ≤y′\ni (resp. yi < y′\ni) (∀i), and, y ⪇y′ means y ≤y′ and y ̸= y′. The\nusual norm and the usual inner product in Rp will be respectively denoted by ∥y∥= √y · y and\ny · y′ = yT y′, where yT stands for the transpose of y.\nThe (MOP) consists in minimizing the vector function F over S in the following Pareto senses:\nDefinition 2.1 A point x∗∈S is said to be\n(a) weakly efficient for (MOP), if ∄x ∈S, F(x) < F(x∗);\n(b) efficient (Pareto optimum) for (MOP), if ∄x ∈S, F(x) ⪇F(x∗);\n(c) properly efficient (in Henig’s sense) for (MOP), if there exists C ⊊Rr a pointed1 convex cone\nsuch that its topological interior int C ⊃Rr\n+\\ {0}, and, ∄x ∈S, F(x) ⪇C F(x∗).2\nThe sets of weakly efficient points, efficient points and properly efficient points will be denoted\nrespectively by Ew, Ee and Ep. To unify the notations, we briefly denote by Eσ the set of all\nσ-efficient solutions depending on the choice of σ ∈{w, e, p}. Recall the following relationships:\nEp ⊆Ee ⊆Ew.\n(1)\n1A pointed cone C means that C ∩−C = {0}.\n2The relation y ⪇C y′ means that y′ −y ∈C\\ {0}.\n\ngeneralized reduced jacobian method\n3\nThe concept of local σ-efficiency is defined by replacing S by S ∩N(x∗) in Definition 2.1, where\nN(x∗) is some neighbourhood of x∗. The set of all locally σ-efficient points will be denoted by\nEloc\nσ . The image set F(Eσ) (resp. F(Eloc\nσ )) is called σ-Pareto front (resp. local σ-Pareto front).\nTaking into account the inclusions (1), the determination of the weakly efficient set will include\nall the other efficient solutions.\nLet us denote by JF(x) =\n\u0000∂fj(x)/∂xi\n\u0001\nj,i the Jacobian matrix of F at x; the jth line of JF(x)\nbeing of course the gradient ∇fj(x) of the jth objective fj at x.\nDefinition 2.2 Let S be a convex set. The vector mapping F : S ⊂Rn →Rr is said to be\n(a) convex on S, if\n∀x, x′ ∈S, ∀α ∈[0, 1] ,\nF(αx + (1 −α)x′) ≤αF(x) + (1 −α)F(x′);\n(b) strictly convex on S, if\n∀x, x′ ∈S, x ̸= x′, ∀α ∈]0, 1[ ,\nF(αx + (1 −α)x′) < αF(x) + (1 −α)F(x′);\n(c) pseudoconvex on S, if it is pseudoconvex at any x ∈S, i.e.,\n∀x′ ∈S,\nF(x′) < F(x) =⇒JF(x)(x′ −x) < 0;\n(d) strictly pseudoconvex on S, if it is strictly pseudoconvex at any x ∈S, i.e.,\n∀x′ ∈S, x′ ̸= x,\nF(x′) ≤F(x) =⇒JF(x)(x′ −x) < 0.\nRemark 2.1 It is obvious that F is componentwise convex, iff F is convex. But the converse does\nnot hold for pseudoconvexity (see [24, Theorem 9.2.3 and Remark 5.3]). The same should hold\ntrue for the strict concepts. As in the scalar case, pseudoconvexity generalizes convexity.\nA multiobjective search direction is defined as follows.\nDefinition 2.3 A vector d ∈Rn is said to be\n(a) a feasible direction at x ∈S, if ∃tf > 0, ∀t ∈]0, tf] , x + td ∈S;\n(b) a tangent direction to S at x ∈S, if ∃(dk)k ⊂Rn, ∃(tk)k ⊂]0, +∞[ such that\nlim\nk→+∞dk = d,\nlim\nk→+∞tk = 0 and x + tkdk ∈S (∀k);\n(c) a descent direction of F at x ∈S, if ∃td > 0, ∀t ∈]0, td] , F(x + td) < F(x).\nThe sets of feasible directions, tangent directions and descent directions are cones which will\nbe denoted respectively by AS(x), TS(x) and DS(x).\nA sufficient condition for d ∈DS(x) is (see [20])\nJF(x)d < 0.\n(2)\nA geometric necessary condition for weak efficiency is obtained using the tangent directions:\nTheorem 2.1 If x∗∈Eloc\nw , then ∄d ∈TS(x∗), JF(x∗)d < 0.\nProof\nLet d ∈TS(x∗). We can suppose that d ̸= 0, otherwise the result is trivial. So, by the very\ndefinition, ∃(dk)k −→d, ∃(tk)k ↘0+ such that\nxk := x∗+ tkdk ∈S\n(∀k).\n(3)\nBy hypothesis, ∃N(x∗) a neighbourhood of x∗such that\n∄x ∈N(x∗) ∩S, F(x) < F(x∗).\n(4)\n\n4\nm. el maghri, y. elboulqe\nFrom (3), we see that xk ∈N(x∗) ∩S for k large enough, and from (4), we deduce that\n∃K ∈N, ∀k ≥K, ∃jk ∈{1, . . . , r} , fjk(xk) ≥fjk(x∗).\n(5)\nWithout loss of generality, we can assume that jk = j0 (a constant index) for k large enough. On\nthe other hand, by differentiability, we have that\nfj0\n\u0000xk\u0001\n= fj0\n\u0000x∗+ tkdk\u0001\n= fj0 (x∗) + tk∇fj0 (x∗)T dk + o\n\u0000tkdk\u0001\n.\nIt follows, from (5), that for k large enough,\ntk∇fj0 (x∗)T dk + o\n\u0000tkdk\u0001\n≥0.\nPassing to the limit, when k ↗+∞, after dividing by tk∥dk∥, we obtain\n∇fj0 (x∗)T d ≥0,\nwhich shows the result of the theorem.\n□\nAs is well known, the tangent cone TS(x) is introduced to handle nonlinear equality constraints,\nsince the cone AS(x) may be empty in such cases (see, e.g., the case where S is a circle). On the\nother hand, the tangent cone TS(x) is always closed and coincides with the closure of AS(x) when\nS is convex. This helps to explain the suitability of using TS(x) instead of AS(x). Furthermore,\nalthough the tangent cone plays a theoretically crucial role for a general feasible set S, its explicit\ndetermination often remains challenging in practice. However, when S is explicitly described by\nequality and/or inequality constraints, the tangent cone can always be expressed in terms of these\nconstraints via the so-called linearized cone. For our (MOP), the latter is formulated at a given\nx ∈S as follows:\nCS(x) = {d ∈Rn : JG(x)d = 0, ∀i ∈Ia(x), di ≥0, ∀i ∈Ib(x), di ≤0} ,\nwhere Ia(x) = {i ∈{1, . . . , n} : xi = ai} and Ib(x) = {i ∈{1, . . . , n} : xi = bi} are the index sets\nof active variables, and, JG(x) is the Jacobian matrix of G at x. In general, one only has that\nTS(x) ⊆CS(x) (see, e.g., [4, Chapter 5]). When equality holds, we say that Abadie’s constraint\nqualification (ACQ) is satisfied at x:\n(ACQ)\nTS(x) = CS(x).\nWith the assumption (ACQ), we can state the following explicit conditions of weak efficiency\nthat will be useful in the sequel.\nTheorem 2.2 Assume that (ACQ) holds at x∗∈S. If x∗∈Eloc\nw , then the following two equivalent\nconditions are satisfied:\n(i) There is no tangent descent direction to S at x∗∈S satisfying (2), i.e.,\n∄d ∈Rn :\n\n\n\n\n\n\n\n\n\nJF(x∗)d < 0,\nJG(x∗)d = 0,\ndi ≥0, ∀i ∈Ia(x∗),\ndi ≤0, ∀i ∈Ib(x∗).\n(ii) The point x∗∈S is Pareto KKT-stationary, i.e.,\n∃(λ, u, v, w) ∈Rr\n+\\ {0} × Rm × Rn\n+ × Rn\n+ :\n\n\n\n\n\nJF (x∗)T λ + JG (x∗)T u −v + w = 0,\nv · (x∗−a) = 0,\nw · (b −x∗) = 0.\nConversely, if condition (i) or (ii) is satisfied at x∗∈S, F is pseudoconvex (resp. strictly pseudo-\nconvex) at x∗and the kernel Ker JG(x∗) contains N(0) ∩(S −x∗) for some neighbourhood N(0)\nof 0, then x∗∈Eloc\nw\n(resp. x∗∈Eloc\ne ).\n\ngeneralized reduced jacobian method\n5\nProof\nNecessity: condition (i) is the result of Theorem 2.1 expressed under (ACQ). Equivalence\nbetween (i) and (ii) comes directly from Motzkin’s alternative theorem (see, e.g., [28]).\nSufficiency: Since it is assumed that N(0) ∩(S −x∗) ⊆Ker JG(x∗), then there exists N(x∗)\na neighbourhood of x∗such that JG(x∗)(x −x∗) = 0 for all x ∈N(x∗) ∩S. Now, by way of\ncontraposition, suppose that x∗̸∈Eloc\nw\n(resp. x∗̸∈Eloc\ne ). Then, there exists x0 ∈S ∩N(x∗)\nsuch that F(x0) < F(x∗) (resp. F(x0) ⪇F(x∗)). By putting d = x0 −x∗, pseudoconvexity (resp.\nstrict pseudoconvexity) of F at x∗implies that JF(x∗)d = JF(x∗)\n\u0000x0 −x∗\u0001\n< 0. On the other\nhand, we also have that JG(x∗)d = JG(x∗)(x0 −x∗) = 0 because x0 ∈N(x∗) ∩S ⊆Ker JG(x∗).\nMoreover,\ndi =\n(\nx0\ni −ai, if i ∈Ia(x∗),\nx0\ni −bi, if i ∈Ib(x∗),\nsatisfying di ≥0, if i ∈Ia(x∗) and di ≤0, if i ∈Ib(x∗), which under the constraint qualification\ncondition (ACQ), shows the existence of a tangent descent direction of F at x∗.\n□\nRemark 2.2 If, in Theorem 2.2, we replace N(0) by Rn, then it is easy to see by the same proof\nthat the local efficiency becomes global, in particular, when G is affine.\n3\nGRJ strategy\nLet A(x) := JG(x) ∈Rm×n be the Jacobian matrix of G at x ∈S such that A(x) is of full rank\nm < n. Then, there exists a subset B ⊂{1, . . . , n}, called basis of A(x), such that the submatrix\nAB(x) is invertible, where (rearranging the columns if necessary) A(x) = [AB(x) AN(x)] with\nAB(x) = JGB(x), AN(x) = JGN(x) and obviously N = {1, . . . , n} \\B. Rearranging also the\ncomponents of x we can write x = (xB, xN), where xB (resp. xN) is the well-known vector of\nbasic (resp. nonbasic) variables. Since G is assumed to be continuously differentiable and AB(x)\nis invertible, then by the implicit function theorem, there exist V ⊂Rm a neighbourhood of xB,\nW ⊂Rn−m a neighbourhood of xN and a unique function ψ : W −→V which is continuously\ndifferentiable such that\n∀x′\nN ∈W,\n\n\n\n\n\nG (ψ(x′\nN), x′\nN) = 0,\n∂ψ\n∂xN\n(x′\nN) = −A−1\nB (x′)AN(x′).\n(6)\nHence, the feasible set S may be expressed only in terms of nonbasic variables:\nx ∈S ⇐⇒a ≤x = (ψ(xN), xN) ≤b.\nA point x ∈S is said to be nondegenerate, if there exists B a basis such that aB < xB < bB. In\nthis case, B is also said to be a nondegenerate basis for x; otherwise x is degenerate. The feasible\nset S is in turn called nondegenerate, if any x ∈S is nondegenerate.\nGiven a vector d ∈Rn that we partition in the same way to d = (dB, dN), then it is clear that\nJG(x)d = 0 ⇐⇒dB = −A−1\nB (x)AN(x)dN.\nSo by also partitioning the Jacobian matrix JF(x) = [JFB(x) JFN(x)], for any d such that\ndB = −A−1\nB (x)AN(x)dN, it follows that\nJF(x)d = UN(x)dN,\nwhere UN(x) is an r × (n −m) matrix, which will be called generalized reduced Jacobian matrix of\nF at x, explicitly given by\nUN(x) := JFN(x) −JFB(x)A−1\nB (x)AN(x).\n(7)\n\n6\nm. el maghri, y. elboulqe\nObserve that the jth row of UN(x) is nothing more than the well-known reduced gradient of the\ncriterion fj at x which is given by\nuj\nN(x) = ∂fj\n∂xN\n(x) −\n\u0000A−1\nB (x)AN(x)\n\u0001T ∂fj\n∂xB\n(x).\nNow, we define a multiobjective reduced descent direction of F at x as being a nonbasic vector\ndN ∈Rn−m satisfying:\nUN(x)dN < 0,\n∀i ∈Ia(x) ∩N, di ≥0,\nand\n∀i ∈Ib(x) ∩N, di ≤0.\n(8)\nThe geometric stationarity stated in Theorem 2.2(i) then can be expressed only in terms of\nnonbasic directions:\nCorollary 3.1 Under (ACQ), a nondegenerate point x ∈S is Pareto KKT-stationary for (MOP),\niff there is no multiobjective reduced descent direction of F at x.\nThe next result shows that given any multiobjective reduced descent direction at a given\nnondegenerate point, feasibility along this direction with an Armijo-like steplength is guaranteed.\nProposition 3.1 Let x ∈S and ψ be the associated implicit function given by (6), and, let dN be\na multiobjective reduced descent direction of F at x. Let t ∈R+ and define\ntN := min\n\u0012\u001aai −xi\ndi\n: di < 0, i ∈N\n\u001b\n∪\n\u001abi −xi\ndi\n: di > 0, i ∈N\n\u001b\u0013\n.\n(9)\nThen,\n(i) tN > 0.\n(ii) t ≤tN ⇐⇒aN ≤xN + tdN ≤bN.\nIf furthermore aB < xB < bB (i.e., x is nondegenerate with B = {1, . . . , n} \\ N), then\n(iii) ∃tf ∈]0, tN], ∀t ∈]0, tf],\n\u0000ψ(xN + tdN), xN + tdN\n\u0001\n∈S\nand\naB < ψ(xN + tdN) < bB.\n(10)\n(iv) ∀β ∈]0, 1[ (fixed), ∃ta ∈]0, tf], ∀t ∈]0, ta],\nF\n\u0000ψ(xN + tdN), xN + tdN\n\u0001\n< F(x) + βtUN(x)dN.\n(11)\nProof\n(i) It is clear that tN exists (since dN ̸= 0).\nThe positivity of tN follows by the very\ndefinition of dN. Indeed, if tN = (ai −xi)/di, then from (8), it follows that ai −xi < 0, and\nconsequently, tN > 0; otherwise, tN = (bi −xi)/di, which in accordance with (8), entails that\nbi −xi > 0, thereby ensuring tN > 0.\n(ii) Let us show the first implication. Let i ∈N. The assertion is obvious if di = 0. Now, if\ndi > 0, then ai ≤xi + tdi and (bi −xi)/di ≥tN ≥t, so that bi ≥xi + tdi. The case di < 0 is\nsimilar. Conversely, since dN ̸= 0, then if di < 0 (i ∈N), the hypothesis ai ≤xi + tdi implies that\nt ≤(ai −xi)/di; and if di > 0, then the hypothesis xi + tdi ≤bi implies that t ≤(bi −xi)/di,\nwhich shows that t ≤tN.\n(iii) By definition of ψ : W −→V (see (6)) and the continuity of t 7−→xN + tdN at 0, there exists\nt1 > 0 such that xN + tdN ∈W for all t ∈[0, t1]. It follows by (6) that\n∀t ∈[0, t1] ,\nG (ψ(xN + tdN), xN + tdN) = 0.\nOn the other hand, by assertion (ii), we have that\n∀t ∈[0, tN] ,\naN ≤xN + tdN ≤bN.\n\ngeneralized reduced jacobian method\n7\nSo, it remains to show that the box-constraint also holds for the basic variables.\nIndeed, by\nnondegeneracy assumption and continuity, it is immediate that\n∃t2 > 0,\n∀t ∈[0, t2] ,\naB < ψ(xN + tdN) < bB.\nThus, by putting tf = min {t1, t2, tN}, we obtain the desired result.\n(iv) To prove the Armijo-like inequality (11), let us consider the reduced function eF(·) := F (ψ(·), ·)\nwhich is continuously differentiable on W ⊂Rn−m. By using the derivative of ψ given by (6), we\nimmediately deduce the Jacobian of eF:\nJ eF(xN) = JFN(x) −JFB(x)AB(x)−1AN(x) = UN(x).\nBy virtue of (iii), xN + tdN ∈W for all t ∈]0, tf]. So, by differentiability, it follows that\neF(xN + tdN)\n=\neF(xN) + tUN(x)dN + o(t)\n=\neF(xN) + tβUN(x)dN + t\n\u0002\n(1 −β)UN(x)dN + 1\nt o(t)\n\u0003\n.\nSince (1 −β)UN(x)dN < 0 and lim\nt→0+\n1\nt o(t) = 0, then\n∃ta ∈]0, tf] , ∀t ∈]0, ta] , (1 −β)UN(x)dN + 1\nt o(t) < 0,\nthus implying that\neF(xN + tdN) < eF(xN) + tβUN(x)dN.\nAs we have eF(xN + tdN) = F\n\u0000ψ(xN + tdN), xN + tdN\n\u0001\nand ψ(xN) = xB, then the result follows\nstraightforwardly.\n□\nAfter having shown that it is possible to obtain feasible steplengths along reduced directions, it\nremains for completing our GRJ scheme to show how to determine such reduced descent directions.\n4\nGRJ search direction\nLet us consider first a positive functional ϕ : R →R+ satisfying ϕ(t) = 0 iff t = 0; e.g., ϕp(t) = |t|p\np\nfor some value of p ∈]0, 1], or, ϕ0(t) = 1R∗the characteristic function of R∗defined by 1R∗(t) = 1 if\nt ̸= 0; 1R∗(0) = 0. Let ⌊a⌋+ := max(0, a) and ⌊a⌋−:= max(0, −a) denote, respectively, the positive\nand negative parts of the scalar a. Then, for x ∈S, we introduce the following finding-direction\nsubproblem:\n(Px)\nMin\nλ∈Λ f(λ, x) := 1\n2\nX\ni∈N\n\u0012\nϕ (bi −xi)\nj \u0000UN(x)T λ\n\u0001\ni\nk2\n−+ ϕ (xi −ai)\nj \u0000UN(x)T λ\n\u0001\ni\nk2\n+\n\u0013\nwhere Λ = {(λ1, . . . , λr) ∈Rr\n+ : Pr\nj=1 λj = 1}, xi (resp. ai and bi) is the ith component of xN\n(resp. aN and bN) and\n\u0000UN(x)T λ\n\u0001\ni is the ith component of UN(x)T λ.\nIt is obvious that (Px) is a convex continuously differentiable problem and always has an\noptimal solution in the compact set Λ. Moreover, (Px) has simplicial constraints and it is easy to\nverify that its objective function has a computable gradient explicitly given by\n∇λf(λ, x) = −UN(x)δN(λ, x),\nwhere\nδN(λ, x) =\n\u0012\nϕ (bi −xi)\nj \u0000UN(x)T λ\n\u0001\ni\nk\n−−ϕ (xi −ai)\nj \u0000UN(x)T λ\n\u0001\ni\nk\n+\n\u0013\ni∈N\n.\n(12)\n\n8\nm. el maghri, y. elboulqe\nThe descent GRJ scheme will consist to take dN := δN(λ(x), x), where λ(x) ∈argmin(Px).\nThen, according to (12), ∀i ∈N,\ndi =\n\n\n\n−ϕ(xi −ai)\n\u0000UN(x)T λ(x)\n\u0001\ni ,\nif\n\u0000UN(x)T λ(x)\n\u0001\ni > 0,\n−ϕ(bi −xi)\n\u0000UN(x)T λ(x)\n\u0001\ni ,\nelse.\n(13)\nThe lemma below characterizes the optimal solutions of (Px), which will be crucial not only\nfor proving the descent properties of the vector dN, but also for the convergence analysis of the\nresulting algorithm. Its proof is omitted, as it is similar to the one of [16, Lemma 3.2].\nLemma 4.1 λ(x) ∈Λ is an optimal solution of (Px), iff\nδN(λ(x), x) ·\n\u0000UN(x)T λ\n\u0001\n≤−2f (λ(x), x)\n(∀λ ∈Λ),\nor equivalently,\nδN(λ(x), x) · uj\nN(x) ≤−2f (λ(x), x)\n(j = 1, . . . , r).\nThe descent properties are stated as follows:\nProposition 4.1\n(i) dN = 0 ⇐⇒f(λ(x), x) = 0.\n(ii) dN ̸= 0 =⇒dN is a multiobjective reduced descent direction of F at x. If moreover (ACQ)\nholds and x is nondegenerate, then x is not Pareto KKT-stationary for (MOP).\n(iii) dN = 0 =⇒x is Pareto KKT-stationary for (MOP).\n(iv) The functional x 7→f(λ(x), x) is continuous on S provided ϕ is continuous.\nProof\n(i) To show the direct implication, observe that as ϕ ≥0, then dN = 0 implies that, ∀i ∈N,\nj\nϕ (bi −xi)\n\u0000UN(x)T λ(x)\n\u0001\ni\nk\n−=\nj\nϕ (xi −ai)\n\u0000UN(x)T λ(x)\n\u0001\ni\nk\n+.\nIf we suppose that the last two terms are not equal to zero, we would have\nϕ (bi −xi)\n\u0000UN(x)T λ(x)\n\u0001\ni < 0\nand\nϕ (xi −ai)\n\u0000UN(x)T λ(x)\n\u0001\ni > 0,\nwhich would mean that ϕ (bi −xi) and ϕ (xi −ai) have not the same sign contradicting ϕ ≥0.\nThus, ∀i ∈N,\nj\nϕ (bi −xi)\n\u0000UN(x)T λ(x)\n\u0001\ni\nk\n−=\nj\nϕ (xi −ai)\n\u0000UN(x)T λ(x)\n\u0001\ni\nk\n+ = 0,\nand this shows that f(λ(x), x) = 0. The reverse implication follows straightforwardly from the\npositivity of ϕ and (13).\n(ii) If dN ̸= 0, then by assertion (i), f(λ(x), x) > 0. It follows, by Lemma 4.1, that\ndN · uj\nN(x) ≤−2f(λ(x), x) < 0\n(j = 1, . . . , r).\nHence, UN(x)dN < 0. On the other hand, for i ∈Ia(x) ∩N, we have that\ndi = ϕ(bi −xi)\nj \u0000UN(x)T λ(x)\n\u0001\ni\nk\n−≥0,\nand, for i ∈Ib(x) ∩N, we have that\ndi = −ϕ(ai −xi)\nj \u0000UN(x)T λ(x)\n\u0001\ni\nk\n+ ≤0.\n\ngeneralized reduced jacobian method\n9\nThis shows, according to the definition (8), that dN is well a multiobjective reduced descent\ndirection. The second part of this assertion follows straightforwardly from Corollary 3.1.\n(iii) Suppose that dN = 0. If we put\nv∗\nN :=\n\u0012j \u0000UN(x)T λ(x)\n\u0001\ni\nk\n+\n\u0013\ni∈N\nand\nw∗\nN :=\n\u0012j \u0000UN(x)T λ(x)\n\u0001\ni\nk\n−\n\u0013\ni∈N\n,\nthen, by (13) and the property that ϕ(t) = 0 iff t = 0, we obtain\n(v∗\nN, w∗\nN) ≥0,\nv∗\nN · (xN −aN) = 0\nand\nw∗\nN · (bN −xN) = 0.\nClearly, for λ∗:= λ(x), u∗:= −(A−1\nB (x))T JFB(x)T λ∗, v∗:= (0, v∗\nN) and w∗:= (0, w∗\nN), it holds\nthat (λ∗, u∗, v∗, w∗) ∈Rr\n+\\ {0} × Rm × Rn\n+ × Rn\n+, JF(x)T λ∗+ JG(x)T u∗−v∗+ w∗= 0 and\nv∗· (x −a) = w∗· (b −x) = 0, which express the Pareto KKT-stationarity for (MOP).\n(iv) Since f(λ(x), x) is the optimal value of (Px), its continuity with respect to x follows from the\ncompactness of the feasible set Λ and the continuity of the function (λ, x) 7→f(λ, x).\n□\n5\nGRJ algorithm and its convergence\n5.1\nThe algorithm\nThe generalized reduced Jacobian algorithm we propose to solve the multicriteria problem (MOP)\nis described below:\nGRJ pseudocode with Armijo line search\nStep 0: Initialization.\n- Select x ∈S, fix the Armijo’s constant β ∈]0, 1[ and choose the functional ϕ.\nStep 1: Nondegenerate basis selection.\n- Identify a basis B such that aB < xB < bB. Set N = {1, . . . , n} \\B.\nStep 2: Generalized reduced Jacobian.\n- Compute UN(x) according to (7).\nStep 3: Reduced direction.\n- Solve (Px) to determine dN according to (13).\nStep 4: Stopping criterion.\n- If min(Px) = 0, then STOP: x is Pareto KKT-stationary for (MOP).\nStep 5: Feasible Armijo line search.\n- Compute tN according to (9).\n- Determine, according to (10)-(11), a steplength t ∈]0, tN] that satisfies:\nt = max\np∈N\n\n\n\n\n\n\n\n\n\n1\n2p\n:\n\u0010\nψ\n\u0000xN + 1\n2p dN\n\u0001\n, xN + 1\n2p dN\n\u0011\n∈S, and\nF\n\u0010\nψ\n\u0000xN + 1\n2p dN\n\u0001\n, xN + 1\n2p dN\n\u0011\n< F(x) + β 1\n2p UN(x)dN\n\n\n\n\n\n\n\n\n\n,\nwhere ψ is the implicit function defined in (6).\nStep 6: Updated point.\n- Set x := (ψ (xN + tdN) , xN + tdN).\nStep 7: Degeneracy test.\n- If B is a degenerate basis for x, go to Step 1; otherwise go to Step 2.\n\n10\nm. el maghri, y. elboulqe\n5.2\nConvergence analysis\nIf the algorithm terminates after a finite number of iterations, then according to Proposition 4.1,\nit terminates at a Pareto KKT-stationary point. So we will assume that the sequence produced by\nthe algorithm is infinite. We also assume that the bases are chosen in such a way that the following\nhypothesis is satisfied:\n(H)\n\n\n\nxk\n−→\nk∈K⊆N x∗non Pareto KKT-stationary,\n∃B basis, ∀k ∈K, aB < xk\nB < bB.\n\n\n=⇒aB < x∗\nB < bB.\nKnown as the basis property, this hypothesis proves to be essential for the global convergence of\nreduced gradient methods (e.g., [16, 18, 30, 37] 3). An example provided in [30] shows that, without\nthis hypothesis, the algorithm may fail to converge even if the constraints are linear. However,\nthere are basis selection procedures that ensure the basis property for both linear and nonlinear\nconstraint cases, such as those mentioned in [18].\nTheorem 5.1 Assume that S is nondegenerate, ϕ is chosen to be continuous and derivable at 0.\nLet (xk)k∈N be the sequence produced by the GRJ algorithm. Then,\n(i) xk ∈S and F(xk+1) < F(xk) for all k ∈N;\n(ii) under the hypothesis (H), any accumulation point x∗of (xk)k is a Pareto KKT-stationary\npoint for (MOP), and,\nlim\nk→+∞F(xk) = F(x∗).\nProof\n(i) This assertion follows directly from Proposition 3.1.\n(ii) We need to prove the following lemma.\nLemma 5.1 If, in addition to the hypotheses of Theorem 5.1, a subsequence (xk)k∈K produced by\nthe GRJ algorithm converges to x∗with constant bases Bk = B for all k ∈K, and\n\u0000λ(xk)\n\u0001\nk∈K\nconverges too, then\n(a) dk\nN := δN(λ(xk), xk) −→\nk∈K d∗\nN := δN(λ(x∗), x∗), where N = {1, . . . , n} \\B;\n(b) ¯t := inf\nk∈K tk\nN > 0, where\ntk\nN := min\n\u0012\u001aai −xk\ni\ndk\ni\n: dk\ni < 0, i ∈N\n\u001b\n∪\n\u001abi −xk\ni\ndk\ni\n: dk\ni > 0, i ∈N\n\u001b\u0013\n.\nProof\n(a) Let λ∗=\nlim\nk∈K\nk→+∞\nλ(xk), then λ∗∈Λ since\n\u0000λ(xk)\n\u0001\nk∈K ⊂Λ which is closed. According\nto the hypothesis and Lemma 4.1, for all k ∈K, we have\nδN(λ(xk), xk) · uj\nN(xk) ≤−2f(λ(xk), xk)\n(j = 1, . . . , r),\nTaking into account that the mappings (λ, x) 7−→f(λ, x) and (λ, x) 7−→δN(λ, x) are continuous,\nby passing onto the limit, when k ↗+∞in the previous inequality, we obtain that\nδN(λ∗, x∗) · uj\nN(x∗) ≤−2f(λ∗, x∗)\n(j = 1, . . . , r),\nSince by closeness of S, x∗remains feasible as (xk)k, this shows applying once again Lemma 4.1\nthat λ∗∈argmin(Px∗), i.e., λ∗= λ(x∗), which proves the assertion.\n(b) To prove this assertion, we shall proceed by a contradiction way by assuming that ¯t = 0\n(because ¯t ≥0). Then, it would exist K0 ⊆K an infinite subset such that tk\nN −→\nk∈K0 0. Since the\n3As pointed out in [18], the hypothesis (H) was inadvertently omitted in the proof of [16, Theorem 4.1].\n\ngeneralized reduced jacobian method\n11\nindex set N is finite, we can assume (without loss of generality) that one of the two following cases\nwould happen:\n∃i0 ∈N,\n∀k ∈K0,\ntk\nN = ai0 −xk\ni0\ndk\ni0\n,\ndk\ni0 < 0,\nor\n∃i0 ∈N,\n∀k ∈K0,\ntk\nN = bi0 −xk\ni0\ndk\ni0\n,\ndk\ni0 > 0.\nIn the first case, following (13), we would have\n∀k ∈K0,\ntk\nN =\n1\n(UN(xk)T λ(xk))i0\n\"\nxk\ni0 −ai0\nϕ(xk\ni0 −ai0)\n#\n.\n(14)\nNow, by assertion (a), dk\ni0 := −ϕ(xk\ni0 −ai0)\n\u0000UN(xk)T λ(xk)\n\u0001\ni0 −→\nk∈K0 d∗\ni0, and, by the contradiction\nhypothesis, tk\nN −→\nk∈K0 0. So we would have that the sequence\n\u0000xk\ni0 −ai0\n\u0001\nk∈K0 also converges to 0;\ni.e., xk\ni0 −→\nk∈K0 ai0. But this would imply that\nϕ(xk\ni0 −ai0) −→\nk∈K0 ϕ(x∗\ni0 −ai0) = ϕ(0) = 0,\nand, by letting k ↗+∞in (14), we would obtain the contradiction\ntk\nN −→\nk∈K0\n1\n(UN(x∗)T λ∗)i0\n×\n1\nϕ′(0) ̸= 0,\nThe second case is similar. Hence, ¯t > 0.\n□\nLet us go back to the proof of Theorem 5.1(ii).\nBy hypothesis, there exists a subsequence (xk)k∈K −→x∗with K ⊆N an infinite subset. By\ncloseness of S, x∗remains feasible as (xk)k. Now, by continuity of F and the assertion (i), it\nfollows that the sequence\n\u0000F(xk)\n\u0001\nk∈N −→F(x∗). On the other hand, since the index set Bk is in\nthe finite set {1, . . . , n}, we can assume (without loss of generality) that Bk = B (hence Nk = N)\nfor all k ∈K. Now, by Armijo’s condition: ∀k ∈N,\nF(xk+1) −F(xk) < βtkUNk(xk)dk\nNk < 0.\nIt follows that\nlim\nk∈K\nk→+∞\ntkUN(xk)dk\nN =\nlim\nk∈K\nk→+∞\ntkUNk(xk)dk\nNk = 0.\n(15)\nSimilarly, as the sequence\n\u0000λ(xk)\n\u0001\nk ⊂Λ which is a compact set, we can assume (without loss of\ngenerality) that λ(xk) −→\nk∈K λ∗∈Λ. Then, by virtue of Lemma 5.1(a), dk\nN −→\nk∈K d∗\nN. If we suppose,\nby a contradiction way, that x∗is not Pareto KKT-stationary, then according to Proposition\n(4.1)(ii)–(iii), d∗\nN would be a multiobjective reduced descent direction of F at x∗∈S. Hence,\naccording to the definition (8),\nUN(x∗)d∗\nN < 0.\n(16)\nWe can see from (15) that this would imply tk −→\nk∈K 0. In particular, for any p ∈N, tk <\n1\n2p for\nany k ∈K sufficiently large. Recall that, by the implicit function theorem applied at x∗, there\nexists a unique mapping ψ∗: W ∗−→V ∗, where W ∗(resp. V ∗) is a neighbourhood of x∗\nN (resp.\nx∗\nB = ψ∗(x∗\nN)) such that G\n\u0000ψ∗(xN), xN\n\u0001\n= 0 for all xN ∈W ∗. This applies to each xk\nN + 1\n2p dk\nN as\n(k, p) →+∞, since xk\nN + 1\n2p dk\nN −→x∗\nN; hence, for all p ∈N and k ∈K, both sufficiently large,\nG\n\u0010\nψ∗\u0000xk\nN + 1\n2p dk\nN\n\u0001\n, xk\nN + 1\n2p dk\nN\n\u0011\n= 0.\n(17)\n\n12\nm. el maghri, y. elboulqe\nOn the other hand, since x∗is assumed to be not Pareto KKT-stationary, then using the hypothesis\n(H), x∗would be nondegenerate, so that aB < x∗\nB = ψ∗(x∗\nN) < bB. Thus, by the continuity of ψ∗,\nwe would also have that, for all p ∈N and k ∈K, both sufficiently large,\naB < ψ∗\u0010\nxk\nN + 1\n2p dk\nN\n\u0011\n< bB.\n(18)\nNow, using Lemma 5.1(b), we have that 0 <\n1\n2p ≤¯t for all p large enough, and this shows, according\nto Proposition 3.1(ii), that for any p ∈N sufficiently large and k ∈K,\naN ≤xk\nN + 1\n2p dk\nN ≤bN.\n(19)\nSubsequently, we would have, following (17)–(19), that for all p ∈N and k ∈K, both sufficiently\nlarge,\n\u0010\nψ∗\u0000xk\nN + 1\n2p dk\nN\n\u0001\n, xk\nN + 1\n2p dk\nN\n\u0011\n∈S.\nHowever, given the previous assertion that tk <\n1\n2p , this would mean that, in Step 5 of the GRJ\nalgorithm, for all p ∈N and k ∈K, both sufficiently large, the Armijo inequality would fail to\nhold; that is,\nF\n\u0010\nψ∗\u0000xk\nN + 1\n2p dk\nN\n\u0001\n, xk\nN + 1\n2p dk\nN\n\u0011\n̸< F(xk) + β 1\n2p UN(xk)dk\nN.\n(20)\nFixing p ∈N sufficiently large and passing onto the limit when k ↗+∞, we would obtain that\nF\n\u0010\nψ∗\u0000x∗\nN + 1\n2p d∗\nN\n\u0001\n, x∗\nN + 1\n2p d∗\nN\n\u0011\n̸< F(x∗) + β 1\n2p UN(x∗)d∗\nN.\nSince p ∈N was arbitrary large enough, this would be in contradiction with Proposition 3.1(iv),\nand this proves that x∗is well Pareto KKT-stationary for (MOP).\n□\n6\nNumerical experiments\n6.1\nTest problems and implementation\nIn this section, we present computational results obtained through the proposed GRJ method.\nWe also provide comparisons with three other multicriteria methods: ZMO (a Zoutendijk-like\nmethod [31]), MOSQP (an SQP-type approach [22]) and the evolutionary NSGA-II method (the\nnondominated sorting genetic algorithm [10]). All codes were implemented on a machine equipped\nwith 1.90 GHz Intel(R) Core(TM) i5 CPU and 16 Go memory. GRJ, ZMO, and MOSQP were\ncoded in SCILAB-2024.1.0, while NSGA-II was executed using MATLAB’s predefined function\n“gamultiobj” (MATLAB R2024b).\nThirty constrained multicriteria optimization test problems, selected from the literature, were\nconsidered in this experiment and are summarized in Table 1, where ‘OV’, ‘L’ and ‘NL’, respectively,\nindicate the number of original variables, the number of linear constraints and the number of\nnonlinear constraints.\nWe selected the ‘Disc Brake’ and ‘Welded Beam’ problems, which are\nderived from real-world engineering design applications [35]. Detailed descriptions of these two\nmodels, along with further analysis of the results, are provided at the end of this section. Also,\nwe introduced the problem ‘EL3’ below showing the importance of the numerical aspect of the\ntangent cone TS(x), in the case where no feasible direction exists, i.e., AS(x) = ∅.\n(EL3)\nMinimize\n\u0012\nx3\n2 + log\n\u0000x2\n1 + 1\n\u0001\n, sin\n\u0010\nx1\nx2 + 2\n\u0011\u0013\n,\nsubject to\nx2\n1 + x2\n2 = 1,\n(21)\n0 ≤xi ≤1,\ni = 1, 2.\n\ngeneralized reduced jacobian method\n13\nProblem\nRef.\nr\nOV\nL\nNL\nProblem\nRef.\nr\nOV\nL\nNL\nABC comp\n[26]\n2\n2\n2\n1\nLAP1\n[5]\n2\n2\n1\n2\nBNH\n[8]\n2\n2\n0\n2\nLAP2\n[5]\n2\n30\n0\n1\nCF11\n[44]\n2\n3\n0\n2\nLIR-CMOP1\n[19]\n2\n30\n0\n2\nCPT1\n[9]\n2\n2\n0\n30\nLIR-CMOP2\n[19]\n2\n30\n0\n2\nDisc Brake\n[35]\n2\n4\n2\n3\nLIR-CMOP3\n[19]\n2\n30\n0\n3\nDTLZ0\n[11]\n3\n3\n0\n2\nliswetm\n[27]\n2\n7\n5\n0\nDTLZ9\n[11]\n3\n30\n0\n2\nMOLPg 001\n[38]\n3\n8\n8\n0\nEL3\n(21)\n2\n2\n0\n1\nMOLPg 002\n[38]\n3\n12\n13\n0\nex001\n[7]\n2\n5\n1\n2\nMOLPg 003\n[38]\n3\n10\n12\n0\nex002\n[41]\n2\n5\n0\n4\nOSY\n[8]\n2\n6\n4\n2\nex003\n[8]\n2\n2\n0\n2\nSRN\n[8]\n2\n2\n1\n1\nGE1\n[14]\n2\n2\n0\n1\nTamaki\n[35]\n3\n3\n0\n1\nGE4\n[14]\n3\n3\n0\n1\nTLK1\n[39]\n2\n2\n4\n0\nHanne4\n[6]\n2\n2\n0\n1\nTNK\n[8]\n2\n2\n0\n2\nhs05x\n[25]\n3\n5\n6\n0\nWelded Beam\n[35]\n2\n4\n1\n3\nTable 1: Linearly and nonlinearly constrained multicriteria test problems\nAll these problems were solved starting with the same population of 200 selected individuals.\nTo ensure that all compared solvers generate the same number of 200 final solutions, and since\nMATLAB’s NSGA-II solver may provide fewer than 200, we set “PopulationSize” option to 571\nfor NSGA-II, while all other NSGA-II options were left with the default settings.\nIn our implementation, the three codes GRJ, ZMO and MOSQP use the value 0.25 as Armijo’s\nconstant, and\nmin (Px) < 10−6\nas the stopping criterion, where Px denotes the direction-finding subproblem specific to each of\nthe three solvers. The latter was solved, for GRJ, by coding a standard RGM based on Wolfe’s\ncontinuous scheme (see, e.g., [18] for details), while for ZMO and MOSQP, it was solved using\nSCILAB’s predefined function “qld” for linear quadratic programming problems.\nStep 5 of the GRJ algorithm, which involves computing feasible Armijo’s steplengths, relies\non the results presented in Proposition 3.1. This step is carried out using the Newton method,\nfollowing the procedure outlined below:\nFeasible Armijo line search pseudocode\nStep 0: Initialization.\n- Fix a tolerance ε > 0 (e.g., ε = 10−6) and a maximum number of iterations L ∈N (e.g., L = 200).\n- Set t = tN, y1\nB = xB and l = 1.\nStep 1: Newton rule.\n- Set yl+1\nB\n= yl\nB −A−1\nB\n\u0010\nyl\nB, xN + tdN\n\u0011\nG\n\u0010\nyl\nB, xN + tdN\n\u0011\n.\nStep 2: Feasible Armijo test.\n- If\n\r\r\rG\n\u0010\nyl+1\nB , xN + tdN\n\u0011\r\r\r < ε,\naB ≤yl+1\nB\n≤bB,\nand\nF\n\u0010\nyl+1\nB , xN + tdN\n\u0011\n< F(x) + βtUNdN,\nthen STOP: the current implicit basic vector is set as ψ (xN + tdN) = yl+1\nB .\nStep 3: Update.\n- If l = L, set t = t\n2, y1\nB = xB and l = 1; otherwise, set l = l + 1.\n- Return to Step 1.\n\n14\nm. el maghri, y. elboulqe\n6.2\nPerformance measures and interpretation\nTo thoroughly analyse and compare the results obtained by the solvers, a set of performance met-\nrics, as suggested in the literature (see, e.g., [2]), may be employed. These metrics essentially assess\ntwo key aspects: the convergence of the approximate solutions towards the Pareto front and their\ndiversity. In this work, we have chosen three performance measures: Purity (P) [3], Spread (∆∗)\n[43] and Generational distance (GD) [40]. These three measures use, as a proxy for the true Pareto\nfront, the so-called reference Pareto front, defined as:\nFp =\n(\ny∗∈\n[\ns∈S\nFp,s : ∄y ∈\n[\ns∈S\nFp,s, y < y∗\n)\n,\nwhere Fp,s stands for the approximated Pareto front of problem p ∈P provided by method s ∈S;\nP denotes the set of tested problems and S the set of considered solvers.\n−Purity (P). This metric consists in measuring the proportion of points in Fp,s admitted in Fp:\nPp,s = |Fp,s ∩Fp|\n|Fp,s|\n.\nClearly Pp,s ∈[0, 1] and the extreme values are significant in the sense that a value Pp,s close\nto 1 indicates better performance, while a value equal to 0 implies that the solver is unable\nto generate any point of Fp.\n−Spread (∆∗). This metric measures the diversity and the dispersion of the points in Fp,s with\nrespect to Fp:\n∆∗\np,s =\nrP\nj=1\nd\n\u0000y∗\nj , Fp,s\n\u0001\n+ P\ny∈Fp\n\f\fd (y, Fp,s \\ {y}) −¯d\n\f\f\nrP\nj=1\nd\n\u0000y∗\nj , Fp,s\n\u0001\n+ |Fp| ¯d\n,\nwhere d(y, A) = mina∈A ∥y −a∥is the Euclidean distance from the vector y to the set A;\nhere, y∗\nj denotes the minimum value of the criterion fj in Fp, and ¯d represents the average\ndistance between each solution y ∈Fp and the set Fp,s \\ {y}. Note that lower values of ∆∗\np,s\nreflects more uniform distributions of the generated solutions.\n−Generational Distance (GD). Measuring the convergence, this metric represents how far Fp,s\nis from Fp:\nGDp,s =\n1\n|Fp,s|\ns X\ny∈Fp,s\nd2 (y, Fp).\nObviously, lower values of GDp,s are requested.\nTable 2 lists the numerical results obtained by the four considered methods.\nThe column\n‘CPU’ indicates the average execution time (in seconds). The best scores are in bold. To analyse\nthese results and look for significant differences between the compared solvers, the performance\nprofile is used (see, e.g., [12]). Recall that a profile is a graphical representation of the (cumulative)\ndistribution function ρ whose outputs are a solver’s scores on the set of test problems against a\nperformance metric. More precisely, given the measure mp,s (set here to 1/P, ∆∗, GD or CPU) by\nsolver s ∈S for solving p ∈P, the corresponding distribution function is given formally by\nρs = |{p ∈P : rp,s ≤α}|\n|P|\n,\nwhere rp,s = mp,s/ min {mp,s : s ∈S}. The purity measure mp,s is set to 1/P in order that all\nthe considered metrics exhibit the same asymptotic behaviour, in a sense that the smaller the\nmeasure, the better the solver.\nNote that at the threshold α = 1, ρs(α) gives us the largest\nnumber of problems among the best solved by s according to the analysed performance. However,\na value ρs(α) attaining 1 means that all the problems p ∈P have been solved by solver s at the\nthreshold α. Thus, the best overall performance of a solver is that which reaches the value 1 for\nthe smallest value of α.\n\ngeneralized reduced jacobian method\n15\nProblems\nGRJ\nZMO\nCPU\nP\n∆⋆\nGD\nCPU\nP\n∆⋆\nGD\nABC comp\n0.015935 0.99\n0.902345 0.0000076\n0.021454\n1\n0.953210\n0\nBNH\n1.436124\n0.915\n0.810000\n0.0000086\n0.004504 1\n0.995432\n0\nCF11\n0.415047\n0.885\n0.920000\n0\n0.058855\n0.61\n0.902345 0\nCPT1\n1.712818\n0.97\n0.765432 0.000003\n0.055902\n0.98\n0.895432\n0.000003\nDisc Brake\n1.242267\n0.69\n0.310123 0.0031797\n1.817904\n0.035\n0.860987\n0.004630\nDTLZ0\n0.410519\n0.805 0.705678\n0.000615\n0.000827 0\n0.765432\n0.001699\nDTLZ9\n0.209932 0.9\n0.254321\n0\n1.004320\n0.08\n0.989765\n0.000092\nEL3\n0.005883 1\n0.285432 0\n0.005889\n1\n0.925678\n0\nex001\n0.112501\n1\n0.415098\n0\n0.020554\n0.33\n0.350987 0.276242\nex002\n0.027517\n0.46\n0.785432 0.0027654\n0.010278 0\n0.825678\n0.042952\nex003\n0.044351\n0.535\n0.811023\n0.0003054\n0.038164 0.565 0.985432\n0.000682\nGE1\n0.018449\n0.995 0.870987\n0\n0.002708 0.995 0.990987\n0\nGE4\n0.048618\n0.985\n0.632098\n0.0000041\n0.037197\n0.99\n0.991098\n0.000004\nHanne4\n0.024853\n0.56\n0.707098 0.0004533\n0.030180\n0.46\n0.998098\n0.000614\nhs05x\n1.008168\n0.975\n0.980987\n0.0006024\n0.006076 0.01\n0.970987\n0.019143\nLAP1\n1.822010\n0.755\n0.505432 0.000108\n0.028570\n0.965 0.640987\n0.000128\nLAP2\n0.087737\n0.87\n0.180123 0.0058377\n0.082840\n1\n0.995432\n0\nLIR-CMOP1\n0.389043\n0.837 0.840987\n0.000219\n0.035736 0\n0.965098\n0.003135\nLIR-CMOP2\n0.135677 0.9\n0.750987 0.000038\n0.135736\n0.02\n0.997453\n0.003135\nLIR-CMOP3\n0.031187\n0.995 0.509871 0\n0.010200 0.62\n0.740987\n0.000065\nliswetm\n0.154443\n0.61\n0.150987 0.000048\n0.015055 0\n0.930987\n0.000815\nMOLPg 001\n0.184877 0.91\n0.955432\n0.000004\n0.550163\n0.81\n0.980987\n0.000005\nMOLPg 002\n1.200010\n0.625\n0.740987\n0.000758\n0.015298 0\n0.720987\n0.040700\nMOLPg 003\n1.005378\n0.99\n0.700987\n0\n0.042475\n0\n0.560987\n0.021736\nOSY\n1.539581\n0.47\n0.923444\n0.003442\n0.061214 0\n0.781341 0.019456\nSRN\n1.937231\n0.945 0.860987\n0\n0.010315 0.9\n0.940987\n0.006855\nTamaki\n1.541475\n0.83\n0.700987\n0\n0.002469 1\n0.530987 0\nTLK1\n0.013301 0.42\n0.840987 0.000101\n0.033947\n0.39\n0.950097\n0.000080\nTNK\n0.459437\n0.395 0.830987\n0.0003194\n0.013540 0.275\n0.760987 0.004407\nWelded Beam 1.137963\n0.91\n0.740987 0.0033562\n13.20645\n0\n0.960987\n1.226983\nProblems\nMOSQP\nNSGA-II\nCPU\nP\n∆⋆\nGD\nCPU\nP\n∆⋆\nGD\nABC comp\n0.020315\n0.99\n0.980987\n0.000017\n0.857500\n0.41\n0.970000\n0.000906\nBNH\n0.163576\n0.105\n0.992109\n0.010358\n0.013100\n0.84\n0.780000 0.001654\nCF11\n0.011990 0.59\n1.000000\n0\n0.021800\n1\n0.994567\n0\nCPT1\n0.017768 0.74\n0.900000\n0.000025\n0.191800\n1\n1.000000\n0\nDisc Brake\n0.021562\n0\n0.360000\n0.006729\n0.0084\n0.84\n0.840012\n0.001951\nDTLZ0\n0.014875\n0.495\n0.960000\n0.000528\n0.0094\n0.485\n0.610000 0.003421\nDTLZ9\n0.395620\n0.93\n0.840000\n0.005980\n0.3825431 0.08\n0.070981 0.000092\nEL3\n0.012636\n0\n0.659765\n0.010208\n0.0143\n0.99\n0.591220\n0.000028\nex001\n0.087176 0\n1.000000\n1.875443\n4.9669\n0\n0.991197\n0\nex002\n0.029159\n0.55\n0.967787\n0.001515\n0.038400\n0.085\n1.000000\n0.021188\nex003\n0.065472\n0.395\n0.793245 0.000364\n0.147100\n0\n0.985023\n0.000061\nGE1\n0.031198\n0.995 1.000000\n0\n0.010200\n0.62\n0.760000 0.000040\nGE4\n0.019889 0.97\n0.992501\n0\n0.01200\n0.87\n0.570000 0.000458\nHanne4\n0.022001\n0.355\n0.980000\n0.000308\n0.014400 0.775 0.900000\n0.000643\nhs05x\n0.056837\n1\n0.990000\n0\n3.059300\n0.941\n0.905401 0.010439\nLAP1\n0.023713\n0.255\n0.970350\n0.000203\n0.010900 0.78\n0.870000\n0.000176\nLAP2\n0.024061\n0.27\n0.790000\n0.006000\n0.016500 0.075\n0.960000\n0.059720\nLIR-CMOP1\n0.059540\n0.8\n0.806701 0.000053\n0.1824925 0.22\n1.000000\n0.000002\nLIR-CMOP2\n0.236534\n0.53\n0.950276\n0.000022\n2.1357355 0.1\n0.993204\n0.041562\nLIR-CMOP3\n0.323865\n0.76\n0.805467\n0.000376\n0.010200 0.62\n0.760000\n0.000065\nliswetm\n0.016368\n0.385\n1.000000\n0.000410\n1.119700\n0.515\n0.830000\n0.000244\nMOLPg 001\n0.1888712 0.96\n0.934124 0.000234\n0.210634\n0.9\n0.993204\n0\nMOLPg 002\n0.017967\n0.68\n0.995119\n0\n1.95200\n0.075\n0.703005 0.075590\nMOLPg 003\n0.032949 0.79\n0.968578\n0.001239\n1.693900\n0.19\n0.503204 0.042700\nOSY\n1.582346\n0.525 0.998123\n0.001645\n1.837200\n0.25\n0.934551\n0.001930\nSRN\n0.036814\n0.945 0.967634\n0\n2.393800\n0.685\n0.773497 0.004229\nTamaki\n0.02418\n0.9\n0.843457\n0\n0.017700\n0.99\n0.952078\n0.000007\nTLK1\n0.019948\n0.225\n0.996780\n0.000042\n0.959700\n0.985 0.990840\n0.000004\nTNK\n1.014764\n0.09\n1.000000\n0.000245\n0.97000\n0.33\n1.000000\n0.003962\nWelded Beam 0.024213 0.05\n0.999732\n0.178934\n0.0875\n0.97\n0.950000\n0.000621\nTable 2: Multiobjective performance measurements\n\n16\nm. el maghri, y. elboulqe\nAs illustrated in Fig. 1, the GRJ method consistently demonstrates good performance with\nrespect to the P metric, compared to the other methods. Indeed, the best value ρ(α) = 1 is achieved\nby GRJ at a relatively small threshold α, whereas the other methods fail to reach this value. This\nobservation is further supported by the results in Tab. 2, where the purity values reported for the\nZMO, MOSQP and NSGA-II methods are in some cases very close or equal to zero. This suggests\nthat these methods fail to reach the reference Pareto front for certain problems.\nIn terms of the ∆∗and GD metrics, the four methods appear to be generally comparable.\nAlthough NSGA-II is widely recognized for its effectiveness in spread, the observed profile indicates\ngood overall performance across all solvers, with a slight advantage noted for GRJ and NSGA-II.\nSimilar observations can be made regarding the GD metric, noting this time a slight advantage\nof the three methods over NSGA-II, which is well known for its limited convergence toward the\nPareto front.\nRegarding CPU time, as shown in Fig. 1 and Tab. 2, the three methods exhibit a slight speed\nadvantage over GRJ. This can be attributed to the inherent characteristics of the GRJ algorithm\nand its operational mechanism. More specifically, the algorithm’s iterative process and heuristic\nstrategies, such as the basis selection procedure in Step 1 and the line search procedure in Step 5\n(as previously described), can be time-consuming. Nevertheless, these components enable efficient\nexploration of the solution space, which helps to explain its performance differences relative to\nthe other methods. The graphical representation of the approximated Pareto fronts illustrated in\nFig. 2–3 also supports these interpretations. Note also that ZMO generally appears to be faster\nthan MOSQP based on the scores given in Tab. 2, except for the case where ZMO relatively took a\nlong time to solve the ’Welded Beam’ design problem. This may explain the influence of MOSQP’s\nCPU performance profile on ZMO observed in Fig. 1. However, as reported in Tab. 2, the execution\ntime remains generally very small and reasonable for all the considered solvers.\nFigure 1: Performance profiles\n\ngeneralized reduced jacobian method\n17\n6.3\nReal-world applications\nAs mentioned earlier, we conclude this section by providing additional details on the two real-world\nengineering problems: ‘Disc Brake’ and ‘Welded Beam’.\n6.3.1\n‘Disc Brake’ design problem\nThis problem consists to minimize simultaneously the mass of the brake and the stopping time.\nThe variables represent the inner radius of the disc, the outer radius, the engaging force, and the\nnumber of friction surfaces. The constraints for the design include a minimum distance between\nthe inner and outer radii, a maximum allowable brake length, and limitations related to pressure,\ntemperature, and torque. The problem is a nonlinearly constrained bicriteria problem (see [35] for\nfurther details), formally defined as follows:\n(Disc Brake)\nMinimize\n \n4.9 × 10−5 \u0000x2\n2 −x2\n1\n\u0001\n(x4 −1) , 9.82 × 106 \u0000x2\n2 −x2\n1\n\u0001\nx3x4 (x3\n2 −x3\n1)\n!\n,\nsubject to\n(x2 −x1) + 20 ≤0,\n2.5 (x4 + 1) −30 ≤0,\nx3\n3.14 (x2\n2 −x2\n1) −0.4 ≤0,\n2.22x3\n\u0000x3\n1 −x3\n2\n\u0001\n103 × (x2\n2 −x2\n1)2 −1 ≤0,\n2.66x3x4\n\u0000x3\n1 −x3\n2\n\u0001\n102 (x2\n2 −x2\n1)2\n+ 900 ≤0,\n55 ≤x1 ≤80,\n75 ≤x2 ≤110,\n103 ≤x3 ≤3 × 103,\n2 ≤x4 ≤20.\nFigure 2: Best Pareto front approximations for the ‘Disc Brake’ design problem\nby GRJ, ZMO, MOSQP and NSGA-II\n\n18\nm. el maghri, y. elboulqe\nThe graphical representation of the approximate Pareto fronts in Fig. 2 obtained by the four\nmethods highlights the superior ability of GRJ and NSGA-II to better explore the Pareto front\nfor the ‘Disc Brake’ problem, in contrast to ZMO and MOSQP, which face serious challenges.\nNevertheless, it is important to note that MOSQP slightly outperforms in this example ZMO, as\nit manages to find some relatively interesting solutions.\n6.3.2\n‘Welded Beam’ design problem\nThis problem involves minimizing both the cost of fabrication and the deflection of the beam under\nan applied load subject to some constraints. The two objectives are inherently conflicting, since\nreducing deflection generally leads to higher manufacturing costs, primarily due to setup, material,\nand welding labor costs. The design involves four decision variables and four nonlinear constraints:\nshear stress, normal stress, weld length, and buckling limitations (see [35] for more details). Its\nformal definition is given as follows:\n(Welded Beam)\nMinimize\n\u0012\n1.10471x2\n1x2 + 0.04811x3x4(14 + x3), 2.1952\nx3\n3x4\n\u0013\n,\nsubject to\nv\nu\nu\ntτ ′(x)2 + τ ′′(x)2 +\nx2τ ′(x)τ ′′(x)\nq\n0.25\n\u0000x2\n2 + (x1 + x3)2\u0001 ≤13600,\n1\nx3\n3x4\n≤\n3\n504, x3x3\n4(0.0282346x3 −1) ≤\n6 × 103\n64746.022,\nx1 ≤x4, 0.125 ≤x1, x4 ≤5,\n0.1 ≤x2, x3 ≤10,\nwhere τ ′(x) =\n6×103\n√\n2x1x2 and τ ′′(x) =\n3×103\n\u0010\n14+0.5x2√\n0.25(x2\n2+(x1+x3)2)\n\u0011\n0.707x1x2\n\u0010\nx2\n2\n12 +0.25(x1+x3)2\n\u0011\n.\nFigure 3: Best Pareto front approximations for the ‘Welded Beam’ design problem\nby GRJ, ZMO, MOSQP and NSGA-II\n\ngeneralized reduced jacobian method\n19\nIn the context of the ‘Welded Beam’ problem, as shown in Fig. 3, all methods succeed in\napproximating significant regions of the Pareto front. However, the resulting dispersions differ\nfrom one method to another, so that none can be objectively favoured in this regard. In terms of\nconvergence, it is also observed that the ZMO method is dominated by the other methods, which\nremain in strong competition with each other. This clearly demonstrates the effectiveness of our\nGRJ method in solving real-world problems. Moreover, the consistent performance of GRJ across\nvarious metrics highlights its robustness and reliability as a practical optimization approach.\nStatements and Declarations.\nThe authors have no pertinent declarations concerning conflicts of\ninterest, financial or non-financial interests, competing interests or other statements to disclose.\nReferences\n[1] Abadie, J., Carpentier, J.: Generalization of the Wolfe reduced gradient method to the case\nof nonlinear constraintes. In: Optimization (R. Fletcher, ed.) Academic Press, London and\nNew York, 37–47 (1969)\n[2] Audet, C., Bigeon, J., Cartier, D., et al.: Performance indicators in multiobjective optimiza-\ntion. Eur. J. Oper. 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McGraw-Hill, New York (1963)\n[43] Zhou, A., Jin, Y., Zhang, Q., Sendhoff, B., Tsang, E.:\nCombining model-based and\ngenetics-based offspring generation for multi-objective optimization using a convergence\ncriterion. In IEEE international conference on evolutionary computation, 892–899 (2006)\nhttps://doi.org/10.1109/CEC.2006.1688406\n[44] Zhou, Y., Xiang, Y., He, X.: Constrained multiobjective optimization: Test problem con-\nstruction and performance evaluations. IEEE Transactions on Evolutionary Computation, 25,\n172–186 (2021)\n"},"cleaned":{"kind":"string","value":"Generalized Reduced Jacobian Method M. El Maghri∗1 and Y. Elboulqe†2 1 ̈ın Chock, Hassan II University, Casablanca, BP. 5366, Morocco 2 . 2390, Morocco October 17, 2025 Abstract In a recent work, we presented the reduced Jacobian method (RJM) as an extension of Wolfe's reduced gradient method to multicriteria (multiobjective) optimization problems dealing with linear constraints. This approach reveals that using a reduction technique of the Jacobian matrix of the objective avoids scalarization. In the present work, we intend to generalize RJM to handle nonlinear constraints too. In fact, we propose a generalized reduced Jacobian (GRJ) method that extends Abadie-Carpentier's approach for single-objective programs. To this end, we adopt a global reduction strategy based on the fundamental theorem of implicit functions. In this perspective, only a reduced descent direction common to all the criteria is computed by solving a simple convex program. After establishing an Armijo-type line search condition that ensures feasibility, the resulting algorithm is shown to be globally convergent, under mild assumptions, to a Pareto critical (KKT-stationary) point. Finally, experimental results are presented, including comparisons with other deterministic and evolutionary approaches. Keywords Multicriteria optimization. Pareto optima. Nonlinear programming. Pareto KKTstationarity. Reduced Jacobian method. Nonlinear constraints. Mathematics Subject Classification (2020) 90C29 · 90C30 · 90C52 1 Introduction In many areas, like economics, medicine, design, transportation and so on, we are faced with multicriteria (or multiobjective) optimization problems (MOP), where several functions must be optimized at the same time. Because of the conflicts which may arise between these functions, almost never a single point will minimize or maximize them at once. So the Pareto optimality concept have to be considered. A point is said to be Pareto optimum or efficient solution, if it cannot be improved with respect to a criterion without degrading at least one of the others. Taking account the principle, Pareto solutions are incomparable and no single point can represent all the others. The problem then consists of determining the set of all Pareto solutions or their objective values called the Pareto front or more generally the set of non-dominated solutions. In recent years, it has been observed that this task has become more tractable with the advent of a new generation of algorithms grounded in the principle of multiobjective search descent directions, which aim to identify a direction that decreases all the criteria simultaneously (e.g. [5, 13, 16, 18, 20, 21, 23, 29, 31, 32, 33, 34]). In most cases, the resulting algorithms are merely direct extensions of well-known nonlinear descent methods, which tackle the given problem without resorting to intermediate transformations or introducing artificial parameters that may be sensitive to the ∗Corresponding author. Email: †Email: 1 16 Oct 2025 2 m. el maghri, y. elboulqe original problem. From a theoretical perspective, convergence toward Pareto-KKT stationarity is well established, and numerically, these approaches have also proven very promising, both in terms of convergence to the true Pareto front and the diversity of non-dominated approximate solutions. For instance, in [16], we proposed a direct extension of the Wolfe reduced gradient method (RGM) [42] (see also [15] for a new RGM variant) to the so-called reduced Jacobian method (RJM) for solving linearly constrained MOPs. We also refer the reader to [18], where we introduced an RJM variant that possesses the full convergence property. In the present study, we build upon our previous work by focusing on the so-called generalized reduced gradient (GRG) method, originally introduced by Abadie and Carpentier [1], which extends Wolfe's reduction strategy to problems with nonlinear constraints. Accordingly, we propose a novel approach, which we term the generalized reduced Jacobian (GRJ) method, to effectively handle nonlinear constraints in multicriteria optimization. The GRG method is widely recognized as one of the fundamental and practical approaches to nonlinear mathematical programming. Its convergence properties were rigorously established in subsequent works by Smeers [37], MokhtarKharroubi [30], and Schittkowski [36]. As far as we are concerned, our investigations cover both the theoretical and algorithmic aspects of this method. Abadie-Carpentier's reduction strategy, based on the well-known implicit function theorem, is adopted in this work, thereby enabling a complete characterization of Pareto-KKT stationarity through multiobjective reduced descent directions. It is subsequently proven that such directions guarantee simultaneous descent of all criteria and ensure the existence of Armijo-like steplengths, while preserving feasibility throughout the process. From a computational point of view, a simple direction-finding subproblem is introduced to iteratively compute the reduced directions. Under mild assumptions, the resulting GRJ algorithm is proven to be globally convergent to a Pareto KKT-stationary point in the sense of accumulation points. Finally, numerical experiments are reported, illustrating the effectiveness of the GRJ method compared with other methods, including a Zoutendijk-like method [31], an SQP-type approach [22], and the well-known evolutionary method NSGA-II [10]. 2 Efficiency and multiobjective descent We are dealing with the following nonlinearly constrained multicriteria optimization problem: (MOP) Min F(x) subject to G(x) = 0, a ≤x ≤b, where F, G : x ∈Rn 7→ F(x), G(x) = f1(x), . . . , fr(x) , g1(x), . . . , gm(x) ∈Rr × Rm are two vector-valued functions continuously differentiable on the feasible set denoted by S, and, a = (a1, . . . , an), b = (b1, . . . , bn) ∈Rn. For y = (y1, . . . , yp) ∈Rp and y′ = (y′ 1, . . . , y′ p) ∈Rp, y ≤y′ (resp. y 0, ∀t ∈]0, tf] , x + td ∈S; (b) a tangent direction to S at x ∈S, if ∃(dk)k ⊂Rn, ∃(tk)k ⊂]0, +∞[ such that lim k→+∞dk = d, lim k→+∞tk = 0 and x + tkdk ∈S (∀k); (c) a descent direction of F at x ∈S, if ∃td > 0, ∀t ∈]0, td] , F(x + td) 0, i ∈N . (9) Then, (i) tN > 0. (ii) t ≤tN ⇐⇒aN ≤xN + tdN ≤bN. If furthermore aB 0; otherwise, tN = (bi -xi)/di, which in accordance with (8), entails that bi -xi > 0, thereby ensuring tN > 0. (ii) Let us show the first implication. Let i ∈N. The assertion is obvious if di = 0. Now, if di > 0, then ai ≤xi + tdi and (bi -xi)/di ≥tN ≥t, so that bi ≥xi + tdi. The case di 0, then the hypothesis xi + tdi ≤bi implies that t ≤(bi -xi)/di, which shows that t ≤tN. (iii) By definition of ψ : W -→V (see (6)) and the continuity of t 7-→xN + tdN at 0, there exists t1 > 0 such that xN + tdN ∈W for all t ∈[0, t1]. It follows by (6) that ∀t ∈[0, t1] , G (ψ(xN + tdN), xN + tdN) = 0. On the other hand, by assertion (ii), we have that ∀t ∈[0, tN] , aN ≤xN + tdN ≤bN. generalized reduced jacobian method 7 So, it remains to show that the box-constraint also holds for the basic variables. Indeed, by nondegeneracy assumption and continuity, it is immediate that ∃t2 > 0, ∀t ∈[0, t2] , aB 0, -φ(bi -xi) UN(x)T λ(x) i , else. (13) The lemma below characterizes the optimal solutions of (Px), which will be crucial not only for proving the descent properties of the vector dN, but also for the convergence analysis of the resulting algorithm. Its proof is omitted, as it is similar to the one of [16, Lemma 3.2]. Lemma 4.1 λ(x) ∈Λ is an optimal solution of (Px), iff δN(λ(x), x) · UN(x)T λ ≤-2f (λ(x), x) (∀λ ∈Λ), or equivalently, δN(λ(x), x) · uj N(x) ≤-2f (λ(x), x) (j = 1, . . . , r). The descent properties are stated as follows: Proposition 4.1 (i) dN = 0 ⇐⇒f(λ(x), x) = 0. (ii) dN ̸= 0 =⇒dN is a multiobjective reduced descent direction of F at x. If moreover (ACQ) holds and x is nondegenerate, then x is not Pareto KKT-stationary for (MOP). (iii) dN = 0 =⇒x is Pareto KKT-stationary for (MOP). (iv) The functional x 7→f(λ(x), x) is continuous on S provided φ is continuous. Proof (i) To show the direct implication, observe that as φ ≥0, then dN = 0 implies that, ∀i ∈N, j φ (bi -xi) UN(x)T λ(x) i k -= j φ (xi -ai) UN(x)T λ(x) i k +. If we suppose that the last two terms are not equal to zero, we would have φ (bi -xi) UN(x)T λ(x) i 0, which would mean that φ (bi -xi) and φ (xi -ai) have not the same sign contradicting φ ≥0. Thus, ∀i ∈N, j φ (bi -xi) UN(x)T λ(x) i k -= j φ (xi -ai) UN(x)T λ(x) i k + = 0, and this shows that f(λ(x), x) = 0. The reverse implication follows straightforwardly from the positivity of φ and (13). (ii) If dN ̸= 0, then by assertion (i), f(λ(x), x) > 0. It follows, by Lemma 4.1, that dN · uj N(x) ≤-2f(λ(x), x) 0, where tk N := min ai -xk i dk i : dk i 0, i ∈N . Proof (a) Let λ∗= lim k∈K k→+∞ λ(xk), then λ∗∈Λ since λ(xk) k∈K ⊂Λ which is closed. According to the hypothesis and Lemma 4.1, for all k ∈K, we have δN(λ(xk), xk) · uj N(xk) ≤-2f(λ(xk), xk) (j = 1, . . . , r), Taking into account that the mappings (λ, x) 7-→f(λ, x) and (λ, x) 7-→δN(λ, x) are continuous, by passing onto the limit, when k ↗+∞in the previous inequality, we obtain that δN(λ∗, x∗) · uj N(x∗) ≤-2f(λ∗, x∗) (j = 1, . . . , r), Since by closeness of S, x∗remains feasible as (xk)k, this shows applying once again Lemma 4.1 that λ∗∈argmin(Px∗), i.e., λ∗= λ(x∗), which proves the assertion. (b) To prove this assertion, we shall proceed by a contradiction way by assuming that ̄t = 0 (because ̄t ≥0). Then, it would exist K0 ⊆K an infinite subset such that tk N -→ k∈K0 0. Since the 3As pointed out in [18], the hypothesis (H) was inadvertently omitted in the proof of [16, Theorem 4.1]. generalized reduced jacobian method 11 index set N is finite, we can assume (without loss of generality) that one of the two following cases would happen: ∃i0 ∈N, ∀k ∈K0, tk N = ai0 -xk i0 dk i0 , dk i0 0. In the first case, following (13), we would have ∀k ∈K0, tk N = 1 (UN(xk)T λ(xk))i0 \" xk i0 -ai0 φ(xk i0 -ai0) # . (14) Now, by assertion (a), dk i0 := -φ(xk i0 -ai0) UN(xk)T λ(xk) i0 -→ k∈K0 d∗ i0, and, by the contradiction hypothesis, tk N -→ k∈K0 0. So we would have that the sequence xk i0 -ai0 k∈K0 also converges to 0; i.e., xk i0 -→ k∈K0 ai0. But this would imply that φ(xk i0 -ai0) -→ k∈K0 φ(x∗ i0 -ai0) = φ(0) = 0, and, by letting k ↗+∞in (14), we would obtain the contradiction tk N -→ k∈K0 1 (UN(x∗)T λ∗)i0 × 1 φ′(0) ̸= 0, The second case is similar. Hence, ̄t > 0. □ Let us go back to the proof of Theorem 5.1(ii). By hypothesis, there exists a subsequence (xk)k∈K -→x∗with K ⊆N an infinite subset. By closeness of S, x∗remains feasible as (xk)k. Now, by continuity of F and the assertion (i), it follows that the sequence F(xk) k∈N -→F(x∗). On the other hand, since the index set Bk is in the finite set {1, . . . , n}, we can assume (without loss of generality) that Bk = B (hence Nk = N) for all k ∈K. Now, by Armijo's condition: ∀k ∈N, F(xk+1) -F(xk) 0 (e.g., ε = 10-6) and a maximum number of iterations L ∈N (e.g., L = 200). - Set t = tN, y1 B = xB and l = 1. Step 1: Newton rule. - Set yl+1 B = yl B -A-1 B yl B, xN + tdN G yl B, xN + tdN . Step 2: Feasible Armijo test. - If G yl+1 B , xN + tdN < ε, aB ≤yl+1 B ≤bB, and F yl+1 B , xN + tdN < F(x) + βtUNdN, then STOP: the current implicit basic vector is set as ψ (xN + tdN) = yl+1 B . Step 3: Update. - If l = L, set t = t 2, y1 B = xB and l = 1; otherwise, set l = l + 1. - Return to Step 1. 14 m. el maghri, y. elboulqe 6.2 Performance measures and interpretation To thoroughly analyse and compare the results obtained by the solvers, a set of performance metrics, as suggested in the literature (see, e.g., [2]), may be employed. These metrics essentially assess two key aspects: the convergence of the approximate solutions towards the Pareto front and their diversity. In this work, we have chosen three performance measures: Purity (P) [3], Spread (∆∗) [43] and Generational distance (GD) [40]. These three measures use, as a proxy for the true Pareto front, the so-called reference Pareto front, defined as: Fp = ( y∗∈ [ s∈S Fp,s : ∄y ∈ [ s∈S Fp,s, y < y∗ ) , where Fp,s stands for the approximated Pareto front of problem p ∈P provided by method s ∈S; P denotes the set of tested problems and S the set of considered solvers. -Purity (P). This metric consists in measuring the proportion of points in Fp,s admitted in Fp: Pp,s = |Fp,s ∩Fp| |Fp,s| . Clearly Pp,s ∈[0, 1] and the extreme values are significant in the sense that a value Pp,s close to 1 indicates better performance, while a value equal to 0 implies that the solver is unable to generate any point of Fp. -Spread (∆∗). This metric measures the diversity and the dispersion of the points in Fp,s with respect to Fp: ∆∗ p,s = rP j=1 d y∗ j , Fp,s + P y∈Fp d (y, Fp,s \\ {y}) - ̄d rP j=1 d y∗ j , Fp,s + |Fp| ̄d , where d(y, A) = mina∈A ∥y -a∥is the Euclidean distance from the vector y to the set A; here, y∗ j denotes the minimum value of the criterion fj in Fp, and ̄d represents the average distance between each solution y ∈Fp and the set Fp,s \\ {y}. Note that lower values of ∆∗ p,s reflects more uniform distributions of the generated solutions. -Generational Distance (GD). Measuring the convergence, this metric represents how far Fp,s is from Fp: GDp,s = 1 |Fp,s| s X y∈Fp,s d2 (y, Fp). Obviously, lower values of GDp,s are requested. Table 2 lists the numerical results obtained by the four considered methods. The column 'CPU' indicates the average execution time (in seconds). The best scores are in bold. To analyse these results and look for significant differences between the compared solvers, the performance profile is used (see, e.g., [12]). Recall that a profile is a graphical representation of the (cumulative) distribution function ρ whose outputs are a solver's scores on the set of test problems against a performance metric. More precisely, given the measure mp,s (set here to 1/P, ∆∗, GD or CPU) by solver s ∈S for solving p ∈P, the corresponding distribution function is given formally by ρs = |{p ∈P : rp,s ≤α}| |P| , where rp,s = mp,s/ min {mp,s : s ∈S}. The purity measure mp,s is set to 1/P in order that all the considered metrics exhibit the same asymptotic behaviour, in a sense that the smaller the measure, the better the solver. Note that at the threshold α = 1, ρs(α) gives us the largest number of problems among the best solved by s according to the analysed performance. However, a value ρs(α) attaining 1 means that all the problems p ∈P have been solved by solver s at the threshold α. Thus, the best overall performance of a solver is that which reaches the value 1 for the smallest value of α. generalized reduced jacobian method 15 Problems GRJ ZMO CPU P ∆⋆ GD CPU P ∆⋆ GD ABC comp 0.015935 0.99 0.902345 0.0000076 0.021454 1 0.953210 0 BNH 1.436124 0.915 0.810000 0.0000086 0.004504 1 0.995432 0 CF11 0.415047 0.885 0.920000 0 0.058855 0.61 0.902345 0 CPT1 1.712818 0.97 0.765432 0.000003 0.055902 0.98 0.895432 0.000003 Disc Brake 1.242267 0.69 0.310123 0.0031797 1.817904 0.035 0.860987 0.004630 DTLZ0 0.410519 0.805 0.705678 0.000615 0.000827 0 0.765432 0.001699 DTLZ9 0.209932 0.9 0.254321 0 1.004320 0.08 0.989765 0.000092 EL3 0.005883 1 0.285432 0 0.005889 1 0.925678 0 ex001 0.112501 1 0.415098 0 0.020554 0.33 0.350987 0.276242 ex002 0.027517 0.46 0.785432 0.0027654 0.010278 0 0.825678 0.042952 ex003 0.044351 0.535 0.811023 0.0003054 0.038164 0.565 0.985432 0.000682 GE1 0.018449 0.995 0.870987 0 0.002708 0.995 0.990987 0 GE4 0.048618 0.985 0.632098 0.0000041 0.037197 0.99 0.991098 0.000004 Hanne4 0.024853 0.56 0.707098 0.0004533 0.030180 0.46 0.998098 0.000614 hs05x 1.008168 0.975 0.980987 0.0006024 0.006076 0.01 0.970987 0.019143 LAP1 1.822010 0.755 0.505432 0.000108 0.028570 0.965 0.640987 0.000128 LAP2 0.087737 0.87 0.180123 0.0058377 0.082840 1 0.995432 0 LIR-CMOP1 0.389043 0.837 0.840987 0.000219 0.035736 0 0.965098 0.003135 LIR-CMOP2 0.135677 0.9 0.750987 0.000038 0.135736 0.02 0.997453 0.003135 LIR-CMOP3 0.031187 0.995 0.509871 0 0.010200 0.62 0.740987 0.000065 liswetm 0.154443 0.61 0.150987 0.000048 0.015055 0 0.930987 0.000815 MOLPg 001 0.184877 0.91 0.955432 0.000004 0.550163 0.81 0.980987 0.000005 MOLPg 002 1.200010 0.625 0.740987 0.000758 0.015298 0 0.720987 0.040700 MOLPg 003 1.005378 0.99 0.700987 0 0.042475 0 0.560987 0.021736 OSY 1.539581 0.47 0.923444 0.003442 0.061214 0 0.781341 0.019456 SRN 1.937231 0.945 0.860987 0 0.010315 0.9 0.940987 0.006855 Tamaki 1.541475 0.83 0.700987 0 0.002469 1 0.530987 0 TLK1 0.013301 0.42 0.840987 0.000101 0.033947 0.39 0.950097 0.000080 TNK 0.459437 0.395 0.830987 0.0003194 0.013540 0.275 0.760987 0.004407 Welded Beam 1.137963 0.91 0.740987 0.0033562 13.20645 0 0.960987 1.226983 Problems MOSQP NSGA-II CPU P ∆⋆ GD CPU P ∆⋆ GD ABC comp 0.020315 0.99 0.980987 0.000017 0.857500 0.41 0.970000 0.000906 BNH 0.163576 0.105 0.992109 0.010358 0.013100 0.84 0.780000 0.001654 CF11 0.011990 0.59 1.000000 0 0.021800 1 0.994567 0 CPT1 0.017768 0.74 0.900000 0.000025 0.191800 1 1.000000 0 Disc Brake 0.021562 0 0.360000 0.006729 0.0084 0.84 0.840012 0.001951 DTLZ0 0.014875 0.495 0.960000 0.000528 0.0094 0.485 0.610000 0.003421 DTLZ9 0.395620 0.93 0.840000 0.005980 0.3825431 0.08 0.070981 0.000092 EL3 0.012636 0 0.659765 0.010208 0.0143 0.99 0.591220 0.000028 ex001 0.087176 0 1.000000 1.875443 4.9669 0 0.991197 0 ex002 0.029159 0.55 0.967787 0.001515 0.038400 0.085 1.000000 0.021188 ex003 0.065472 0.395 0.793245 0.000364 0.147100 0 0.985023 0.000061 GE1 0.031198 0.995 1.000000 0 0.010200 0.62 0.760000 0.000040 GE4 0.019889 0.97 0.992501 0 0.01200 0.87 0.570000 0.000458 Hanne4 0.022001 0.355 0.980000 0.000308 0.014400 0.775 0.900000 0.000643 hs05x 0.056837 1 0.990000 0 3.059300 0.941 0.905401 0.010439 LAP1 0.023713 0.255 0.970350 0.000203 0.010900 0.78 0.870000 0.000176 LAP2 0.024061 0.27 0.790000 0.006000 0.016500 0.075 0.960000 0.059720 LIR-CMOP1 0.059540 0.8 0.806701 0.000053 0.1824925 0.22 1.000000 0.000002 LIR-CMOP2 0.236534 0.53 0.950276 0.000022 2.1357355 0.1 0.993204 0.041562 LIR-CMOP3 0.323865 0.76 0.805467 0.000376 0.010200 0.62 0.760000 0.000065 liswetm 0.016368 0.385 1.000000 0.000410 1.119700 0.515 0.830000 0.000244 MOLPg 001 0.1888712 0.96 0.934124 0.000234 0.210634 0.9 0.993204 0 MOLPg 002 0.017967 0.68 0.995119 0 1.95200 0.075 0.703005 0.075590 MOLPg 003 0.032949 0.79 0.968578 0.001239 1.693900 0.19 0.503204 0.042700 OSY 1.582346 0.525 0.998123 0.001645 1.837200 0.25 0.934551 0.001930 SRN 0.036814 0.945 0.967634 0 2.393800 0.685 0.773497 0.004229 Tamaki 0.02418 0.9 0.843457 0 0.017700 0.99 0.952078 0.000007 TLK1 0.019948 0.225 0.996780 0.000042 0.959700 0.985 0.990840 0.000004 TNK 1.014764 0.09 1.000000 0.000245 0.97000 0.33 1.000000 0.003962 Welded Beam 0.024213 0.05 0.999732 0.178934 0.0875 0.97 0.950000 0.000621 Table 2: Multiobjective performance measurements 16 m. el maghri, y. elboulqe As illustrated in Fig. 1, the GRJ method consistently demonstrates good performance with respect to the P metric, compared to the other methods. Indeed, the best value ρ(α) = 1 is achieved by GRJ at a relatively small threshold α, whereas the other methods fail to reach this value. This observation is further supported by the results in Tab. 2, where the purity values reported for the ZMO, MOSQP and NSGA-II methods are in some cases very close or equal to zero. This suggests that these methods fail to reach the reference Pareto front for certain problems. In terms of the ∆∗and GD metrics, the four methods appear to be generally comparable. Although NSGA-II is widely recognized for its effectiveness in spread, the observed profile indicates good overall performance across all solvers, with a slight advantage noted for GRJ and NSGA-II. Similar observations can be made regarding the GD metric, noting this time a slight advantage of the three methods over NSGA-II, which is well known for its limited convergence toward the Pareto front. Regarding CPU time, as shown in Fig. 1 and Tab. 2, the three methods exhibit a slight speed advantage over GRJ. This can be attributed to the inherent characteristics of the GRJ algorithm and its operational mechanism. More specifically, the algorithm's iterative process and heuristic strategies, such as the basis selection procedure in Step 1 and the line search procedure in Step 5 (as previously described), can be time-consuming. Nevertheless, these components enable efficient exploration of the solution space, which helps to explain its performance differences relative to the other methods. The graphical representation of the approximated Pareto fronts illustrated in Fig. 2-3 also supports these interpretations. Note also that ZMO generally appears to be faster than MOSQP based on the scores given in Tab. 2, except for the case where ZMO relatively took a long time to solve the 'Welded Beam' design problem. This may explain the influence of MOSQP's CPU performance profile on ZMO observed in Fig. 1. However, as reported in Tab. 2, the execution time remains generally very small and reasonable for all the considered solvers. Figure 1: Performance profiles generalized reduced jacobian method 17 6.3 Real-world applications As mentioned earlier, we conclude this section by providing additional details on the two real-world engineering problems: 'Disc Brake' and 'Welded Beam'. 6.3.1 'Disc Brake' design problem This problem consists to minimize simultaneously the mass of the brake and the stopping time. The variables represent the inner radius of the disc, the outer radius, the engaging force, and the number of friction surfaces. The constraints for the design include a minimum distance between the inner and outer radii, a maximum allowable brake length, and limitations related to pressure, temperature, and torque. The problem is a nonlinearly constrained bicriteria problem (see [35] for further details), formally defined as follows: (Disc Brake) Minimize 4.9 × 10-5 x2 2 -x2 1 (x4 -1) , 9.82 × 106 x2 2 -x2 1 x3x4 (x3 2 -x3 1) ! , subject to (x2 -x1) + 20 ≤0, 2.5 (x4 + 1) -30 ≤0, x3 3.14 (x2 2 -x2 1) -0.4 ≤0, 2.22x3 x3 1 -x3 2 103 × (x2 2 -x2 1)2 -1 ≤0, 2.66x3x4 x3 1 -x3 2 102 (x2 2 -x2 1)2 + 900 ≤0, 55 ≤x1 ≤80, 75 ≤x2 ≤110, 103 ≤x3 ≤3 × 103, 2 ≤x4 ≤20. Figure 2: Best Pareto front approximations for the 'Disc Brake' design problem by GRJ, ZMO, MOSQP and NSGA-II 18 m. el maghri, y. elboulqe The graphical representation of the approximate Pareto fronts in Fig. 2 obtained by the four methods highlights the superior ability of GRJ and NSGA-II to better explore the Pareto front for the 'Disc Brake' problem, in contrast to ZMO and MOSQP, which face serious challenges. Nevertheless, it is important to note that MOSQP slightly outperforms in this example ZMO, as it manages to find some relatively interesting solutions. 6.3.2 'Welded Beam' design problem This problem involves minimizing both the cost of fabrication and the deflection of the beam under an applied load subject to some constraints. The two objectives are inherently conflicting, since reducing deflection generally leads to higher manufacturing costs, primarily due to setup, material, and welding labor costs. The design involves four decision variables and four nonlinear constraints: shear stress, normal stress, weld length, and buckling limitations (see [35] for more details). Its formal definition is given as follows: (Welded Beam) Minimize 1.10471x2 1x2 + 0.04811x3x4(14 + x3), 2.1952 x3 3x4 , subject to v u u tτ ′(x)2 + τ ′′(x)2 + x2τ ′(x)τ ′′(x) q 0.25 x2 2 + (x1 + x3)2 ≤13600, 1 x3 3x4 ≤ 3 504, x3x3 4(0.0282346x3 -1) ≤ 6 × 103 64746.022, x1 ≤x4, 0.125 ≤x1, x4 ≤5, 0.1 ≤x2, x3 ≤10, where τ ′(x) = 6×103 √ 2x1x2 and τ ′′(x) = 3×103 14+0.5x2√ 0.25(x2 2+(x1+x3)2) 0.707x1x2 x2 2 12 +0.25(x1+x3)2 . Figure 3: Best Pareto front approximations for the 'Welded Beam' design problem by GRJ, ZMO, MOSQP and NSGA-II generalized reduced jacobian method 19 In the context of the 'Welded Beam' problem, as shown in Fig. 3, all methods succeed in approximating significant regions of the Pareto front. However, the resulting dispersions differ from one method to another, so that none can be objectively favoured in this regard. In terms of convergence, it is also observed that the ZMO method is dominated by the other methods, which remain in strong competition with each other. This clearly demonstrates the effectiveness of our GRJ method in solving real-world problems. Moreover, the consistent performance of GRJ across various metrics highlights its robustness and reliability as a practical optimization approach. Statements and Declarations. The authors have no pertinent declarations concerning conflicts of interest, financial or non-financial interests, competing interests or other statements to disclose. References [1] Abadie, J., Carpentier, J.: Generalization of the Wolfe reduced gradient method to the case of nonlinear constraintes. In: Optimization (R. Fletcher, ed.) Academic Press, London and New York, 37-47 (1969) [2] Audet, C., Bigeon, J., Cartier, D., et al.: Performance indicators in multiobjective optimization. Eur. J. Oper. Res. 292, 397-422 (2021) [3] Bandyopadhyay, S., Pal, S.K., Aruna, B.: Multiobjective GAs, quantitative indices, and pattern classification. IEEE Trans . Syst. Man Cyber. 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In IEEE international conference on evolutionary computation, 892-899 (2006) https://doi.org/10.1109/CEC.2006.1688406 [44] Zhou, Y., Xiang, Y., He, X.: Constrained multiobjective optimization: Test problem construction and performance evaluations. IEEE Transactions on Evolutionary Computation, 25, 172-186 (2021)"}}},{"rowIdx":198,"cells":{"id":{"kind":"string","value":"2510.14780"},"raw":{"kind":"string","value":"Causal Discovery for Linear DAGs with Dependent\nLatent Variables via Higher-order Cumulants\nMing Caia, Penggang Gaoa, Hisayuki Haraa,b\naGraduate School of Informatics, Kyoto University, Yoshida\nKonoe-cho, Kyoto, 606-8501, Japan\nbInstitute for Liberal Arts and Sciences, Kyoto University, Yoshida\nNihonmatsu-cho, Kyoto, 606-8501, Japan\nAbstract\nThis paper addresses the problem of estimating causal directed acyclic graphs\nin linear non-Gaussian acyclic models with latent confounders (LvLiNGAM).\nExisting methods assume mutually independent latent confounders or cannot\nproperly handle models with causal relationships among observed variables.\nWe propose a novel algorithm that identifies causal DAGs in LvLiNGAM,\nallowing causal structures among latent variables, among observed variables,\nand between the two. The proposed method leverages higher-order cumu-\nlants of observed data to identify the causal structure. Extensive simulations\nand experiments with real-world data demonstrate the validity and practical\nutility of the proposed algorithm.\nKeywords:\ncanonical model, causal discovery, cumulants, DAG, latent\nconfounder, Triad constraints\n1. Introduction\nEstimating causal directed acyclic graphs (DAGs) in the presence of la-\ntent confounders has been a major challenge in causal analysis. Conven-\ntional causal discovery methods, such as the Peter–Clark (PC) algorithm [1],\nGreedy Equivalence Search (GES) [2], and the Linear Non-Gaussian Acyclic\nModel (LiNGAM) [3, 4], focus solely on the causal model without latent\nconfounders.\nFast Causal Inference (FCI) [1] extends the PC algorithm to handle latent\nvariables, recovering a partial ancestral graph (PAG) under the faithfulness\nassumption. However, FCI is computationally intensive and, moreover, often\narXiv:2510.14780v1 [cs.LG] 16 Oct 2025\n\nfails to determine the causal directions. Really Fast Causal Inference (RFCI)\n[5] trades off some independence tests for speed, at the loss of estimating\naccuracy. Greedy Fast Causal Inference (GFCI) [6] hybridizes GES and FCI\nbut inherits the limitation of FCI.\nThe assumption of linearity and non-Gaussian disturbances in the causal\nmodel enables the identification of causal structures beyond the PAG. The\nlinear non-Gaussian acyclic model with latent confounders (LvLiNGAM) is\nan extension of LiNGAM that incorporates latent confounders. Hoyer et al.\n[7] demonstrated that LvLiNGAM can be transformed into a canonical model\nin which all latent variables are mutually independent and causally precede\nthe observed variables. They proposed estimating the canonical models using\novercomplete ICA [8], assuming that the number of latent variables is known.\nOvercomplete ICA can identify the causal DAG only up to permutations and\nscaling of the variables. Thus, substantial computational effort is required\nto identify the true causal DAG from the many candidate models. Another\nlimitation of overcomplete ICA is its tendency to converge to local optima.\nSalehkaleybar et al. [9] improved the algorithm by reducing the candidate\nmodels.\nOther methods for estimating LvLiNGAM, based on linear regression\nanalysis and independence testing, have also been developed.[10, 11, 12, 13].\nFurthermore, Multiple Latent Confounders LiNGAM (ML-CLiNGAM) [14]\nand FRITL [15] initially identify the causal skeleton using a constraint-based\nmethod, and then estimate the causal directions of the undirected edges in the\nskeleton using linear regression and independence tests. While these methods\ncan identify structures among observed variables that are not confounded\nby latent variables, they cannot necessarily determine the causal direction\nbetween two variables confounded by latent variables.\nMore recently, methods using higher-order cumulants have led to new\ndevelopments in the identification of canonical LvLiNGAMs. Cai et al. [16]\nassume that each latent variable has at least three observed children, and that\nthere exists a subset of these children that are not connected by any other\nobserved or latent variables. Then, cumulants are employed to identify one-\nlatent-component structures and latent influences are recursively removed to\nrecover the underlying causal relationships. Chen et al. [17] show that if\ntwo observed variables share one latent confounder, the causal direction be-\ntween them can be identified by leveraging higher-order cumulants. Schkoda\net al. [18] introduced ReLVLiNGAM, a recursive approach that leverages\nhigher-order cumulants to estimate canonical LvLiNGAM with multiple la-\n2\n\ntent parents. One strength of ReLVLiNGAM is that it does not require prior\nknowledge of the number of latent variables.\nThe methods reviewed so far are estimation methods for the canonical\nLvLiNGAM. A few methods, however, have been proposed to estimate the\ncausal DAG of LvLiNGAM when latent variables exhibit causal relation-\nships.\nA variable is said to be pure if it is conditionally independent of\nother observed variables given its latent parents; otherwise, it is called im-\npure.\nSilva et al.\n[19] showed that the latent DAG is identifiable under\nthe assumption that each latent variable has at least three pure children,\nby employing tetrad conditions on the covariance of the observed variables.\nCai et al. [20] proposed a two-phase algorithm, LSTC (learning the struc-\nture of latent variables based on Triad Constraints), to identify the causal\nDAG where each latent variable has at least two children, all of which are\npure, and each observed variable has a single latent parent. Xie et al. [21]\ngeneralized LSTC and defined the linear non-Gaussian latent variable model\n(LiNGLaM), where observed variables may have multiple latent parents but\nno causal edges among them, and proved its identifiability. In [20] and [21],\ncausal clusters are defined as follows:\nDefinition 1.1 (Causal cluster [20, 21]). A set of observed variables that\nshare the same latent parents is called a causal cluster.\nTheir methods consist of two main steps: identifying causal clusters and\nthen recovering the causal order of latent variables. LSTC and the algorithm\nfor LiNGLaM estimate clusters of observed variables by leveraging the Triad\nconstraints or the generalized independence noise (GIN) conditions. It is also\npossible to define clusters in the same manner as Definition 1.1 for models\nwhere causal edges exist among observed variables. However, when impure\nobserved variables exist, their method might fail to identify the clusters, re-\nsulting in an incorrect estimation of both the number of latent variables and\nlatent DAGs. Several recent studies have shown that LvLiNGAM remains\nidentifiable even when some observed variables are impure [22, 23, 24]. How-\never, these methods still rely on the existence of at least some pure observed\nvariables in each cluster.\n1.1. Contributions\nIn this paper, we relax the pure observed children assumption of Cai et\nal. [20] and investigate the identifiability of the causal DAG for an extended\nmodel that allows causal structures both among latent variables and among\n3\n\nobserved variables. Using higher-order cumulants of the observed data, we\nshow the identifiability of the causal DAG of a class of LvLiNGAM and pro-\npose a practical algorithm for estimating the class. The proposed method first\nestimates clusters using the approaches of [20, 21]. When causal edges exist\namong observed variables, the clusters estimated by using Triad constraints\nor GIN conditions may be over-segmented compared to the true clusters. The\nproposed method leverages higher-order cumulants of observed variables to\nrefine these clusters, estimates causal edges within clusters, determines the\ncausal order among latent variables, and finally estimates the exact causal\nstructure among latent variables.\nIn summary, our main contributions are as follows:\n1. Demonstrate identifiability of causal DAGs in a class of LvLiNGAM,\nallowing causal relationships among latent and observed variables.\n2. Extend the causal cluster estimation methods of [20] and [21] to han-\ndle cases where directed edges exist among observed variables within\nclusters.\n3. Propose a top-down algorithm using higher-order cumulants to infer\nthe causal order of latent variables.\n4. Develop a bottom-up recursive procedure to reconstruct the latent\ncausal DAG from latent causal orders.\nThe rest of this paper is organized as follows. Section 2 defines the class\nof LvLiNGAM considered in this study. In Section 2, we also summarize\nsome basic facts on higher-order cumulants. Section 3 describes the proposed\nmethod in detail. Section 4 presents numerical simulations to demonstrate\nthe effectiveness of the proposed method. Section 5 evaluates the usefulness of\nthe proposed method by applying it to the Political Democracy dataset [25].\nFinally, Section 6 concludes the paper. All proofs of theorems, corollaries,\nand lemmas in the main text are provided in the Appendices.\n2. Preliminaries\n2.1. LvLiNGAM\nLet X = (X1, . . . , Xp)⊤and L = (L1, . . . , Lq)⊤be vectors of observed\nand latent variables, respectively. In this paper, we identify these vectors\nwith the corresponding set of variables. Define V = X ∪L = {V1, . . . , Vp+q}.\nLet G = (V , E) be a causal DAG. Vi →Vj denotes a directed edge from Vi\n4\n\nto Vj. Anc(Vi), Pa(Vi), and Ch(Vi) are the sets of ancestors, parents, and\nchildren of Vi, respectively. We use Vi ≺Vj to indicate that Vi precedes Vj\nin a causal order.\nThe LvLiNGAM considered in this paper is formulated as\n\u0014 L\nX\n\u0015\n=\n\u0014 A\n0\nΛ\nB\n\u0015 \u0014 L\nX\n\u0015\n+\n\u0014 ϵ\ne\n\u0015\n,\n(2.1)\nwhere A = {aji}, B = {bji}, and Λ = {λji} are matrices of causal co-\nefficients, while ϵ and e denote vectors of independent non-Gaussian dis-\nturbances associated with L and X, respectively. Let aji, λji, and bji be\nthe causal coefficients from Li to Lj, from Li to Xj, and from Xi to Xj,\nrespectively.\nDue to the arbitrariness of the scale of latent variables, we\nmay, without loss of generality, set one of the coefficients λji to 1 for some\nXj ∈Ch(Li). Hereafter, such a normalization will often be used.\nA and B can be transformed into lower triangular matrices by row and\ncolumn permutations. We assume that the elements of ϵ and e are mutu-\nally independent and follow non-Gaussian continuous distributions. Let MG\ndenote the LvLiNGAM defined by G. As shown in (2.1), we assume in this\npaper that all observed variables are not ancestors of any latent variables.\nConsider the following reduced form of (2.1),\n\u0014 L\nX\n\u0015\n=\n\u0014\n(Iq −A)−1\n0\n(Ip −B)−1Λ(Iq −A)−1\n(Ip −B)−1\n\u0015 \u0014 ϵ\ne\n\u0015\n.\nLet αll\nji, αol\nji, and αoo\nji represent the total effects from Li to Lj, Li to Xj, and\nXi to Xj, respectively. Thus, (Iq −A)−1 = {αll\nji}, (Ip −B)−1Λ(Iq −A)−1 =\n{αol\nji}, and (Ip −B)−1 = {αoo\nji }. The total effect from Vi to Vj is denoted by\nαji, with the superscript omitted.\nM := [(Ip −B)−1Λ(Iq −A)−1, (Ip −B)−1] is called a mixing matrix of\nthe model (2.1). Denote u = (ϵ⊤, e⊤)⊤. Then, X is written as\nX = Mu,\n(2.2)\nwhich conforms to the formulation of the overcomplete ICA problem [8, 26, 7].\nM is said to be irreducible if every pair of columns is linearly independent.\nG is said to be minimal if and only if M is irreducible. If G is not minimal,\nsome latent variables can be absorbed into other latent variables, resulting\nin a minimal graph [9].\nMG is called the canonical model when A = 0\n5\n\nand M is irreducible. Hoyer et al. [7] showed that any LvLiNGAM can be\ntransformed into an observationally equivalent canonical model. For exam-\nple, the LvLiNGAM defined by the DAG in Figure 2.1 (a) is the canonical\nmodel of the LvLiNGAM defined by the DAG in Figure 2.1 (b). Hoyer et al.\n[7] also demonstrated that, when the number of latent variables is known,\nthe canonical model can be identified up to observational equivalent models\nusing overcomplete ICA.\nSalehkaleybar et al. [9] showed that, even when A ̸= 0, the irreducibil-\nity of M is a necessary and sufficient condition for the identifiability of the\nnumber of latent variables.\nHowever, they did not provide an algorithm\nfor estimating the number of latent variables. Schkoda et al. [18] proposed\nReLVLiNGAM to estimate the canonical model with generic coefficients even\nwhen the number of latent variables is unknown. However, the canonical\nmodel derived from an LvLiNGAM with A ̸= 0 lies in a measure-zero sub-\nset of the parameter space, which prevents ReLVLiNGAM from accurately\nidentifying the number of latent confounders between two observed variables\nin such cases. For example, ReLVLiNGAM may not identify the canonical\nmodel in Figure 2.1 (a) from data generated by the LvLiNGAM in Figure 2.1\n(b).\nCai et al. [20] and Xie et al. [21] demonstrated that within LvLiNGAMs\nwhere all the observed children of latent variables are pure, there exists a\nclass, such as the models shown in Figure 2.1 (b), in which the causal order\namong latent variables is identifiable. They proposed algorithms for esti-\nmating the causal order. However, the complete causal structure cannot be\nidentified solely from the causal order, and their algorithm cannot be gener-\nL1\nL2\nL3\nX1\nX2\nX3\nX4\nX5\nX6\n(a) An example of canonical LvLiNGAM\nL1\nL2\nL3\nX1\nX4\nX2\nX5\nX3\nX6\n(b) An LvLiNGAM that can be identified by\n[20, 21]\nFigure 2.1: Examples of LvLiNGAMs\n6\n\nalized to cases where causal edges exist among observed variables or where\nlatent variables do not have sufficient pure children.\nIn this paper, we introduce the following class of models, which generalizes\nthe class of models in Cai et al. [20] by allowing causal edges among the\nobserved variables, and consider the problem of identifying the causal order\namong observed variables within each cluster as well as the causal structure\namong the latent variables.\nA1. Each observed variable has only one latent parent.\nA2. Each latent variable has at least two children, at least one of which is\nobserved.\nA3. There are no direct causal paths between causal clusters.\nA4. The model satisfies the faithfulness assumption.\nA5. The higher-order cumulant of each component of the disturbance u is\nnonzero.\nIn Section 3, we demonstrate that the causal structure of latent variables\nand the causal order of observed variables for the LvLiNGAM that satisfies\nAssumption A1-A5 are identifiable, and we provide an algorithm for esti-\nmating the causal DAG for this class. The proposed method enables the\nidentification not only of the causal order among latent variables but also of\ntheir complete causal structure.\nUnder Assumption A1, every observed variable is assumed to have one\nlatent parent. However, even if there exist observed variables without latent\nparents, the estimation problem can sometimes be reduced to a model sat-\nisfying Assumption A1 by applying ParceLiNGAM [11] or repetitive causal\ndiscovery (RCD) [12, 13] as a preprocessing step of the proposed method.\nDetails are provided in Appendix E.\n2.2. Cumulants\nThe proposed method leverages higher-order cumulants of observed data\nto identify the causal structure among latent variables. In this subsection,\nwe summarize some facts on higher-order cumulants. First, we introduce the\ndefinition of a higher-order cumulant.\nDefinition 2.1 (Cumulants [27]). Let i1, . . . , ik ∈{1, . . . , p}. The k-th order\ncumulant of random vector (Xi1, . . . , Xik) is\nc(k)\ni1,...,ik = cum(k)(Xi1, . . . , Xik)\n7\n\n=\nX\n(I1,...,Ih)\n(−1)h−1(h −1)!E\n\"Y\nj∈I1\nXj\n#\n· · · E\n\"Y\nj∈Ih\nXj\n#\n,\nwhere the sum is taken over all partitions (I1, . . . , Ih) of (i1, . . . , ik).\nIf i1 = · · · = ik = i, we write cum(k)(Xi) to denote cum(k)(Xi, . . . , Xi).\nThe k-th order cumulants of the observed variables of LvLiNGAM satisfy\nc(k)\ni1,i2,...,ik = cum(k)(Xi1, . . . , Xik)\n=\nq\nX\nj=1\nαol\ni1j · · · αol\nikjcum(k)(ϵj) +\np\nX\nj=1\nαoo\ni1j · · · αoo\nikjcum(k)(ej)\nWe consider an LvLiNGAM in which all variables except Xi and Xj\nare regarded as latent variables. We refer to the canonical model that is\nobservationally equivalent to this model as the canonical model over Xi and\nXj. Let Conf(Xi, Xj) = {L′\n1, L′\n2, · · · , L′\nℓ} be the set of latent confounders\nin the canonical model over Xi and Xj, where all L′\nh ∈Conf(Xi, Xj) are\nmutually independent.\nWithout loss of generality, we assume that Xj /∈\nAnc(Xi). Then, Xi and Xj are expressed as\nXi =\nℓ\nX\nh=1\nα′\nihL′\nh + vi,\nXj =\nℓ\nX\nh=1\nα′\njhL′\nh + αoo\nji vi + vj,\n(2.3)\nwhere vi and vj are disturbances, and α′\nih and α′\njh are total effects from L′\nh\nto Xi and Xj, respectively, in the canonical model over them. We note that\nthe model (2.3) is a canonical model with generic parameters, and that ℓis\nequal to the number of confounders in the original model MG.\nSchkoda et al. [18] proposed an algorithm for estimating the canonical\nmodel with generic parameters by leveraging higher-order cumulants. Several\nof their theorems concerning higher-order cumulants are also applicable to\nthe canonical model over Xi and Xj. They define a\n\u0010Pk2−k1+1\ni=0\ni\n\u0011\n×k1 matrix\nA(k1,k2)\n(Xi→Xj) as follows:\n8\n\nA(k1,k2)\n(Xi→Xj) =\n\n\nc(k1)\ni,i,...,i\nc(k1)\ni,i,...,j\n. . .\nc(k1)\ni,i,...,j\nc(k1+1)\ni,i,i,...,i\nc(k1+1)\ni,i,i,...,j\n. . .\nc(k1+1)\ni,i,j,...,j\nc(k1+1)\nj,i,i,...,i\nc(k1+1)\nj,i,i,...,j\n. . .\nc(k1+1)\nj,i,j,...,j\n...\n...\n...\n...\nc(k2)\ni,...,i,i,i,...,i,i\nc(k2)\ni,...,i,i,i,...,i,j\n. . .\nc(k2)\ni,...,i,i,j,...,j,j\n...\n...\n...\n...\nc(k2)\nj,...,j,i,i,...,i,i\nc(k2)\nj,...,j,i,i,...,i,j\n. . .\nc(k2)\nj,...,j,i,j,...,j,j\n\n\n,\n(2.4)\nwhere k1 < k2. A(k1,k2)\n(Xj→Xi) is defined similarly by swapping the indices i and j\nin A(k1,k2)\n(Xi→Xj). Proposition 2.2 enables the identification of ℓin (2.3) and the\ncausal order between Xi and Xj.\nProposition 2.2 (Theorem 3 in [18]). For two observed variables Xi and Xj\nwhere Xj /∈Anc(Xi). Let m := min(Pk2−k1+1\ni=1\ni, k1). Then,\n1. A(k1,k2)\n(Xi→Xj) generically has rank min(ℓ+ 1, m).\n2. If αoo\nji ̸= 0, A(k1,k2)\n(Xi→Xj) generically has rank min(ℓ+ 2, m).\n3. If αoo\nji = 0, A(k1,k2)\n(Xi→Xj) generically has rank min(ℓ+ 1, m).\nDefine A(ℓ)\n(Xi→Xj) as A(k1,k2)\n(Xi→Xj) for the case where k1 = ℓ+ 2 and k2 is the\nsmallest possible choice, and let ˜A(ℓ)\n(Xi→Xj) be the matrix obtained by adding\nthe row vector (1, α, . . . , αℓ+1) as the first row of A(ℓ)\n(Xi→Xj).\nProposition 2.3 (Theorem 4 in [18]). Consider the determinant of an (ℓ+\n2) × (ℓ+ 2) minor of ˜A(ℓ)\n(Xi→Xj) that contains the first row and treat it as a\npolynomial in α. Then, the roots of this polynomial are αoo\nji , αol\nj1, · · · , αol\njℓ.\nProposition 2.3 enables the identification of αoo\nji , αol\nj1, · · · , αol\njℓup to per-\nmutation.\nThe following proposition plays a crucial role in this paper in\nidentifying both the number of latent variables and the true clusters.\n9\n\nProposition 2.4 (Lemma 5 in [18]). For two observed variables Xi and Xj,\nαoo\nji , αol\nj1, · · · , αol\njℓare the roots of the polynomial in Proposition 2.3. Then\n\n\n1\n1\n. . .\n1\nαoo\nji\nαol\nj1\n. . .\nαol\njℓ\n...\n...\n...\n...\n(αoo\nji )k−1\n(αol\nj1)k−1\n. . .\n(αol\njℓ)k−1\n\n\n\n\ncum(k)(vi)\ncum(k)(L′\n1)\n...\ncum(k)(L′\nℓ)\n\n=\n\n\nc(k)\ni,i,...,i\nc(k)\ni,i,...,j\n...\nc(k)\ni,j,...,j\n\n\n(2.5)\n(2.5) is generically uniquely solvable if k ≥ℓ+ 1.\nIn the following, let c(k)\n(Xi→Xj)(L′\nh), where h = 1, . . . , ℓ, denote the solution\nof cum(k)(L′\nh) in (2.5).\n3. Proposed Method\nIn this section, we propose a three-stage algorithm for identifying LvLiNGAM\nthat satisfy Assumptions A1–A5. In the first stage, leveraging Cai et al. [20]’s\nTriad constraints and Proposition 2.2, the method estimates over-segmented\ncausal clusters and assigns a latent parent to each cluster. In this stage,\nthe ancestral relationships among observed variables are also estimated. In\nthe second stage, Proposition 2.3 is employed to identify latent sources re-\ncursively and, as a result, the causal order among the latent variables is\nestimated. When multiple latent variables are found to have identical cumu-\nlants, their corresponding clusters are merged, enabling the identification of\nthe true clusters. In general, even if the causal order among latent variables\ncan be estimated, the causal structure among them cannot be determined.\nThe final stage identifies the exact causal structure among latent variables\nin a bottom-up manner.\n3.1. Stage I: Estimating Over-segmented Clusters\nFirst, we introduce the Triad constraint proposed by Cai et al. [20], which\nalso serves as a key component of our method in this stage.\nDefinition 3.1 (Triad constraint [20]). Let Xi, Xj, and Xk be observed\nvariables in the LvLiNGAM and assume that Cov(Xj, Xk) ̸= 0. Define Triad\nstatistic e(Xi,Xj|Xk) by\ne(Xi,Xj|Xk) := Xi −Cov(Xi, Xk)\nCov(Xj, Xk)Xj.\n(3.1)\n10\n\nL1\nL2\nL3\nX1\nX2\nX3\nX4\nX5\n(a) An example of LvLiNGAM with impure chil-\ndren (1)\nL1\nL2\nL3\nX1\nX3\nX2\nX5\nX4\nX6\n(b) An example of LvLiNGAM with impure\nchildren (2)\nFigure 3.1: Two examples of LvLiNGAM with impure children\nIf e(Xi,Xj|Xk) ⊥⊥Xk, we say that {Xi, Xj} and Xk satisfy the Triad constraint.\nThe following propositions are also provided by Cai et al. [20].\nProposition 3.2 ([20]). Assume that all observed variables are pure, and\nXi and Xj are dependent. If {Xi, Xj} and all Xk ∈X \\ {Xi, Xj} satisfy the\nTriad constraint, then Xi and Xj form a cluster.\nProposition 3.3 ([20]). Let ˆC1 and ˆC2 be two clusters estimated by using\nTriad constraints. If ˆC1 and ˆC2 satisfy ˆC1 ∩ˆC2 ̸= ∅, ˆC1 ∪ˆC2 also forms a\ncluster.\nWhen all observed variables are pure, as in the model shown in Fig. 2.1\n(b), the correct clusters can be identified in two steps: first, apply Proposition\n3.2 to find pairs of variables in the same cluster; then, merge them using\nProposition 3.3. However, when impure observed variables are present, the\nclusters obtained using this method become over-segmented relative to the\ntrue clusters.\nThe correct clustering for the model in Figure 3.1 (a) is {X1}, {X2},\n{X3, X4, X5}, and the correct clustering for the model in Figure 3.1 (b) is\n{X1}, {X2}, {X3, X4, X5, X6}. However, the above method incorrectly par-\ntitions the variables into {X1}, {X2}, {X3, X5}, {X4} for (a), and {X1},\n{X2}, {X3}, {X4}, {X5}, {X6} for (b), respectively. As in Figure 3.1 (b),\nwhen three or more variables in the same cluster form a complete graph, no\npair of these observed variables satisfies the Triad constraint.\nHowever, even for models in which there exist causal edges among ob-\nserved variables within the same cluster, it can be shown that a pair of\n11\n\nAlgorithm 1 Estimating over-segmented clusters\nInput: X = (X1, . . . , Xp)⊤\nOutput: Estimated clusters ˆC and AO = {Anc(Xi) | Xi ∈X}\n1: Initialize ˆC ←{{X1}, . . . , {Xp}}, Anc(Xi) ←∅for i = 1, . . . , p\n2: for all pairs (Xi, Xj) do\n3:\nif Xi, Xj satisfy Theorem 3.2 or have an ancestral relationship by Proposition 2.2\nthen\n4:\nMerge {Xi} and {Xj}\n5:\nUpdate ˆC and AO\n6:\nend if\n7: end for\n8: Merge clusters in ˆC and update ˆC by applying Proposition 3.3\n9: return ˆC, AO\nvariables satisfying the Triad constraint is a sufficient condition for them to\nbelong to the same cluster.\nTheorem 3.4. Assume the model satisfies Assumptions A1-A4. If two de-\npendent observed variables Xi and Xj satisfy the Triad constraint for all\nXk ∈X \\ {Xi, Xj}, they belong to the same cluster.\nUnder Assumption A3, the presence of ancestral relationships between\ntwo observed variables implies that they belong to the same cluster. Proposi-\ntion 2.2 allows us to determine ancestral relationships between two observed\nvariables. Using Proposition 2.2, it is possible to identify X3 ∈Anc(X5)\nin the model of Figure 3.1(a) and X4 ∈Anc(X5), X4 ∈Anc(X6), and\nX5 ∈Anc(X6) in the model of Figure 3.1(b).\nMoreover, it follows that Proposition 3.3 also holds for the models consid-\nered in this paper. By applying it, the model in Figure 3.1(a) is clustered into\n{X1}, {X2}, {X3, X5}, {X4}, while the model in Figure 3.1(b) is clustered\ninto {X1}, {X2}, {X3}, and {X4, X5, X6}.\nEven when Theorem 3.4 and Proposition 3.2 are applied, the resulting\nclusters are generally over-segmented. To obtain the correct clusters, it is\nnecessary to merge some of them. The correct clustering is obtained in the\nsubsequent stage.\nThe algorithm for Stage I is presented in Algorithm 1.\n3.2. Stage II: Identifying the Causal Order among Latent Variables\nIn this section, we provide an algorithm for estimating the correct clusters\nand the causal order among latent variables. Suppose that, as a result of\n12\n\napplying Algorithm 1, K clusters ˆC = { ˆC1, . . . , ˆCK} are estimated. Associate\na latent variable Li with each cluster ˆCi for i = 1, . . . , K, and define ˆL =\n{L1, . . . , LK}. As stated in the previous section, K ≥q. When K > q, some\nclusters must be merged to recover the true clustering.\nX can be partitioned into maximal subsets of mutually dependent vari-\nables. Each observed variable in these subsets has a corresponding latent\nparent. If the causal order of the latent parents within each subset is deter-\nmined, then the causal order of the entire latent variable set ˆL is uniquely\ndetermined. Henceforth, we assume, without loss of generality, that X itself\nforms one such maximal subset.\n3.2.1. Determining the Source Latent Variable\nSince we assume that X consists of mutually dependent variables, G\ncontains only one source node among the latent variables.\nTheorem 3.5\nprovides the necessary and sufficient condition for a latent variable to be a\nsource node.\nTheorem 3.5. Let Xi denote the observed variable with the highest causal\norder among ˆCi. Then, Li is generically a latent source in G if and only if\nConf(Xi, Xj) are identical across all Xj ∈X \\ {Xi} such that Xi ̸⊥⊥Xj in\nthe canonical model over Xi and Xj, with their common value being {Li}.\nNote that in Stage I, the ancestral relationships among the observed vari-\nables are determined. Hence, the causal order within each cluster can also be\ndetermined. Let Xj be the observed variable with the highest causal order\namong ˆCj for j = 1, . . . , K and define Xoc = {X1, . . . , XK}. When | ˆCi| ≥2,\nlet Xi′ be any element in ˆCi \\ {Xi}. Define Xi by\nXi =\n\u001a {Xi′},\nif | ˆCi| ≥2\n∅,\nif | ˆCi| = 1.\n(3.2)\nLet L(i,j) denote a latent confounder of Xi and Xj in the canonical model\nover them.\nIn the implementation, we verify whether the conditions of Theorem 3.5\nare satisfied by using Corollary 3.6.\nCorollary 3.6. Assume k ≥3. Li is generically a latent source in G if and\nonly if one of the following two cases holds:\n1. Xi = ∅and |Xoc \\ {Xi}| = 1\n13\n\nL1\nX1\nX2\nX3\n(a)\nL1\nL2\nX1\nX2\nX3\n(b)\nFigure 3.2: An example of merging clusters in Stage II\n2. |(Xoc ∪Xi) \\ {Xi}| ≥2 and the following all hold:\n(a) In the canonical model over Xi and Xj, |Conf(Xi, Xj)| = 1 for\nXj ∈(Xoc ∪Xi) \\ {Xi} such that Xi ̸⊥⊥Xj.\n(b) c(k)\n(Xi→Xj)(L(i,j)) are identical for Xj ∈(Xoc ∪Xi) \\ {Xi}.\nWhen Xi = ∅and |Xoc \\ {Xi}| = 1, it is trivial by Assumption A2 that\nLi is a latent source. Otherwise, for Li to be a latent source, it is necessary\nthat |Conf(Xi, Xj)| = 1 for all Xj ∈(Xoc ∪Xi) \\ {Xi}. This can be verified\nby using Condition 1 of Proposition 2.2. In addition, if c(k)\n(Xi→Xj)(L(i,j)) for\nXj ∈(Xoc ∪Xi) \\ {Xi} are identical, Li can be regarded as a latent source.\nWhen Xi ∈Anc(Xi′), the equation (2.5) yields two distinct solutions,\nc(k)\n(Xi→Xi′)(L(i,i′)) = c(k)\n(Xi→Xi′)(Li) and c(k)\n(Xi→Xi′)(ei), that are identifiable only\nup to a permutation of the two.\nIf either of these two solutions equals\nc(k)\n(Xi→Xj)(L(i,j)) for all Xj ∈Xoc \\ {Xi}, then Li can be identified as the\nlatent source.\nExample 3.7. Consider the models in Figure 3.2. For both models (a) and\n(b), the clusters estimated in Stage I are ˆC1 = {X1} and ˆC2 = {X2, X3},\nand let L1 and L2 be the latent parents assigned to ˆC1 and ˆC2, respectively.\nThen, Xoc = {X1, X2}. In the model (a), we can assume λ11 = 1 without\nloss of generality. Then, the model (a) is expressed as\nX1 = ϵ1 + e1,\nX2 = λ21ϵ1 + e2,\nX3 = (λ21b32 + λ31)ϵ1 + b32e2 + e3.\nBy Proposition 2.4 and assuming k ≥2, we can obtain\nc(k)\n(X1→X2)(L(1,2)) = cum(k)(ϵ1),\nc(k)\n(X2→X1)(L(2,1)) = c(k)\n(X2→X3)(L(2,3)) = cum(k)(λ21ϵ1).\n14\n\nAlgorithm 2 Finding latent sources\nInput: Mutually dependent Xoc, ˆC, and AL\nOutput: Xoc, ˆC, and a set of ancestral relationships between latent variables AL\n1: Each cluster is assigned one latent parent and let ˆL be the set of latent parents\n2: Apply Corollary 3.6 to find the latent sources Ls\n3: Assume Ls ∈Ls and ˆCs ∈ˆC\n4: if |Ls| ≥2 then\n5:\nMerge the corresponding clusters into ˆCs and update ˆC and Xoc\n6:\nIdentify all latent parents in Ls with Ls\n7: end if\n8: for all Li ∈ˆL \\ Ls do\n9:\nAnc(Li) ←{Ls}\n10: end for\n11: Xoc ←Xoc \\ {Xs}\n12: AL ←AL ∪{Anc(Li) | Li ∈ˆL \\ Ls} ∪{Anc(Ls) = ∅}\n13: return Xoc, ˆC, and AL\nSince |Xoc \\{X1}| = |{X2}| = 1 and c(k)\n(X2→X1)(L(2,1)) = c(k)\n(X2→X3)(L(2,3)), both\nL1 and L2 are determined as latent sources. The dependence between X1 and\nX2 leads to L1 and L2 being regarded as a single latent source, resulting in\nthe merging of ˆC1 and ˆC2.\nIn the model (b), we can assume λ11 = λ22 = 1 without loss of generality.\nThen, the model (b) is described as\nX1 = ϵ1 + e1,\nX2 = (a21ϵ1 + ϵ2) + e2,\nX3 = (b32 + λ31)(a21ϵ1 + ϵ2) + b32e2 + e3.\nThen,\nc(k)\n(X1→X2)(L(1,2)) = cum(k)(ϵ1),\nc(k)\n(X2→X1)(L(2,1)) = cum(k)(a21ϵ1) ̸= c(k)\n(X2→X3)(L(2,3)) = cum(k)(a21ϵ1 + ϵ2).\nTherefore, L1 is a latent source, while L2 is not.\nAs in model (a), multiple latent variables may also be identified as latent\nsources. In such cases, their observed children are merged into a single cluster.\nOnce Li is established as a latent source, it implies that Li is an ancestor of\nthe other elements in ˆL. The procedure of Section 3.2.1 is summarized in\nAlgorithm 2.\n15\n\n3.2.2. Determining the Causal Order of Latent Variables\nNext, we address the identification of subsequent latent sources after find-\ning L1 in the preceding procedure. If the influence of the latent source can\nbe removed from the observed descendant, the subsequent latent source may\nbe identified through a procedure analogous to the one previously applied.\nThe statistic ˜e(Xi,Xh), defined below, serves as a key quantity for removing\nsuch influence.\nDefinition 3.8. Let Xi and Xh be two observed variables. Define ˜e(Xi,Xh) as\n˜e(Xi,Xh) = Xi −ρ(Xi,Xh)Xh,\nwhere\nρ(Xi,Xh) =\n\n\n\n\n\n\n\ncum(Xi, Xi, Xh, Xh)\ncum(Xi, Xh, Xh, Xh)\nXi ̸⊥⊥Xh,\n0\nXi ⊥⊥Xh.\nUnder Assumption 5, when Xi ̸⊥⊥Xh, ρ(Xi,Xh) is shown to be generically\nfinite and non-zero. See Lemma A.2 in the Appendix for details. Let Lh be\nthe latent source, and let Xh be its observed child with the highest causal\norder. When there is no directed path between Xi and Xh, ˜e(Xi,Xh) can be\nregarded as Xi after removing the influence of Lh.\nExample 3.9. Consider the model in Figure 3.3 (a). We can assume λ22 =\nλ33 = 1 without loss of generality. Then, X1, X2 and X3 are described as\nX1 = ϵ1 + e1,\nX2 = a21ϵ1 + ϵ2 + e2,\nX3 = a31ϵ1 + ϵ3 + e3.\nWe can easily show that ρ(X2,X1) = a21. Hence, we have\n˜e(X2,X1) = −a21e1 + ϵ2 + e2.\nIt can be seen that ˜e(X2,X1) does not depend on L1 = ϵ1, and that ˜e(X2,X1) and\nX3 are mutually independent.\nExample 3.10. Consider the model in Figure 3.3 (b). We can assume that\nλ11 = λ22 = λ33 = 1 without loss of generality. Then, the model is described\nas\nX1 = ϵ1 + e1,\nX2 = a21ϵ1 + ϵ2 + e2,\n16\n\nL1\nL2\nL3\nX1\nX3\nX2\n(a)\nL1\nL2\nL3\nX1\nX2\nX3\nX4\nX5\n(b)\nFigure 3.3: Examples of LvLiNGAMs\nX3 = a32a21ϵ1 + a32ϵ2 + ϵ3 + e3,\nX4 = λ42(a21ϵ1 + ϵ2) + e4,\nX5 = (λ53 + b53)(a32a21ϵ1 + a32ϵ2 + ϵ3) + b53e3 + e5.\nWe can easily show that ρ(X2,X1) = a21 and ρ(X3,X1) = a32a21. Hence, we have\n˜e(X2,X1) = −a21e1 + ϵ2 + e2,\n˜e(X3,X1) = −a32a21e1 + a32ϵ2 + ϵ3 + e3.\nIt can be seen that ˜e(X2,X1) and ˜e(X3,X1) are obtained by replacing L1 = ϵ1 with\n−e1. The model for (˜e(X2,X1), X3) and (˜e(X2,X1), X5) are described by canonical\nmodels with Conf(˜e(X2,X1), X3) = Conf(˜e(X2,X1), X5) = {ϵ2}, respectively. The\nmodel for (˜e(X3,X1), X2) and (˜e(X3,X1), X5) are described by canonical models\nwith Conf(˜e(X3,X1), X2) = {ϵ2} and Conf(˜e(X3,X1), X5) = {a32ϵ2 + ϵ3, e3}, re-\nspectively. X5 contains {ϵ1, ϵ2, ϵ3, e3, e5}, and ˜e(X3,X1) contains {ϵ2, ϵ3, e1, e3}.\nSince these sets are not in an inclusion relationship, it follows from Lemma 5\nof Salehkaleybar et al. [9] that there is no ancestral relationship between\n˜e(X3,X1) and X5.\nIt is noteworthy that ˜e(X3,X1) and X5 share two latent confounders, and\nthat no ancestral relationship exists between them even though X3 ∈Anc(X5)\nin the original graph.\nLet L1 be the current latent source identified by the preceding procedure.\nLet G−({L1}) be the subgraph of G induced by V \\ ({L1} ∪ˆC1). By gen-\neralizing the discussions in Examples 3.9 and 3.10, we obtain the following\ntheorems.\nTheorem 3.11. For Xi, Xj ∈Xoc \\ {X1} and their respective latent parent\nLi and Lj, Li ⊥⊥Lj | L1 if and only if ˜e(Xi,X1) ⊥⊥Xj.\n17\n\nTheorem 3.12. Let Li denote the latent parent of Xi ∈Xoc \\ {X1}. If\n| ˆCi| ≥2, let Xi′ be an element of ˆCi \\ {Xi}.\nXi is defined in the same\nmanner as (3.2).\nThen, Li is generically a source in G−({L1}) if and only if the following\ntwo conditions hold:\n1. Conf(˜e(Xi,X1), Xj) are identical for all Xj ∈Xoc \\ {X1, Xi} such that\n˜e(Xi,X1) ̸⊥⊥Xj, with their common value being {ϵi}.\n2. If Xi ̸= ∅, Conf(˜e(Xi,X1), Xi′) ∩Conf(˜e(Xi,X1), Xj) are identical for all\nXj ∈Xoc \\ {X1, Xi}, with their common value being {ϵi}.\nBy applying Theorem 3.11, we can obtain the family of maximal depen-\ndent subsets of Xoc \\ {X1} in the conditional distribution given L1. Theo-\nrem 3.12 allows us to verify whether Li is a latent source in G−({L1}).\nBy recursively iterating such a procedure, the ancestral relationships\namong the latent variables can be identified. To achieve this, it is necessary\nto generalize ˜e(Xi,X1) defined as in Definition 3.13. Let G−({L1, . . . , Ls−1})\ndenote the subgraph of G induced by V except for {L1, . . . , Ls−1} and their\nobserved children and L1, . . . , Ls−1 be latent sources in\nG, G−({L1}), G−({L1, L2}), . . . , G−({L1, . . . , Ls−2}),\nrespectively. Then, {L1, . . . , Ls−1} has a causal order L1 ≺· · · ≺Ls−1.\nDefinition 3.13. For i ≥s, ˜e(Xi,˜es) is defined as follows.\n˜e(Xi,˜es) =\n\u001a Xi\ns = 1,\nXi −Ps−1\nh=1 ρ(Xi,˜e(Xh,˜eh))˜e(Xh,˜eh)\ns > 1,\nwhere ˜es = (˜e(X1,˜e1), . . . , ˜e(Xs−1,˜es−1)).\n˜e(Xi,˜es) can be regarded as a statistic with the information of L1, . . . , Ls−1\neliminated from Xi. The following lemma shows that ˜e(Xi,˜es) is obtained by\nreplacing the information of ϵ1, . . . , ϵs−1 with that of e1, . . . , es−1.\nLemma 3.14. Let X1, . . . , Xs−1, and Xi be the observed children with the\nhighest causal order of L1, . . . , Ls−1, and Li, respectively. ˜e(Xi,˜es) can be ex-\npressed as\n˜e(Xi,˜es) =\n\n\n\nϵi + U[i]\ni = s\nϵi + Pi−1\nh=s αll\nihϵh + ei\ni > s and s = 1\nϵi + Pi−1\nh=s αll\nihϵh + U[s−1] + ei\ni > s and s > 1,\n18\n\nwhere U[i] and U[s−1] are linear combinations of {e1, . . . , ei} and {e1, . . . , es−1},\nrespectively.\nBy using ˜e(Xi,˜es) in Definition 3.13, we obtain Theorems 3.15 and 3.16,\nwhich generalize Theorems 3.11 and 3.12, respectively.\nTheorem 3.15. For Xi, Xj ∈Xoc \\ {X1, . . . , Xs−1} and their respective\nlatent parent Li and Lj, Li ⊥⊥Lj | {L1, . . . , Ls−1} if and only if ˜e(Xi,˜es) ⊥⊥Xj.\nTheorem 3.16. Let Li be the latent parent of Xi ∈Xoc \\ {X1, . . . , Xs−1}.\nIf | ˆCi| ≥2, let Xi′ be an element of ˆCi \\ {Xi}. Xi is defined in the same\nmanner as (3.2).\nThen, Li is generically a latent source in G−({L1, . . . , Ls−1}) if and only\nif the following two conditions hold:\n1. Conf(˜e(Xi,˜es), Xj) are identical for all Xj ∈Xoc \\ {X1, . . . , Xs−1, Xi}\nsuch that ˜e(Xi,˜es) ̸⊥⊥Xj, with their common value being {ϵi}.\n2. When Xi ̸= ∅, Conf(˜e(Xi,˜es), Xi′) ∩Conf(˜e(Xi,˜es), Xj) are identical for\nall Xj ∈Xoc \\ {X1, . . . , Xs−1, Xi} such that ˜e(Xi,˜es) ̸⊥⊥Xj, with their\ncommon value being {ϵi}.\nAs in Theorem 3.11, by applying Theorem 3.15, we can identify the fam-\nily of maximal dependent subsets of Xoc \\ {X1, . . . , Xs−1} in the conditional\ndistribution given {L1, . . . , Ls−1}. For each maximal dependent subset, we\ncan apply Theorem 3.16 to identify the next latent source. In the implemen-\ntation, we verify whether the conditions of Theorem 3.16 are satisfied using\nCorollary 3.17, which generalizes Corollary 3.6.\nCorollary 3.17. Assume k ≥3. Li is generically a latent source in\nG−({L1, . . . , Ls−1}) if and only if one of the following two cases holds:\n1. Xi = ∅and |Xoc \\ {X1, . . . , Xs−1, Xi}| = 1.\n2. |(Xoc ∪Xi) \\ {X1, . . . , Xs−1, Xi}| ≥2, and the following all hold:\n(a) In the canonical model over ˜e(Xi,˜es) and Xj, |Conf(˜e(Xi,˜es), Xj)| =\n1 for all Xj ∈Xoc \\ {X1, . . . , Xs−1, Xi} such that ˜e(Xi,˜es) ̸⊥⊥Xj.\n(b) c(k)\n(˜e(Xi,˜es)→Xj)(L(i,j)) are identical for all Xj ∈Xoc\\{X1, . . . , Xs−1, Xi}\nsuch that ˜e(Xi,˜es) ̸⊥⊥Xj, where L(i,j) is the unique latent confounder\nin the canonical model over ˜e(Xi,˜es) and Xj.\n19\n\n(c) ˜e(Xi,˜es) and Xi′ has a latent confounder L(i,i′) in the canonical\nmodel over them that satisfies c(k)\n(˜e(Xi,˜es)→Xi′)(L(i,i′)) = c(k)\n(˜e(Xi,˜es)→Xj)(L(i,j))\nfor all Xj ∈Xoc\\{X1, . . . , Xs−1, Xi} such that ˜e(Xi,˜es) ̸⊥⊥Xj, when\nXi ̸= ∅.\nTo determine whether Li is a latent source of G−({L1, . . . , Ls−1}), we first\nexamine, using Condition 1 of Proposition 2.2, whether |Conf(˜e(Xi,˜es), Xj)| =\n1, as in Section 3.2.1. If c(k)\n(˜e(Xi,˜es)→Xj)(L(i,j)) are identical for Xj ∈(Xoc ∪\nXi)\\{X1, . . . , Xs−1, Xi}, Li is identified as a latent source. As in the previous\ncase, when Xi ̸= ∅and Xi ∈Anc(Xi′), the equation (2.5) yields two distinct\nsolutions for the higher-order cumulants of latent confounders. Here, we de-\ntermine that Li is a latent source in G−({L1, . . . , Ls−1}) if either of two solu-\ntions of (2.5) equals to c(k)\n(˜e(Xi,˜es)→Xj)(L(i,j)) for Xj ∈Xoc \\{X1, . . . , Xs−1, Xi}.\nIf multiple latent sources are identified for any element in a mutually de-\npendent maximal subset of Xoc \\ {X1, . . . , Xs−1}, the corresponding clusters\nmust be merged. As latent sources are successively identified, the correct set\nof latent variables L, the ancestral relationships among L, and the correct\nclusters are also successively identified.\nThe procedure of Section 3.2.2 is presented in Algorithm 3. Algorithm 4\ncombines Algorithms 2 and 3 to provide the complete procedure for Stage II.\nExample 3.18. For the model in Figure 3.1 (a), the estimated clusters ob-\ntained in Stage I are {X1}, {X2}, {X3, X5}, and {X4}, with their corre-\nsponding latent parents denoted as L1, L2, L3, and L4, respectively.\nSet\nXoc = {X1, X2, X3, X4}.\nOnly X1 satisfies Corollary 3.6, and thus L1 is identified as the initial\nlatent source.\nThen, we remove X1 from Xoc and update it to Xoc =\n{X2, X3, X4}.\nNext, since it can be shown that only L2 satisfies Corol-\nlary 3.17, i.e.,\nc(3)\n(˜e(X2,X1)→X3)(L(2,3)) = c(3)\n(˜e(X2,X1)→X4)(L(2,4)),\nit follows that L2 is the latent source of G−({L1}). Similarly, we remove X2\nfrom the current Xoc and update it to Xoc = {X3, X4}.\nLet X3′ = X5. In G−({L1, L2}), we compute ˜e(X3,˜e3) and ˜e(X4,˜e3), and find\nthat\nc(3)\n(˜e(X3,˜e3)→X4)(L(3,4)) = c(3)\n(˜e(X3,˜e3)→X5)(L(3,5)),\n|Xoc ∪∅\\ {X4}| = 1,\n20\n\nAlgorithm 3 Finding subsequent latent sources\nInput: Xoc, ˆC, and AL\nOutput: ˆC and AL\n1: Apply Corollary 3.17 to find the set of latent sources L0 in G−(Ls)\n2: if L0 = ∅then\n3:\nreturn ˆC and AL\n4: end if\n5: if |L0| ≥2 then\n6:\nfor all pairs Xi, Xj ∈\n\u0010S\nk:Lk∈L0 ˆCk\n\u0011\n∩Xoc do\n7:\nif ˜e(Xi,˜es) ̸⊥⊥Xj then\n8:\nMerge ˆCj into ˆCi\n9:\nˆC ←ˆC \\ { ˆCj}, ˆL ←ˆL \\ {Lj}, Xoc ←Xoc \\ {Xj}, AL ←AL \\ {Anc(Lj)}\n10:\nend if\n11:\nend for\n12: end if\n13: for all Xi ∈\n\u0010S\nk:Lk∈L0 ˆCk\n\u0011\n∩Xoc do\n14:\nX(i)\noc ←∅\n15:\nfor all Xj ∈Xoc \\ {Xi} do\n16:\nif Xj ̸⊥⊥˜e(Xi,˜es) then\n17:\nAnc(Lj) ←Anc(Lj) ∪{Li}, X(i)\noc ←X(i)\noc ∪{Xj}\n18:\nend if\n19:\nend for\n20:\nˆC, AL ←Algorithm 3 (X(i)\noc , ˆC, AL)\n21: end for\n22: return ˆC and AL\nindicating both L3 and L4 are latent sources by Corollary 3.17. Furhtermore,\nwe conclude that {X3, X5} and {X4} should be merged into one cluster con-\nfounded by L3.\n3.3. Stage III: Identifying Causal Structure among Latent Variables\nBy the end of Stage II, the clusters of observed variables have been\nidentified, as well as the ancestral relationships among latent variables and\namong observed variables. The ancestral relationships among L alone do not\nuniquely determine the complete causal structure of L. Here, we propose a\nbottom-up algorithm to estimate the causal structure of the latent variables.\nNote that if the ancestral relationships among L are known, a causal order\nof L can also be obtained. Theorem 3.19 provides an estimator of the causal\ncoefficients between latent variables.\n21\n\nAlgorithm 4 Finding the ancestral relationships between latent variables\nInput: X, AO, and ˆC\nOutput: ˆC and AL\n1: AL →∅\n2: for all mutually dependent Xoc do\n3:\nXoc, ˆC, AL ←Algorithm 2 (Xoc, ˆC, AL)\n4:\nˆC, AL ←Algorithm 3 (Xoc, ˆC, AL)\n5: end for\n6: return ˆC and AL\nTheorem 3.19. Assume that Anc(Li) = {L1, . . . , Li−1} with the causal order\nL1 ≺· · · ≺Li−1. Let X1, . . . , Xi be the observed children of L1, . . . , Li with\nthe highest causal order, respectively. Define ˜ri,k−1 as\n˜ri,k−1 =\n\u001a Xi,\nk = 1\nXi −Pi−1\nh=i−(k−1) aihXh,\nk ≥2\nWhen we set λ11 = · · · = λii = 1, ai,i−k = ρ(˜ri,k−1,˜e(Xi−k,˜ei−k)) generically holds.\nIn addition, under Assumption A4, it holds generically that ai,i−k = 0 if and\nonly if ˜ri,k−1 ⊥⊥˜e(Xi−k,˜ei−k).\nIf the only information available is the ancestral relationships among\n{L1, . . . , Li}, we cannot determine whether there is an edge Li−k →Li in G.\nHowever, according to Theorem 3.19, if ˜ri,k−1 ⊥⊥˜e(Xi−k,˜ei−k), then ai,i−k = 0,\nand thus it follows that Li−k →Li does not exist.\nAlgorithm 5 describes how Theorem 3.19 is applied to estimate the causal\nstructure among L.\nExample 3.20. For the model in Figure 3.1 (a), the estimated causal order\nof latent variables is L1 ≺L2 ≺L3 with Xoc = {X1, X2, X3}.\nAssume\ninitially that L1, L2, and L3 form a complete graph. Then X1, X2, X3, and\n˜e(X2,˜e2) are\nX1 = ϵ1 + e1,\nX2 = a21ϵ1 + ϵ2 + e2,\nX3 = (a21a32 + a31)ϵ1 + a32ϵ2 + ϵ3 + e3,\n˜e(X2,˜e2) = ˜e(X2,X1) = ϵ2 + e2 −a21e1.\nWe estimate a32 and a31 using Theorem 3.19 as follows:\na32 = ρ(X3,˜e(X2,˜e2)),\n22\n\n˜r31 = X3 −a32X2 = a31ϵ1 + ϵ3 −a32e2 + e3,\na31 = ρ(˜r31,X1).\nThus, if ˜r31 ⊥⊥X1, then a31 = 0. In this case, we can conclude that L1 →L3\ndoes not exist.\nAlgorithm 5 Finding causal structure among latent variables\nInput: Xoc, L, AL\nOutput: An adjacency matrix Aadj of L\n1: function Adjacency(Xoc, Li, Lopen, Aadj, ˜ri)\n2:\nif |Lopen| = 0 then\n3:\nreturn Aadj\n4:\nend if\n5:\nInitialize Lnext ←∅\n6:\nfor all Lj ∈Lopen do\n7:\nˆaij ←0, Lnext ←Lnext ∪Pa(Lj)\n8:\nif ∃{Lk, Lh} ⊂Lnext s.t. Lk ∈Anc(Lh) then\n9:\nLnext ←Lnext \\ {Lk}\n10:\nend if\n11:\nif ˜ri ̸⊥⊥˜e(Xj,˜ej) then\n12:\nˆaij ←an empirical counterpart of aij\n13:\nend if\n14:\n˜ri ←˜ri −ˆaijXj\n15:\nif ˆaji ̸= 0 then\n16:\nAadj[i, j] ←1\n17:\nend if\n18:\nend for\n19:\nLopen ←Lnext\n20:\nAadj ←Adjacency(Xoc, Li, Lopen, Aadj, ˜ri)\n21: end function\n22: function Main(Xoc, L, AL)\n23:\nInitialize Aadj ←{0}|L|×|L|\n24:\nfor all Li ∈L do\n25:\n˜ri ←Xi, Lopen ←Pa(Li)\n26:\nif ∃{Lk, Lh} ⊂Lopen s.t. Lk ∈Anc(Lh) then\n27:\nLopen ←Lopen \\ {Lk}\n28:\nend if\n29:\nAadj ←Adjacency(Xoc, Li, Lopen, Aadj, ˜ri)\n30:\nend for\n31:\nreturn Aadj\n32: end function\n23\n\n3.4. Summary\nThis section integrates Algorithms 1, 4, and 5 into Algorithm 6, which\nidentifies the clusters of observed variables, the causal structure between\nlatent variables, and the ancestral relationships between observed variables\nunder the assumptions A1-A5. Since the causal clusters ˆC have been correctly\nidentified, the directed edges from L to X are also identified.\nAlthough\nthe ancestral relationships among observed variables can be identified, their\nexact causal structure remains undetermined. In conclusion, we obtain the\nfollowing result:\nTheorem 3.21. Given observed data generated from an LvLiNGAM MG in\n(2.1) that satisfies the assumptions A1-A5, the proposed method can identify\nthe latent causal structure among L, causal edges from L to X, and ancestral\nrelationships among X.\nAlgorithm 6 Identify the Causal Structure among Latent Variables\nInput: X = (X1, . . . , Xp)⊤\nOutput: AO, ˆC, and Aadj\n1: ˆC, AO ←Algorithm 1 (X)\n▷Estimate over-segmented clusters\n2: AL, ˆC ←Algorithm 4 (X, AO, ˆC) ▷Identify the causal order among latent variables\n3: Aadj ←Algorithm 5 (Xoc, L, AL) ▷Find the causal structure among latent variables\n4: return AO, ˆC, and Aadj\n4. Simulations\nIn this section, we assess the effectiveness of the proposed method by\ncomparing it with the algorithms proposed by Xie et al. [21] for estimating\nLiNGLaM and by Xie et al. [23] for estimating LiNGLaH, as well as with\nReLVLiNGAM [18], which serves as the estimation method for the canonical\nmodel with generic parameters. For convenience, we hereafter refer to both\nthe model class introduced by Xie et al. [21] and its estimation algorithm as\nLiNGLaM, and likewise use LiNGLaH to denote both the model class and\nthe estimation algorithm proposed by Xie et al. [23].\n4.1. Settings\nIn the simulation, the true models are set to six LvLiNGAMs defined by\nthe DAGs shown in Figures 4.1 (a)-(f). We refer to these models as Models\n(a)-(f), respectively. All these models satisfy Assumptions A1-A3.\n24\n\nAll disturbances are assumed to follow a log-normal distribution, ui ∼\nLognormal(−1.1, 0.8), shifted to have zero mean by subtracting its expected\nvalue. The coefficient λii from Li to Xi is fixed at 1. Other coefficients in Λ\nand A are drawn from Uniform(1.1, 1.5), while those in B are drawn from\nUniform(0.5, 0.9). When all causal coefficients are positive, the faithfulness\ncondition is satisfied. The higher-order cumulant of a log-normal distribution\nis non-zero.\nNone of the models (a)-(f) is LiNGLaM or LiNGLaH. The models (a)\nand (b) are generic canonical models, whereas the canonical models derived\nfrom Figures 4.1 (c)-(f) do not satisfy the genericity assumption of Schkoda\net al. [18].\nThe sample sizes N are set to 1000, 2000, 4000, 8000, and 16000. The\nnumber of iterations is set to 100. We evaluate the performance of the pro-\nposed method and other methods using the following metrics.\n• Ncl, Nls, Nos, and Ncs: The counts of iterations in which the resulting\nclusters, the latent structures, the ancestral relationships among X,\nand the latent structure and the ancestral relationships among X are\ncorrectly estimated, respectively.\n• PREll, RECll, and F1ll: Averages of Precision, Recall, and F1-score of\nthe estimated edges among latent variables, respectively, when clusters\nare correctly estimated.\n• PREoo, RECoo, and F1oo: Averages of Precision, Recall, and F1-score of\nthe estimated causal ancestral relationships among observed variables,\nrespectively, when clusters are correctly estimated.\nLiNGLaM and LiNGLaH assume that each cluster contains at least two\nobserved variables. When a cluster includes only a single observed variable,\nthese methods may fail to assign it to any cluster, resulting in it being left\nwithout an associated latent parent. Here, we treat such variables as indi-\nvidual clusters and assign each a latent parent.\n4.2. Implementation\nHilbert–Schmidt independence criterion (HSIC) [28] is employed for the\nindependence tests in the proposed method. As HSIC becomes computation-\nally expensive for large sample sizes, we randomly select 2,000 samples for\nHSIC when N ≥2000. The significance level of HSIC is set to αind = 0.05.\nWhen estimating the number of latent variables and the ancestral rela-\ntionships among the observed variables, we apply Proposition 2.2. Following\n25\n\nL1\nX1\nX2\nX3\n(a)\nL1\nX1\nX2\nX3\n(b)\nL1\nL2\nX1\nX2\nX3\n(c)\nL1\nL2\nX1\nX2\nX3\nX4\n(d)\nL1\nL2\nL3\nX1\nX2\nX3\nX4\n(e)\nL1\nL2\nL3\nX1\nX2\nX3\nX4\n(f)\nFigure 4.1: Six models for simulations\nSchkoda et al. [18], the rank of A(k1,k2)\n(Xi→Xj) is determined from its singular val-\nues. Let σr denote the r-th largest singular value of A(k1,k2)\n(Xi→Xj) and let τs be\na predefined threshold. If σr/σ1 ≤τs, we set σr to zero. To ensure termina-\ntion in the estimation of the number of confounders between two observed\nvariables, we impose an upper bound on the number of latent variables, fol-\nlowing Schkoda et al. [18]. In this experiment, we set the upper bound on\nthe number of latent variables to two in both our proposed method and\nReLVLiNGAM.\nWhen estimating latent sources, we use Corollaries 3.6 and 3.17. To check\nwhether |Conf(Xi, Xj)| = 1 in the canonical model over Xi and Xj, one\npossible approach is to apply Proposition 2.2. Theorem A.7 in the Appendix\nshows that |Conf(Xi, Xj)| = 1 is equivalent to\n(c(6)\ni,i,i,j,j,j)2 = c(6)\ni,i,i,i,j,jc(6)\ni,i,j,j,j,j.\nBased on this fact, one can alternatively check whether Conf(Xi, Xj) = 1 by\nusing the criterion\n|(c(6)\ni,i,i,j,j,j)2 −c(6)\ni,i,i,i,j,jc(6)\ni,i,j,j,j,j|\nmax\n\u0000(c(6)\ni,i,i,j,j,j)2, |c(6)\ni,i,i,i,j,jc(6)\ni,i,j,j,j,j|\n\u0001 < τo,\n(4.1)\n26\n\nwhere τo is a predefined threshold. In this experiment, we compared these\ntwo approaches.\nTo check condition (b) of Corollary 3.6 and conditions (b) and (c) of\nCorollary 3.17, we use the empirical counterpart of c(k)\n(Xi→Xj)(L(i,j)). In this\nexperiment, we set k = 3. We consider the situation of estimating the first\nlatent source using Corollary 3.6. Let c(3)\nXi be the set of c(3)\n(Xi→Xj)(L(i,j)) for\nXj ∈(Xoc ∪Xi) \\ {Xi}. To show that Li is a latent source, it is necessary to\ndemonstrate that all c(3)\n(Xi→Xj)(L(i,j)) ∈c(3)\nXi are identical. Let ¯ci be\n¯ci :=\n1\n|c(3)\nXi|\nX\nc∈c(3)\nXi\nc\nand s2\ni be the empirical counterpart of\n1\n|c(3)\nXi|\nX\nc∈c(3)\nXi\n(c −¯ci)2.\nThen, we regard Li as a latent source if s2\ni is smaller than a given threshold\nτm1.\nAs mentioned previously, when Xi ̸= ∅, c(3)\n(Xi→Xi′)(L(i,i′)) cannot be\ndetermined, since (2.5) yields two distinct solutions. In this case, we compute\ns2\ni for the two solutions, and if the smaller one is less than τm1, we regard Li\nas a latent source.\nThe estimation of the second and subsequent latent sources using Corol-\nlary 3.17 proceeds analogously, provided that c(3)\nXi is defined as the set of\nc(3)\n(˜e(Xi,˜ei)→Xj)(L(i,j)) for Xj ∈(Xoc ∪Xi) \\ {Xi}. However, for the threshold\napplied to s2\ni , we use τm2, which is larger than τm1. This is because, as the\niterations proceed, |c(3)\nXi| decreases, and hence the variance of s2\ni tends to\nincrease. It would be desirable to increase the threshold gradually as the\niterations proceed. However, in this experiment, we used the same τm2 from\nthe second iteration onward.\nIn this experiment, (τo, τm1, τm2) = (0.001, 0.001, 0.01). For the model in\nFigure 4.1 (a)-(c), τs was set to 0.001, and for the models (d)-(f) τs was set\nto 0.005.\nAll experiments were conducted on a workstation with a 3.0 GHz Core\ni9 processor and 256 GB memory.\n27\n\n4.3. Results and Discussions\nTable 4.1 reports Ncl, Nls, Nos, and Ncs, and Table 4.2 reports PREll,\nRECll, F1ll, PREoo, RECoo, and F1oo for both the proposed and existing\nmethods.\nSince Models (a)-(f) do not satisfy the assumptions of LiNGLaM and\nLiNGLaH, the results of them in Table 4.2 are omitted. The canonical mod-\nels derived from Models (c)–(f) are measure-zero exceptions of the generic\ncanonical models addressed by ReLVLiNGAM and thus cannot be identified,\nso the results of ReLVLiNGAM for Models (c)–(f) are not reported. Models\n(a) and (b) each involve only a single latent variable without latent–latent\nedges, so PREll, RECll, and F1ll are not reported.\nOverall, the proposed method achieves superior accuracy in estimating\nclusters, causal relationships among latent variables, and ancestral relation-\nships among observed variables, with the accuracy improving as the sample\nsize increases. Only the proposed method correctly estimates both the struc-\nture of latent variables and the causal relationships among observed variables\nfor all models. Moreover, it can be confirmed that the proposed method also\ncorrectly distinguishes the difference in latent structures between Models (e)\nand (f). While Models (a) and (b) are identifiable by ReLVLiNGAM, the\nproposed method achieves higher accuracy in both estimations for clusters.\nWhile the proposed method shows lower performance than ReLVLiNGAM in\nestimating ancestral relationships among observed variables for Model (b),\nits performance gradually approaches that of ReLVLiNGAM as the sample\nsize increases.\nIn addition, when comparing the proposed method with and without\nTheorem A.7, the version incorporating Theorem A.7 outperforms the one\nwithout it in most cases.\nAlthough Models (a) and (b) do not satisfy the assumptions of LiNGLaM\nand LiNGLaH, and thus, in theory, these methods cannot identify the mod-\nels, Table 4.1 shows that they occasionally recover the single-cluster structure\nwhen the sample size is relatively small. It can also be seen from Table 4.1\nthat the ancestral relationships among the observed variables are not esti-\nmated correctly at all.\nAs mentioned above, in the original LiNGLaM and LiNGLaH, clusters\nconsisting of a single observed variable are not output and are instead treated\nas ungrouped variables. In this experiment, by regarding such ungrouped\nvariables as clusters, higher clustering accuracy is achieved in Models (c),\n28\n\nTable 4.1: The performance in terms of Ncl, Nls, Nos, and Ncs\nModel\nMethod\nNcl\nNls\nNos\nNcs\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n(a)\nProposed (A.7)\n60\n56\n73\n74\n78\n60\n56\n73\n74\n78\n45\n53\n70\n68\n73\n45\n53\n70\n68\n73\nProposed\n60\n56\n73\n74\n78\n60\n56\n73\n74\n78\n39\n45\n67\n64\n72\n39\n45\n67\n64\n72\nLiNGLaM\n10\n1\n0\n0\n0\n10\n1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n59\n29\n5\n8\n7\n59\n29\n5\n8\n7\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nReLVLiNGAM\n47\n50\n49\n55\n64\n47\n50\n49\n55\n64\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(b)\nProposed (A.7)\n62\n75\n86\n92\n93\n62\n75\n86\n92\n93\n11\n23\n34\n53\n60\n11\n23\n34\n53\n60\nProposed\n61\n75\n86\n92\n93\n61\n75\n86\n92\n93\n11\n23\n34\n53\n60\n11\n23\n34\n53\n60\nLiNGLaM\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n1\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nReLVLiNGAM\n54\n60\n78\n79\n74\n54\n60\n78\n79\n74\n32\n41\n55\n65\n68\n32\n41\n55\n65\n68\n(c)\nProposed (A.7)\n76\n78\n79\n88\n93\n76\n78\n79\n88\n93\n53\n69\n77\n87\n93\n53\n69\n77\n87\n93\nProposed\n76\n78\n79\n88\n94\n76\n78\n79\n88\n94\n47\n55\n63\n79\n78\n47\n55\n63\n79\n78\nLiNGLaM\n87\n90\n90\n93\n90\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n98\n99\n97\n99\n99\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(d)\nProposed (A.7)\n44\n24\n38\n32\n63\n44\n24\n38\n32\n63\n10\n22\n30\n24\n58\n10\n22\n30\n24\n58\nProposed\n48\n26\n49\n55\n71\n48\n26\n49\n55\n71\n8\n8\n20\n19\n21\n8\n8\n20\n19\n21\nLiNGLaM\n38\n14\n9\n8\n8\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(e)\nProposed (A.7)\n37\n44\n68\n75\n88\n27\n39\n57\n72\n83\n36\n42\n62\n69\n80\n26\n37\n52\n66\n75\nProposed\n37\n33\n51\n84\n86\n21\n23\n49\n73\n83\n21\n17\n27\n30\n32\n12\n11\n26\n25\n30\nLiNGLaM\n96\n90\n91\n94\n87\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(f)\nProposed (A.7)\n30\n47\n52\n71\n76\n12\n34\n38\n70\n76\n30\n47\n52\n67\n74\n12\n34\n38\n66\n74\nProposed\n18\n46\n45\n57\n72\n5\n35\n41\n54\n72\n17\n34\n39\n46\n67\n4\n27\n35\n44\n67\nLiNGLaM\n92\n88\n93\n87\n92\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\nLiNGLaH\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(e), and (f). Theoretically, it can also be shown that LiNGLaM is able to\nidentify the clusters in Models (c), (e), and (f), while LiNGLaH can iden-\ntify the clusters in Model (c). However, Table 4.1 clearly shows that neither\nLiNGLaM nor LiNGLaH can correctly estimate the causal structure among\nlatent variables or the ancestral relationships among observed variables. On\nthe other hand, Table 4.1 also shows that LiNGLaM and LiNGLaH fail to cor-\nrectly estimate the clusters in Models (a), (b), and (d). This result suggests\nthat the clustering algorithms of LiNGLaM and LiNGLaH are not applicable\nto all models in this paper.\n4.4. Additional Experiments with Small Sample Sizes\nIn the preceding experiments, the primary objective was to examine the\nidentifiability of the proposed method, and hence the sample size was set to\nbe sufficiently large. However, in practical applications, it is also crucial to\nevaluate the estimation accuracy when the sample size is limited. When the\nsample size is not large, the Type II error rate of HSIC increases, which in\nturn raises the risk of misclassifying clusters. Moreover, with small samples,\nthe variability of the left-hand side of (4.1) becomes larger, thereby affect-\ning the accuracy of Corollaries 3.6 and 3.17. To address this, we investigate\n29\n\nTable 4.2: The performances in terms of PREll, RECll, F1ll, PREoo, RECoo, and F1oo\nModel\nMethod\nPREll\nRECll\nF1ll\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n(c)\nProposed (A.7)\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\nProposed\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n(d)\nProposed (A.7)\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\nProposed\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n1.000\n(e)\nProposed (A.7)\n0.869\n0.962\n0.946\n0.987\n0.981\n0.905\n1.000\n1.000\n1.000\n1.000\n0.884\n0.977\n0.968\n0.992\n0.989\nProposed\n0.824\n0.884\n0.987\n0.956\n0.988\n0.865\n0.955\n1.000\n1.000\n1.000\n0.837\n0.912\n0.992\n0.974\n0.993\n(f)\nProposed (A.7)\n1.000\n1.000\n1.000\n1.000\n1.000\n0.800\n0.908\n0.910\n0.995\n1.000\n0.880\n0.945\n0.946\n0.997\n1.000\nProposed\n1.000\n1.000\n1.000\n1.000\n1.000\n0.759\n0.920\n0.970\n0.982\n1.000\n0.856\n0.952\n0.982\n0.989\n1.000\nModel\nMethod\nPREoo\nRECoo\nF1oo\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n1K\n2K\n4K\n8K\n16K\n(a)\nProposed (A.7)\n0.825\n0.964\n0.970\n0.957\n0.962\n0.900\n0.982\n0.986\n1.000\n1.000\n0.850\n0.970\n0.975\n0.971\n0.972\nProposed\n0.733\n0.821\n0.929\n0.903\n0.949\n0.817\n0.839\n0.945\n0.946\n0.987\n0.761\n0.827\n0.934\n0.917\n0.959\nReLVLiNGAM\n0.262\n0.273\n0.320\n0.321\n0.323\n0.787\n0.820\n0.959\n0.964\n0.969\n0.394\n0.410\n0.480\n0.482\n0.484\n(b)\nProposed (A.7)\n0.895\n0.951\n0.984\n1.000\n1.000\n0.586\n0.702\n0.756\n0.855\n0.878\n0.687\n0.790\n0.837\n0.912\n0.926\nProposed\n0.902\n0.951\n0.984\n1.000\n1.000\n0.590\n0.702\n0.756\n0.855\n0.878\n0.692\n0.790\n0.837\n0.912\n0.926\nReLVLiNGAM\n0.827\n0.872\n0.880\n0.941\n0.973\n0.827\n0.872\n0.880\n0.941\n0.973\n0.827\n0.872\n0.880\n0.941\n0.973\n(c)\nProposed (A.7)\n0.697\n0.885\n0.975\n0.989\n1.000\n0.697\n0.885\n0.975\n0.989\n1.000\n0.697\n0.885\n0.975\n0.989\n1.000\nProposed\n0.618\n0.705\n0.797\n0.898\n0.830\n0.618\n0.705\n0.797\n0.898\n0.830\n0.618\n0.705\n0.797\n0.898\n0.830\n(d)\nProposed (A.7)\n0.392\n0.931\n0.816\n0.818\n0.944\n0.614\n0.958\n0.868\n0.906\n0.968\n0.456\n0.938\n0.829\n0.844\n0.952\nProposed\n0.167\n0.308\n0.408\n0.345\n0.296\n0.167\n0.308\n0.408\n0.345\n0.296\n0.167\n0.308\n0.408\n0.345\n0.296\n(e)\nProposed (A.7)\n0.973\n0.955\n0.912\n0.920\n0.909\n0.973\n0.955\n0.912\n0.920\n0.909\n0.973\n0.955\n0.912\n0.920\n0.909\nProposed\n0.568\n0.515\n0.529\n0.357\n0.372\n0.568\n0.515\n0.529\n0.357\n0.372\n0.568\n0.515\n0.529\n0.357\n0.372\n(f)\nProposed (A.7)\n1.000\n1.000\n1.000\n0.944\n0.974\n1.000\n1.000\n1.000\n0.944\n0.974\n1.000\n1.000\n1.000\n0.944\n0.974\nProposed\n0.944\n0.739\n0.867\n0.807\n0.931\n0.944\n0.739\n0.867\n0.807\n0.931\n0.944\n0.739\n0.867\n0.807\n0.931\nTable 4.3: The performances of the proposed method in Ncs with small sample sizes\nN\n50\n100\n200\n400\nτo\nαind\n0.01\n0.05\n0.1\n0.2\n0.3\n0.01\n0.05\n0.1\n0.2\n0.3\n0.01\n0.05\n0.1\n0.2\n0.3\n0.01\n0.05\n0.1\n0.2\n0.3\n0.001\n0\n0\n1\n2\n6\n0\n3\n4\n10\n9\n0\n6\n9\n13\n18\n5\n11\n16\n12\n11\n0.01\n0\n1\n1\n4\n4\n0\n1\n2\n3\n6\n1\n7\n7\n11\n11\n2\n7\n19\n16\n15\n0.1\n0\n1\n2\n5\n3\n0\n0\n4\n9\n7\n1\n4\n6\n11\n13\n2\n15\n17\n18\n10\nwhether the estimation accuracy of the model can be improved in small-\nsample settings by employing relatively larger values of the significance level\nαind for HSIC and the threshold τo than those used in the previous experi-\nments.\nWe conduct additional experiments under small-sample settings using\nModel (f) in Figure 4.1.\nThe sample sizes N are set to 50, 100, 200,\nand 400. In these experiments, only Ncs is used as the evaluation metric.\nThe parameters (τs, τm1, τm2) are set to (0.005, 0.001, 0.01), while the sig-\nnificance level of HSIC is chosen from αind ∈{0.01, 0.05, 0.1, 0.2, 0.3}, and\nτo ∈{0.001, 0.01, 0.1}.\nTable 4.3 reports the values of Ncs for each combination of αind and τo.\nThe values in bold represent the best performances with fixed N and τo,\nand those in italic represent the best performances with fixed N and αind.\nAlthough the estimation accuracy is not satisfactory when the sample size\nis small, the results in Table 4.3 suggest that relatively larger settings of\n30\n\nαind and τo tend to yield higher accuracy. The determination of appropriate\nthreshold values for practical applications remains an important issue for\nfuture work.\n5. Real-World Example\nWe applied the proposed method to the Political Democracy dataset [25],\na widely used benchmark in structural equation modeling (SEM). Originally\nintroduced by Bollen [25], this dataset was designed to examine the relation\nbetween the level of industrialization and the level of political democracy\nacross 75 countries in 1960 and 1965. It includes indicators for both indus-\ntrialization and political democracy in each year, and is typically modeled\nusing confirmatory factor analysis (CFA) as part of a structural equation\nmodel. In the standard SEM formulation, the model consists of three la-\ntent variables: ind60, representing the level of industrialization in 1960; and\ndem60 and dem65, representing the level of political democracy in 1960\nand 1965, respectively. ind60 is measured by per capita GNP (X1), per\ncapita energy consumption (X2), and the percentage of the labor force in\nnonagricultural sectors (X3). dem60 and dem65 are each measured by four\nindicators: press freedom (Y1, Y5), freedom of political opposition (Y2, Y6),\nfairness of elections (Y3, Y7), and effectiveness of the elected legislatures (Y4,\nY8). The SEM in Bollen [25] specifies paths from ind60 to both dem60 and\ndem65, and from dem60 to dem65.\nThe marginal model for X1, X2 and Y3, . . . , Y6 in the model in Bollen [25]\nis as shown in Figure 5.1 (a). This marginal model satisfies the assumptions\nA1-A3, as well as those of LiNGLaM [21] and LiNGLaH [23]. We examined\nwhether the proposed method, LiNGLaM, and LiNGLaH can recover the\nmodel in Figure 5.1 (a) from observational data X1, X2 and Y3, . . . , Y6. We\nset (τs, τm1, τm2) = (0.005, 0.001, 0.01).\nSince the sample size is as small\nas N = 75, we set (τo, αind) = (0.1, 0.2), which are relatively large values,\nfollowing the discussion in Section 4.4. The upper bound on the number of\nlatent variables is set to 2.\nThe resulting DAGs obtained by each method are shown in Figure 5.1 (b)–(d).\nAmong them, the proposed method estimates the same DAG as in Bollen [25].\nLiNGLaM fails to estimate the correct clusters and the causal structure\namong the latent variables. LiNGLaH incorrectly clusters all observed vari-\nables into two clusters. This result indicates that the proposed method not\nonly outperforms existing methods such as LiNGLaM and LiNGLaH for mod-\n31\n\nind60\ndem60\ndem65\nX1\nX2\nY3\nY4\nY5\nY6\n(a) Model in Bollen [25]\nL1\nL2\nL3\nX1\nX2\nY3\nY4\nY5\nY6\n(b) The proposed method\nL1\nL2\nL3\nY6\nX1\nX2\nY3\nY4\nY5\n(c) LiNGLaM\nL1\nX1\nX2\nY3\nY4\nY5\nY6\nL2\n(d) LiNGLaH\nFigure 5.1: The application on the political democracy dataset.\nels to which those methods are applicable, but is also effective even when the\nsample size is not large.\n6. Conclusion\nIn this paper, we propose a novel algorithm for estimating LvLiNGAM\nmodels in which causal structures exist both among latent variables and\namong observed variables. Causal discovery for such a class of LvLiNGAM\nhas not been completely addressed in any previous studies.\nThrough numerical experiments, we also confirmed the consistency of\nthe proposed method with the theoretical results on its identifiability. Fur-\nthermore, by applying the proposed method to the Political Democracy\n32\n\ndataset [25], a standard benchmark in structural equation modeling, we con-\nfirmed its practical usefulness.\nHowever, the class of models to which our proposed method can be applied\nremains limited. In particular, the assumptions that each observed variable\nhas at most one latent parent and that there are no edges between clusters\nare restrictive. As mentioned in Section 2.1, there exist classes of models\nthat can be identified by the proposed method even when some variables\nhave no latent parents. For further details, see Appendix E. However, even\nso, the proposed method cannot be applied to many generic canonical mod-\nels. 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Roweis (Eds.), Advances in Neural Information Processing Systems,\nVol. 20, Curran Associates, Inc., 2007, pp. 1–8.\n[29] G. Darmois, Analyse générale des liaisons stochastiques: etude partic-\nulière de l’analyse factorielle linéaire, Revue de l’Institut International\nde Statistique / Review of the International Statistical Institute 21 (1/2)\n(1953) 2–8.\n[30] V. P. Skitovich, On a property of the normal distribution, Doklady\nAkademii Nauk 89 (1953) 217–219.\n[31] M. Cai, P. Gao, H. Hara, Learning linear acyclic causal model including\ngaussian noise using ancestral relationships (2024).\n36\n\nA. Some Theorems and Lemmas for Proving Theorems in the Main\nText\nIn this section, we present several theorems and lemmas that are required\nfor the proofs of the theorems in the main text. In the following sections, we\nassume that the coefficient from each latent variable Li to its observed child\nXi with the highest causal order is normalized to λii = 1.\nTheorem A.1 (Darmois-Skitovitch theorem [29, 30]). Define two random\nvariables X1 and X2 as linear combinations of independent random variables\nu1, . . . , um:\nX1 =\nm\nX\ni=1\nα1iui,\nX2 =\nm\nX\ni=1\nα2iui\nThen, if X1 and X2 are independent, all variables ui for which α1iα2i ̸= 0\nare Gaussian.\nLemma A.2. Let X1 and X2 be mutually dependent observed variables in\nLvLiNGAM in (2.2) with mutually independent and non-Gaussian distur-\nbances u. Under Assumptions A4 and A5, cum(4)(X1, X1, X2, X2) and\ncum(4)(X1, X1, X1, X2) are generically non-zero.\nProof. Let X1 and X2 be linear combinations of u1, . . . , up+q:\nX1 =\np+q\nX\ni=1\nα1iui,\nX2 =\np+q\nX\ni=1\nα2iui.\nWhen X1 ̸⊥⊥X2, there must be uj with α1jα2j ̸= 0 by Theorem A.1. There-\nfore, generically\ncum(4)(X1, X1, X2, X2) =\np+q\nX\ni=1\nα2\n1iα2\n2icum(4)(ui, ui, ui, ui) ̸= 0\nA similar proof shows that generically\ncum(4)(X1, X1, X1, X2) ̸= 0.\nLemma A.3. For Vi, Vj ∈V , let αji be the total effect from Vi to Vj. As-\nsume that Vi ∈Anc(Vj). Then, it holds generically that Vi and Vj are not\nconfounded if and only if αjk = αji·αik generically holds for all Vk ∈Anc(Vi).\n37\n\nProof. Please refer to Lemma A.2 in Cai et al. [31] for the proof of sufficiency.\nWe prove the necessity by contrapositive. Suppose that Vi and Vj are\nconfounded. We can arbitrarily choose a Vk as their backdoor common an-\ncestor and assume αjk = αji · αik. From the faithfulness condition, it follows\nthat αji ̸= 0, αik ̸= 0, and αjk ̸= 0. Let Vk ≺Vk+1 ≺· · · ≺Vi−1 ≺Vi ≺\nVi+1 ≺· · · ≺Vj−1 ≺Vj be one possible causal order consistent with the\nmodel. Define\nPjk :=\n\n\n−bk+1,k\n1\n· · ·\n0\n0\n0\n· · ·\n0\n...\n...\n...\n...\n...\n...\n· · ·\n0\n−bi−1,k\n−bi−1,k+1\n· · ·\n1\n0\n0\n· · ·\n0\n−bi,k\n−bi,k+1\n· · ·\n−bi,i−1\n1\n0\n· · ·\n0\n−bi+1,k\n−bi+1,k+1\n· · ·\n−bi+1,i−1\n−bi+1,i\n1\n· · ·\n0\n...\n...\n...\n...\n...\n...\n...\n−bj−1,k\n−bj−1,k+1\n· · ·\n−bj−1,i−1\n−bj−1,i\n−bj−1,i+1\n· · ·\n1\n−bj,k\n−bj,k+1\n· · ·\n−bj,i−1\n−bj,i\n−bj,i+1\n· · ·\n−bj,j−1\n\n\n.\nPji and Pik are defined in the same way. Then,\nαjk = (−1)k+j · |Pjk|,\nαji = (−1)i+j · |Pji|,\nαik = (−1)k+i · |Pik|\nTherefore, αjk = αji · αik implies\n|Pji| · |Pik| = |Pjk|.\n(A.1)\nThe left-hand side of (A.1) equals the determinant of Pjk, in which the (i, i)-\nentry replaced by 0, which implies that (i, i) minor of Pjk vanishes, that\nis,\n\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f\n−bk+1,k\n1\n· · ·\n0\n0\n· · ·\n0\n...\n...\n...\n...\n...\n· · ·\n0\n−bi−1,k\n−bi−1,k+1\n· · ·\n1\n0\n· · ·\n0\n−bi+1,k\n−bi+1,k+1\n· · ·\n−bi+1,i−1\n1\n· · ·\n0\n...\n...\n...\n...\n...\n...\n−bj−1,k\n−bj−1,k+1\n· · ·\n−bj−1,i−1\n−bj−1,i+1\n· · ·\n1\n−bj,k\n−bj,k+1\n· · ·\n−bj,i−1\n−bj,i+1\n· · ·\n−bj,j−1\n\f\f\f\f\f\f\f\f\f\f\f\f\f\f\f\n= 0.\nThe space of brc, r = k+1, . . . , j, c = k, . . . , j−1 satisfying the above equation\nis a real algebraic set and constitutes a measure-zero subset of the parameter\nspace. Hence, generically |Pji| · |Pik| ̸= |Pjk|.\n38\n\nLemma A.4. Let Li and Lj be the latent parents of Xi and Xj, respectively.\nUnder Assumptions A1 and A4,\nXi ⊥⊥Xj ⇔Li ⊥⊥Lj.\nProof. The proof is trivial.\nLemma A.5. Let Xi and Xj be the observed variables with the highest\ncausal order within the clusters formed by the observed children of their la-\ntent parents, Li and Lj, respectively.\nUnder Assumptions A1 and A3, if\nConf(Li, Lj) = ∅,\n1. Li ⊥⊥Lj ⇒Conf(Xi, Xj) = ∅.\n2. Li ∈Anc(Lj) ⇒Conf(Xi, Xj) = {Li}.\nProof. When Li ⊥⊥Lj, Xi ⊥⊥Xj according to Lemma A.4. Therefore, no la-\ntent confounder exists between Xi and Xj.\nSuppose Li ∈Anc(Lj). Under Assumptions A1 and A3, Conf(Xi, Xj) ⊂\nAnc(Xi) ∩Anc(Xj) = Anc(Li) ∪{Li}.\nEvery path originating from the\nvariables in Anc(Li) to Xi and Xj passes through Li. Therefore, the only\npossible latent confounder of Xi and Xj is Li.\nLemma A.6. Let Xi and Xj be the observed variables with the highest causal\norder within the clusters formed by the observed children of their latent par-\nents, Li and Lj, respectively. Under Assumption A1, if Xi and Xj have only\none latent confounder, Li and Lj do not have multiple confounders.\nProof. We will prove this lemma by contrapositive.\nAssume that |Conf(Li, Lj)| ≥2. There exist two distinct nodes Lk, Lk′ ∈\nConf(Li, Lj) such that there are directed paths from Lk and Lk′ to Li and Lj,\nrespectively, and the two paths share no node other than their starting points.\nTherefore, by Assumption A1, two directed paths also exist from Lk to Xi\nand Xj, sharing no node other than Lk. Hence, Conf(Li, Lj) ⊂Conf(Xi, Xj),\nwhich implies that |Conf(Xi, Xj)| ≥2.\nTheorem A.7. Vi, Vj ∈V are two confounded observed variables. Assume\nthat all sixth cross-cumulants of u are non-zero. Then\n(c(6)\ni,i,i,j,j,j)2 = c(6)\ni,i,i,i,j,jc(6)\ni,i,j,j,j,j\nif and only if the following two conditions hold simultaneously:\n39\n\n1. There exists no direct path between Vi and Vj.\n2. Vi and Vj share only one (latent or observed) confounder in the canon-\nical model over Vi and Vj.\nProof. Without loss of generality, assume that Vj /∈Anc(Vi). Define IA, IB,\nand IC by\nIA ={k | Vk ∈Anc(Vi) ∩Anc(Vj)},\nIB ={k | Vk ∈Anc(Vi) \\ Anc(Vj)},\nIC ={k | Vk ∈Anc(Vj) \\ (Anc(Vi) ∪{Vi})}.\nThen, Vi and Vj are expressed as\nVi =\nX\nk∈IA\nαikuk +\nX\nk∈IB\nαikuk + ui,\nVj =\nX\nk∈IA\n(αjk + αjiαik)uk +\nX\nk∈IB\nαjkuk +\nX\nk∈IC\nαjkuk + αjiui + uj.\nSince Vk /∈Conf(Vi, Vj) for all k ∈IB, we have\nX\nk∈IB\nαjkuk = αji\nX\nk∈IB\nαikuk,\nby Lemma A.3. Let ˜ui and ˜uj be\n˜ui =\nX\nk∈IB\nαikuk + ui,\n˜uj =\nX\nk∈IC\nαjkuk + uj.\nThen,\nVi =\nX\nk∈IA\nαikuk + ˜ui,\nVj =\nX\nk∈IA\n(αjk + αjiαik)uk + αji˜ui + ˜uj.\n(A.2)\nNecessity: We assume that αji = 0 and |Conf(Vi, Vj)| = 1. Denote the\ndisturbance of the unique confounder of Vi and Vj by uc. Then Vi and Vj are\nexpressed as\nVi = ˜ui + αicuc,\nVj = ˜uj + αjcuc.\n40\n\nThe sixth cross-cumulants of Vi and Vj is obtained by direct computation as\nfollows:\nc(6)\ni,i,i,j,j,j = α3\nicα3\njccum(6)(uc),\nc(6)\ni,i,j,j,j,j = α2\nicα4\njccum(6)(uc),\nc(6)\ni,i,i,i,j,j = α4\nicα2\njccum(6)(uc).\nTherefore we have\n\u0010\nc(6)\ni,i,i,j,j,j\n\u00112\n= c(6)\ni,i,j,j,j,jc(6)\ni,i,i,i,j,j.\nSufficiency: According to Hoyer et al. [7], uk, k ∈IA can be merged as one\nconfounder, that is, |Conf(Vi, Vj)| = 1 in the canonical model over Vi and Vj.\nFrom (A.2), we have\nc(6)\ni,i,i,j,j,j =\nX\nk∈IA\nα3\nik(αjk + αjiαik)3cum(6)(uk) + (αji)3cum(6)(˜ui),\nc(6)\ni,i,i,i,j,j =\nX\nk∈IA\nα4\nik(αjk + αjiαik)2cum(6)(uk) + (αji)2cum(6)(˜ui),\nc(6)\ni,i,j,j,j,j =\nX\nk∈IA\nα2\nik(αjk + αjiαik)4cum(6)(uk) + (αji)4cum(6)(˜ui).\nFor notational simplicity, we denote the first terms in the right-hand side of\nthe three equations by A33, A42, and A24, respectively. When\n(c(6)\ni,i,i,j,j,j)2 = c(6)\ni,i,j,j,j,jc(6)\ni,i,i,i,j,j,\nwe have\nA2\n33 + 2A33(αji)3cum(6)(˜ui) + (αji)6cum(6)(˜ui)2\n= A42A24 + (αji)2A24cum(6)(˜ui) + A42(αji)4cum(6)(˜ui) + (αji)6cum(6)(˜ui)2,\nwhich is equivalent to\n(2(αji)3A33 −(αji)2A24 −(αji)4A42)cum(6)(˜ui) + (A2\n33 −A42A24) = 0.\nThis implies\n2(αji)2A33 −(αji)A24 −(αji)3A42 = 0,\nA2\n33 −A42A24 = 0.\n(A.3)\n41\n\nWe note that\nA2\n33 = A42A24 ⇔\n X\nk∈IA\nα3\nik(αjk + αjiαik)3cum(6)(uk)\n!2\n=\n X\nk∈IA\nα4\nik(αjk + αjiαik)2cum(6)(uk)\n! X\nk∈IA\nα2\nik(αjk + αjiαik)4cum(6)(uk)\n!\n.\nBy Lagrange’s identity\n∀k ∈IA,\n(αjk + αjiαik)\nαik\n= c ⇔αjk\nαik\n= c −αji\nfor a constant c.\nFor the first equation in (A.3), we have\n2(αji)2A33 −αjiA24 −(αji)3A42\n= αji\n\"\u0012\nαji −A33\nA42\n\u00132\n−A2\n33\nA2\n42\n+ A24\nA42\n#\n= 0.\nSince A2\n33 = A42A24 and (αjk + αjiαik) = c · αik,\nαji\n\"\u0012\nαji −A33\nA42\n\u00132\n−A2\n33\nA2\n42\n+ A24\nA42\n#\n= αji\n\"\nαji −\nP\nk∈IA α3\nik(αjk + αjiαik)3cum(6)(uk)\nP\nk∈IA α4\nik(αjk + αjiαik)2cum(6)(uk)\n#2\n= αji\n\"\nαji −c3 P\nk∈IA α6\nikcum(6)(uk)\nc2 P\nk∈IA α6\nikcum(6)(uk)\n#2\n= αji(αji −c)2 = 0.\nThus, αji = 0 or c. αji = c implies that αjk = 0, which contradicts the\nfaithfulness assumption. Therefore, we conclude that αji = 0, which implies\nthat there is no directed path from Vi to Vj.\nLemma A.8. Assume that Xi and Xj belong to distinct clusters, that they\nare the children with the highest causal order of Li and Lj, respectively, and\nthat Lj /∈Anc(Li).\nUnder Assumptions A1 and A3, if Xi and Xj have\nonly one latent confounder Lc in the canonical model over them, one of the\nfollowing conditions generically holds:\n42\n\n1. Conf(Li, Lj) = ∅. Then, Li and Lc are identical, and\nc(k)\n(Xi→Xj)(Lc) = c(k)\n(Xi→Xj)(Li) = cum(k)(Li),\nc(k)\n(Xj→Xi)(Lc) = c(k)\n(Xj→Xi)(Li) = cum(k)(αll\nji · Li).\n2. Conf(Li, Lj) = {Lc}. Then,\nc(k)\n(Xi→Xj)(Lc) = cum(k)(αll\nic · Lc),\nc(k)\n(Xj→Xi)(Lc) = cum(k)(αll\njc · Lc).\nProof. According to Lemma A.6, |Conf(Li, Lj)| = 0 or 1. Since Xi and Xj\nare confounded by Lc, Xi ̸⊥⊥Xj, which implies Li ̸⊥⊥Lj by Lemma A.4.\nFirst, consider the case where Conf(Li, Lj) = ∅. According to Lemma\nA.5, when Li ̸⊥⊥Lj, the only possible latent confounder of Xi and Xj is Li.\nFurthermore, there is at least one causal path from Li to Lj.\nDefine IA and IB by\nIA = {k | Lk ∈Anc(Li)},\nIB = {k | Lk ∈Anc(Lj) \\ (Anc(Li) ∪{Li})}.\nThen, Xi and Xj are written as\nXi =\n X\nk∈IA\nαll\nikϵk + ϵi\n!\n+ ei,\n(A.4)\nXj =\n X\nk∈IA\nαll\njkϵk +\nX\nk∈IB\nαll\njkϵk + αll\njiϵi + ϵj\n!\n+ ej.\nFrom Lemma A.3, we have\nX\nk∈IA\nαll\njkϵk = αll\nji ·\nX\nk∈IA\nαll\nikϵk.\nLetting vj = P\nk∈IB αll\njkϵk + ϵj + ej, Xj is rewritten as\nXj = αll\nji(\nX\nk∈IA\nαll\nikϵk + ϵi) + vj.\n(A.5)\nNote that Li, ei, and vj are mutually independent. From Proposition 2.3\nwith ℓ= 1, the roots of the polynomial on α\n\f\f\f\f\f\f\f\n1\nα\nα2\nc(3)\ni,i,i\nc(3)\ni,i,j\nc(3)\ni,j,j\nc(4)\ni,i,i,i\nc(4)\ni,i,i,j\nc(4)\ni,i,j,j\n\f\f\f\f\f\f\f\n= 0\n43\n\nare αoo\nji and αol\nji. From (A.4) and (A.5), we have\n\f\f\f\f\f\f\f\n1\nα\nα2\nc(3)\ni,i,i\nc(3)\ni,i,j\nc(3)\ni,j,j\nc(4)\ni,i,i,i\nc(4)\ni,i,i,j\nc(4)\ni,i,j,j\n\f\f\f\f\f\f\f\n=\n\u0000(αll\nji)3cum(3)(Li)cum(4)(Li) −(αll\nji)3cum(3)(Li)cum(4)(Li)\n\u0001\n−α\n\u0000(αll\nji)2 · (cum(3)(ei)cum(4)(Li) −cum(3)(Li)cum(4)(ei))\n\u0001\n+ α2 \u0000(αll\nji) · (cum(3)(ei)cum(4)(Li) −cum(3)(Li)cum(4)(ei))\n\u0001\n= 0,\nwhich is generically equivalent to\n−α · (αll\nji)2 + α2 · αll\nji = 0.\n(A.6)\nThe roots of (A.6) are α = 0, αll\nji. Since Xi and Xj belong to different clusters,\nαoo\nji = 0 and hence αol\nji = λjjαll\nji = αll\nji.\nFrom Proposition 2.4,\n\u0014 1\n1\n0\nαll\nji\n\u0015 \"\nc(k)\n(Xi→Xj)(ei)\nc(k)\n(Xi→Xj)(Lc)\n#\n=\n\"\nc(k)\ni,...,i,i\nc(k)\ni,...,i,j\n#\n=\n\u0014 cum(k)(ei) + cum(k)(Li)\ncum(k)(αll\nji · Li)\n\u0015\n.\nThen, we have c(k)\n(Xi→Xj)(ei) = cum(k)(ei) and c(k)\n(Xi→Xj)(Lc) = cum(k)(Li). In\nthe same way, we can obtain c(k)\n(Xj→Xi)(Lc) = cum(k)(αll\nji · Li).\nNext, we consider the case where Conf(Li, Lj) = {Lc}. Then, only Lc\nhas outgoing directed paths to Xi and Xj that share no latent variable other\nthan Lc.\nDefine IA and IB by\nIA = {k | Lk ∈Anc(Li) \\ (Anc(Lc) ∪{Lc})},\nIB = {k | Lk ∈Anc(Lj) \\ (Anc(Lc) ∪Anc(Li) ∪{Lc, Li})}.\nFollowing Salehkaleybar et al. [9], Xi and Xj are expressed as\nXi = (\nX\nk∈IA\nαll\nikϵk + αll\nicLc + ϵi) + ei,\nXj = (\nX\nk∈IA\nαll\njkϵk +\nX\nk∈IB\nαll\njkϵk + αll\njcLc + αll\njiϵi + ϵj) + ej.\n44\n\nSince\nX\nk∈IA\nαll\njkϵk = αll\nji ·\nX\nk∈IA\nαll\nikϵk,\nby Lemma A.3, Xj is rewritten as\nXj = αll\nji(\nX\nk∈IA\nαll\nikϵk + ϵi) +\nX\nk∈IB\nαll\njkϵk + αll\njcLc + ϵj + ej.\nThe cumulants c(6)\ni,i,i,j,j,j, c(6)\ni,i,i,i,j,j, and c(6)\ni,i,j,j,j,j are written as follows:\nc(6)\ni,i,i,j,j,j = (αll\nji)3 X\nk∈IA\n(αll\nik)6cum(6)(ϵk) + (αll\nji)3cum(6)(ϵi)\n+ (αll\nic)3(αll\njc)3cum(6)(Lc),\nc(6)\ni,i,i,i,j,j = (αll\nji)2 X\nk∈IA\n(αll\nik)6cum(6)(ϵk) + (αll\nji)2cum(6)(ϵi)\n+ (αll\nic)4(αll\njc)2cum(6)(Lc),\nc(6)\ni,i,j,j,j,j = (αll\nji)4 X\nk∈IA\n(αll\nik)6cum(6)(ϵk) + (αll\nji)4cum(6)(ϵi)\n+ (αll\nic)2(αll\njc)4cum(6)(Lc).\nSince Xi and Xj have only one confounder, (c(6)\ni,i,i,j,j,j)2 = c(6)\ni,i,i,i,j,jc(6)\ni,i,j,j,j,j holds\nfrom Theorem A.7, which implies\n2(αll\nji)3(αll\nic)3(αll\njc)3 = (αll\nji)2(αll\nic)2(αll\njc)4 + (αll\nji)4(αll\nic)4(αll\njc)2\n⇔(αll\nji)2\n \nαll\nji −αll\njc\nαll\nic\n!2\n= 0.\nWhen αll\nji = αll\njc/αll\nic, all directed paths from Lc to Lj pass through Li by\nLemma A.3, and then Lc is not a confounder between Li and Lj, which leads\nto a contradiction. Therefore, αll\nji = 0. Letting vi = P\nk∈IA αll\nikϵk + ϵi + ei\nand vj = P\nk∈IB αll\njkϵk + ϵj + ej, Xi and Xj are rewritten as\nXi = αll\nicLc + vi,\nXj = αll\njcLc + vj.\nTherefore, we find that the unique confounder of Xi and Xj is Lc.\n45\n\nIn the same way as (A.6), we have generically\n\f\f\f\f\f\f\f\n1\nα\nα2\nc(3)\ni,i,i\nc(3)\ni,i,j\nc(3)\ni,j,j\nc(4)\ni,i,i,i\nc(4)\ni,i,i,j\nc(4)\ni,i,j,j\n\f\f\f\f\f\f\f\n= 0 ⇔−α · αll\njc + α2 · αll\nic = 0\n(A.7)\nThen, α = 0, αll\njc/αll\nic.\nDue to the assumption A3, αoo\nji = 0.\nTherefore,\nαol\njc = αll\njc/αll\nic from Proposition 2.3. According to Proposition 2.4\n\" 1\n1\n0\nαll\njc\nαll\nic\n# \"\nc(k)\n(Xi→Xj)(ei)\nc(k)\n(Xi→Xj)(Lc)\n#\n=\n\"\nc(k)\ni,...,i,i\nc(k)\ni,...,i,j\n#\n=\n\"\ncum(k)(vi) + cum(k)(αll\nicLc)\nαll\njc\nαll\nic · cum(k)(αll\nicLc)\n#\nSolving this equation yields c(k)\n(Xi→Xj)(ei) = cum(k)(vi) and c(k)\n(Xi→Xj)(Lc) =\ncum(k)(αll\nic·Lc). In the same way, we can obtain c(k)\n(Xj→Xi)(Lc) = cum(k)(αll\njcLc).\nLemma A.9. Let Xi and Xj be two dependent observed variables. Assume\nthat Conf(Xi, Xj) = {Lc} and that there is no directed path between Xi and\nXj in the canonical model over them. Then, ˜e(Xi,Xj) ⊥⊥Lc.\nProof. Let vi and vj be the disturbances of Xi and Xj, respectively, in the\ncanonical model over Xi and Xj. Xi and Xj are expressed as\nXi = αol\nicLc + vi,\nXj = αol\njcLc + vj.\nThen, ˜e(Xi,Xj) is given by\n˜e(Xi,Xj) = αol\nicLc + vi −(αol\nic)2(αol\njc)2cum(4)(Lc)\n(αol\nic)(αol\njc)3cum(4)(Lc) (αol\njcLc + vj) = vi −αol\nic\nαol\njc\nvj,\nwhich shows that ˜e(Xi,Xj) ⊥⊥Lc.\nB. Proofs of Theorems in Section 3.1\nB.1. The proof of Theorem 3.4\nProof. We prove this theorem by contrapositive. Let Li and Lj be the re-\nspective latent parents of Xi and Xj, assuming that Lj /∈Anc(Li).\nWe divide the proof into four cases.\n46\n\n1. Xi and Xj are pure:\n1-1. The number of observed children of Lj is greater than one:\nThere exists another observed child Xk of Lj such that Pa(Xk) =\n{Lj}. Then Xi, Xj and Xk are expressed as\nXi = Li + ei,\nXj = Lj + ej,\nXk = λkjLj + ek,\nand e(Xi,Xj|Xk) is given by\ne(Xi,Xj|Xk) = (Li + ei) −Cov(Li, λkjLj)\nCov(Lj, λkjLj)(Lj + ej).\nSince λkj ̸= 0 and Cov(Li, Lj) ̸= 0 generically, both e(Xi,Xj|Xk)\nand Xk contain terms of ϵj, implying that Xk ̸⊥⊥e(Xi,Xj|Xk) from\nTheorem A.1.\n1-2. The number of observed children of Lj is one:\nAccording to Assumption A2, Lj must have a latent child, denoted\nby Lk, and let Xk be the child of Lk with the highest causal order.\nThen\nXi = Li + ei,\nXj = Lj + ej,\nXk = Lk + ek = akjLj +\nX\nh:Lh∈Pa(Lk)\\{Lj}\nakhLh + ϵk + ek.\nand e(Xi,Xj|Xk) is given by\ne(Xi,Xj|Xk) = Xi −Cov(Xi, Xk)\nCov(Xj, Xk)(Lj + ej).\nSince akj ̸= 0 and Cov(Xi,Xj)\nCov(Xj,Xk) ̸= 0 generically, e(Xi,Xj|Xk) ̸⊥⊥Xk from\nTheorem A.1.\n2. At least one of Xi, Xj is impure: Assume that Xi is impure and that\na directed edge exists between Xi and Xk. The proof proceeds analo-\ngously when Xj is impure.\n2-1. Xi ∈Pa(Xk):\nXi, Xj and Xk are expressed as\nXi = Li + ei,\nXj = Lj + ej,\nXk = (λki + bki)Li + bkiei + ek,\n47\n\nrespectively, and e(Xi,Xj|Xk) is given by\ne(Xi,Xj|Xk) = (Li + ei) −(λki + bki)Var(Li) + bkiVar(ei)\n(λki + bki)Cov(Li, Lj)\n(Lj + ej).\nSince both e(Xi,Xj|Xk) and Xk contain ei, e(Xi,Xj|Xk) ̸⊥⊥Xk from\nTheorem A.1.\n2-2. Xk ∈Pa(Xi):\nXi, Xj and Xk are expressed as\nXi = (λii + bik)Li + bikek + ei,\nXj = Lj + ej,\nXk = Li + ek.\nrespectively, and e(Xi,Xj|Xk) is given by\ne(Xi,Xj|Xk) = (λii + bik)Li + bikek + ei\n−(bik + λii)Var(Li) + bikVar(ek)\nCov(Li, Lj)\n(Lj + ej).\nSince bik ̸= 0, and both e(Xi,Xj|Xk) and Xk contain terms about ek,\ne(Xi,Xj|Xk) ̸⊥⊥Xk is shown from Theorem A.1.\nC. Proofs of Theorems and Lemmas in Section 3.2\nC.1. The proof of Theorem 3.5\nProof. Sufficiency: If Li is a latent source in G, then no confounder ex-\nists between Li and any other latent variable Lj.\nBy Lemma A.5, when\nXi and Xj belong to distinct clusters, we have Conf(Xi, Xj) = {Li}, since\nLi ̸⊥⊥Lj. If Xi and Xj belong to the same cluster confounded by Li, then\nagain Conf(Xi, Xj) = {Li}.\nNecessity: Note that Xoc ⊂X \\ {Xi}. We will prove the necessity by\nshowing that if Li is not a latent source, there exists some Xj ∈Xoc \\ {Xi}\nsuch that Xi ̸⊥⊥Xj and Conf(Xi, Xj) ̸= {Li} in the canonical model over\n{Xi, Xj}.\nLet Ls be a latent source and let Xs be the child of Ls with the highest\ncausal order. By Lemma A.5 and the fact that Li ̸⊥⊥Ls, we have Conf(Xi, Xs) =\n{Ls} ̸= {Li}.\n48\n\nC.2. The proof of Corollary 3.6\nProof. According to Theorem 3.5, sufficiency is immediate.\nWe therefore\nprove only necessity by showing that if Li is not a latent source, then neither\ncase 1 nor case 2 holds.\nIf Li is not a latent source, then Xi ̸= ∅or |Xoc \\ {Xi}| ≥2, and there-\nfore case 1 is not satisfied. We now consider case 2 and show that either\ncondition (a) or (b) is not satisfied. First, note that condition (a) does not\nhold whenever there exists Xj ∈Xoc \\ {Xi} with |Conf(Xi, Xj)| ̸= 1. Hence,\nassume that condition (a) holds.\nLet Ls be the latent source in G and Xs be its observed child with the\nhighest causal order among ˆCs. Let Xj ∈Xoc \\ {Xi} have a latent parent\nLj, and let Lc be the unique confounder between Xi and Xj, and Xc be its\nobserved child with the highest causal order. We have Xc, Xs ∈Xoc \\ {Xi},\nand |Conf(Xi, Xs)| = |Conf(Xi, Xj)| = 1.\nNext, we divide the following\ndiscussion into two cases depending on whether Li has a latent child:\n1. If Li ∈Pa(Lj), then by Lemma A.8,\nc(k)\n(Xi→Xj)(L(i,j)) = cum(k)(Li),\nc(k)\n(Xi→Xs)(L(i,s)) = cum(k)(αll\nis · Ls).\nThus, c(k)\n(Xi→Xj)(L(i,j)) ̸= c(k)\n(Xi→Xs)(L(i,s)) generically, and condition (b)\nis not satisfied.\n2. If Li has no latent children, then Xi ̸= ∅, and\nc(k)\n(Xi→Xi′)(L(i,i′)) = cum(k)(Li).\nAlso c(k)\n(Xi→Xi′)(L(i,i′)) ̸= c(k)\n(Xi→Xs)(L(i,s)) generically, so condition (b) is\nnot satisfied.\nC.3. The proof of Theorem 3.11\nProof. We define sets\nIA = {h | Lh ∈Anc(Li) ∩Anc(Lj) \\ {L1}},\nIB = {h | Lh ∈Anc(Li) \\ (IA ∪{L1})},\nIC = {h | Lh ∈Anc(Lj) \\ (IA ∪{L1})}.\n49\n\nThen,\nX1 = ϵ1 + e1,\nLi = αll\ni1ϵ1 +\nX\nh∈IA\nαll\nihϵh +\nX\nh∈IB\nαll\nihϵh + ϵi,\nXi = Li + ei,\nLj = αll\nj1ϵ1 +\nX\nh∈IA\nαll\njhϵh +\nX\nh∈IC\nαll\njhϵh + ϵj,\nXj = Lj + ej,\nWe can easily show that\ncum(Xi, Xi, X1, X1)\ncum(Xi, X1, X1, X1) = αll\ni1,\nand hence, we have\n˜e(Xi,X1) =\nX\nh∈IA\nαll\nihϵh +\nX\nh∈IB\nαll\nihϵh + ϵi + ei −αll\ni1e1.\nSufficiency: Assume Li ⊥⊥Lj in the submodel induced by G−({L1}), which\nimplies that IA = ∅. Therefore, ˜e(Xi,X1) and Xj can be written as\nXj = αll\nj1ϵ1 +\nX\nh∈IC\nαll\njhϵh + ϵj + ej,\n˜e(Xi,X1) =\nX\nh∈IB\nαll\nihϵh + ϵi + ei −αll\ni1e1.\nThus, we conclude that Xj ⊥⊥˜e(Xi,X1).\nSimilarly, we can also show that\nXi ⊥⊥˜e(Xj,X1).\nNecessity: Assume Li ̸⊥⊥Lj in the submodel induced by G−({L1}), which\nimplies that IA ̸= ∅.\nSince neither αll\nih nor αll\njh for h ∈IA equals zero,\nXj ̸⊥⊥˜e(Xi,X1) by the contrapositive of Theorem A.1. Similarly, we can also\nshow that Xi ̸⊥⊥˜e(Xj,X1).\nC.4. The proof of Theorem 3.12\nAccording to Lemma A.9, ˜e(Xi,X1) can be regarded as a statistic obtained\nby removing the influence of L1 from Xi. Based on this observation, we now\nprovide the proof of Theorem 3.12.\n50\n\nProof. Let L1 and Li be two latent variables, and define the set\nIA = {h | Lh ∈Anc(Li) \\ {L1}}.\nThen, X1, Xi, and ˜e(Xi,X1) are represented as\nX1 = ϵ1 + e1,\nXi = αll\ni1ϵ1 +\nX\nh∈IA\nαll\nihϵh + ϵi + ei,\n˜e(Xi,X1) =\nX\nh∈IA\nαll\nihϵh + ϵi + ei −αll\ni1e1,\nrespectively.\nSufficiency: If Li is the latent source in G−({L1}), IA = ∅. Hence, we have\nXi = αll\ni1ϵ1 + ϵi + ei,\n˜e(Xi,X1) = ϵi + ei −αll\ni1e1.\nAssume that Xj ∈Xoc \\ {X1, Xi}. Define IB by\nIB = {h | Lh ∈Anc(Lj) \\ {L1, Li}}.\nThen, ˜e(Xi,X1) and Xj are written as\n˜e(Xi,X1) = ϵi + (ei −αll\ni1e1),\nXj = αll\njiϵi + αll\nj1ϵ1 +\nX\nh∈IB\nαll\njhϵh + ϵj + ej,\nwhich shows that Conf(˜e(Xi,X1), Xj) = {ϵi}.\nNext, assume that Xi ̸= ∅and Xi′ ∈Xi. We divide the discussion into\nthe following two cases.\n1. Xi /∈Anc(Xi′). Define the set\nIC = {k | Xk ∈Anc(Xi′) ∩ˆCi}.\nWe note that i /∈IA, and rewrite Xi′ as\n˜e(Xi,X1) = ϵi + (ei −αll\ni1e1),\nXi′ = αol\ni′iLi +\nX\nk∈IC\nαoo\ni′kek + ei′ = αol\ni′i(αll\ni1ϵ1 + ϵi) +\nX\nk∈IC\nαoo\ni′kek + ei′,\nhence, Conf(˜e(Xi,X1), Xi′) = {ϵi}.\n51\n\n2. Xi ∈Anc(Xi′). Define the set\nIC = {k | Xk ∈(Anc(Xi′) ∩ˆCi) \\ {Xi}}.\n˜e(Xi,X1) and Xi′ are written as\n˜e(Xi,X1) = ϵi + (ei −αll\ni1e1),\nXi′ = αol\ni′i(αll\ni1ϵ1 + ϵi) + αol\ni′iei +\nX\nk∈IC\nαoo\ni′kek + ei′.\nBoth ei and ϵi appear in ˜e(Xi,X1) and Xi′. Since only ˜e(Xi,X1) contains\ne1 while only Xi′ contains ei′, there is no ancestral relation between\nthem in their canonical model, according to Lemma 5 of Salehkaleybar\net al. [9]. Hence, Conf(˜e(Xi,X1), Xi′) = {ϵi, ei}, and we have\nConf(˜e(Xi,X1), Xi′) ∩Conf(˜e(Xi,X1), Xj) = {ϵi}.\nNecessity: By contrapositive, we aim to show that if Li is not a latent\nsource, then either condition 1 or 2 does not hold. Assume that Ls is the\nlatent source in G−({L1}), and that Xs is its observed child with the highest\ncausal order. Then, we have\n˜e(Xi,X1) = αll\nisϵs +\nX\nh∈IA\\{s}\nαll\nihϵh + ϵi + ei −αll\ni1e1\nXs = as1ϵ1 + ϵs + es,\nimplying that Conf(˜e(Xi,X1), Xs) = {ϵs} ̸= {ϵi}. Thus, condition 1 is not\nsatisfied.\nC.5. The Proof of Lemma 3.14\nProof. Since it is trivial that\n˜e(Xi,˜es) = ϵi +\ni−1\nX\nk=s\nαll\nihϵh + ei,\nwhen i > s and s = 1, we only discuss the remaining two cases.\nWe first prove the case where s = i by induction on i. When i = 1,\n˜e(X1,˜e1) = X1 = ϵ1 + e1,\n52\n\nwhere U[1] = ϵ1.\nAssume that the inductive assumption holds up to i. Then,\n˜e(Xh,˜eh) = ϵh + U[h],\n1 ≤h ≤i.\nSince Xi+1 is expressed as\nXi+1 = ϵi+1 + ei+1 +\ni\nX\nh=1\nαll\ni+1,hϵh,\nwe have\nρ(Xi+1,˜e(Xh,˜eh)) = αll\ni+1,h\nfor h = 1, . . . , i, according to Definition 3.13. Hence, we have\n˜e(Xi+1,˜ei+1) = Xi+1 −\ni\nX\nh=1\nρ(Xi+1,˜e(Xh,˜eh))˜e(Xh,˜eh)\n= ϵi+1 + ei+1 +\ni\nX\nh=1\nαll\ni+1,hϵh −\ni\nX\nh=1\nαll\ni+1,h(ϵh + U[h])\n= ϵi+1 + ei+1 −\ni\nX\nh=1\nαll\ni+1,hU[h] = ϵi+1 + U[i+1],\nwhere U[i+1] = ei+1 −Pi\nh=1 αll\ni+1,hU[h]. Thus, the claim holds for all i by\ninduction.\nNext, we discuss the case where i > s and s > 1. According to Definition\n3.13,\n˜e(Xi,˜es) = Xi −\ns−1\nX\nh=1\nρ(Xi,˜e(Xh,˜eh))˜e(Xh,˜eh),\nρ(Xi,˜e(Xh,˜eh)) = αll\nih.\nUsing the conclusion of the case where i = s, we obtain\n˜e(Xi,˜es) = ϵi +\ni−1\nX\nh=1\nαll\nihϵh + ei −\ns−1\nX\nh=1\nαll\nih\n\u0000ϵh + U[h]\n\u0001\n= ϵi +\ni−1\nX\nh=s\nαll\nihϵh + U[s−1] + ei.\n53\n\nC.6. The proof of Theorem 3.15\nProof. The proof of this theorem follows similarly to that of Theorem 3.11.\nC.7. The proof of Theorem 3.16\nProof. The proof of this theorem follows similarly to that of Theorem 3.12.\nC.8. The proof of Corollary 3.17\nAccording to Theorem 3.16, sufficiency is immediate. We therefore prove\nonly necessity by showing that if Li is not a latent source in G−({L1, . . . , Ls−1}),\nthen neither case 1 nor case 2 holds.\nIf Li is not a latent source, Xi ̸= ∅or |Xoc \\ {X1, . . . , Xs−1, Xi}| ≥2, and\ntherefore case 1 is not satisfied. We will show that one of the conditions (a),\n(b), and (c) is not satisfied. First, note that condition (a) does not hold when-\never there exists Xj ∈Xoc \\ {X1, . . . , Xs−1, Xi} with |Conf(˜e(Xi,˜es), Xj)| ̸= 1.\nHence, assume that condition (a) holds.\nAssume that Ls is the latent source of G−({L1, . . . , Ls−1}), and that Xs\nis its observed child with the highest causal order.\nLet Lj be the latent\nparent of Xj ∈Xoc \\ {X1, . . . , Xs−1, Xi}, respectively.\nSince the condi-\ntion (a) holds, |Conf(˜e(Xi,˜es), Xs)| = |Conf(˜e(Xi,˜es), Xj)| = 1, where ˜es =\n(˜e(X1,˜e1), . . . , ˜e(Xs−1,˜es−1)).\nThen, Xs and ˜e(Xi,˜es) are written as\nXs =\ns−1\nX\nh=1\nαll\nshϵh + ϵs + es,\n˜e(Xi,˜es) = ϵi +\ni−1\nX\nh=s\nαll\nihϵh + U[s−1] + ei,\naccording to Lemma 3.14. Hence, we have Conf(˜e(Xi,˜es), Xs) = {ϵs}.\nAssume that Li has a latent child Lj and that none of the descendants of\nLi are parents of Lj. Xj is expressed by\nXj =\nj−1\nX\nh=1\nαll\njhϵh + ϵj + ej.\n54\n\nBoth ˜e(Xi,˜es) and Xj involve linear combinations of ϵs, . . . , ϵi. Since {ϵs, . . . , ϵi}\nare mutually independent and |Conf(˜e(Xi,˜es), Xj)| = 1, αll\njh = αll\njiαll\nih accord-\ning to Hoyer et al. [7], and then Xj can be rewritten as\nXj =\n\n\n\n\n\n\n\n\n\n\n\ns−1\nX\ni=1\nαll\njhϵh + αll\nji\n \nϵi +\ni−1\nX\nh=s\nαll\nihϵh\n!\n+ ϵj + ej,\nj = i + 1,\ns−1\nX\ni=1\nαll\njhϵh + αll\nji\n \nϵi +\ni−1\nX\nh=s\nαll\nihϵh\n!\n+\nj−1\nX\nh=i+1\nαll\njhϵh + ϵj + ej,\nj > i + 1.\nTherefore,\nc(k)\n(˜e(Xi,˜es)→Xj)(L(i,j)) = cum(k)\n \nϵi +\ni−1\nX\nh=s\nαll\nihϵh\n!\n,\nc(k)\n(˜e(Xi,˜es)→Xs)(L(i,s)) = cum(k)(αll\nisϵs),\naccording to Lemma A.8. Therefore, we conclude that c(k)\n(˜e(Xi,˜es)→Xj)(L(i,j)) ̸=\nc(k)\n(˜e(Xi,˜es)→Xs)(L(i,s)) generically, and condition (b) is not satisfied. Next, we\nassume that Li does not have latent children, so that Xi ̸= ∅. Assume that\nXi′ ∈Xi and it can be expressed as\nXi′ =\ni−1\nX\nh=1\nαll\nihϵh + ϵi + bi′iei + ei′.\nThen,\nConf(˜e(Xi,˜es), Xi′) =\n\n\n\nn\nϵi + Pi−1\nh=s αll\nihϵh\no\n,\nbi′i = 0,\nn\nϵi + Pi−1\nh=s αll\nihϵh, ei\no\n,\nbi′i ̸= 0.\nIn either case, there is one latent L(i,i′) satisfies that\nc(k)\n(˜e(Xi,˜es)→Xi′)(L(i,i′)) = cum(k)\n \nϵi +\ni−1\nX\nh=s\nαll\nihϵh\n!\n,\nwhich generically implies\nc(k)\n(˜e(Xi,˜es)→Xi′)(L(i,i′)) ̸= c(k)\n(˜e(Xi,˜es)→Xs)(L(i,s)).\nThus, the condition (c) is not satisfied.\n55\n\nD. Proofs of Theorems in Section 3.3\nD.1. Theorem 3.19\nProof. From Lemma 3.14,\n˜e(Xi−k,˜ei−k) = ϵi−k + U[i−k].\n(D.1)\nBy definition, ˜ri,k−1 is written as\n˜ri,k−1 = Xi −\ni−1\nX\nh=i−(k−1)\naihXh\n=\ni−1\nX\nh=1\naihLh + ϵi + ei −\ni−1\nX\nh=i−(k−1)\naih(Lh + eh)\n=\ni−(k+1)\nX\nh=1\naihLh + ai,i−kϵi−k + ϵi + ei −\ni−1\nX\nh=i−(k−1)\naiheh\n= V[i−(k+1)] + ai,i−kϵi−k + ϵi + U[i]\\[i−k],\n(D.2)\nwhere V[i−(k+1)] is a linear combination of {ϵ1, . . . , ϵi−(k+1)} and U[i]\\[i−k] is a\nlinear combination of {ei−(k−1), . . . , ei}. From (D.1) and (D.2), we can show\nthat\n˜ri,k−1 ⊥⊥˜e(Xi−k,˜ei−k) ⇔ai,i−k = 0,\nand otherwise\nρ(Xi,˜e(Xi−1,˜ei−1)) = a2\ni,i−1 cum(4)(ϵi−1)\nai,i−1 cum(4)(ϵi−1) = ai,i−1\ngenerically holds.\nE. Reducing an LvLiNGAM\nIn this paper, we have discussed the identifiability of LvLiNGAM under\nthe assumption that each observed variable has exactly one latent parent.\nHowever, even when some observed variables do not have latent parents, by\niteratively marginalizing out sink nodes and conditioning on source nodes to\nremove such variables one by one, the model can be progressively reduced\nto one in which each observed variable has a single latent parent. This can\n56\n\nbe achieved by first estimating the causal structure involving the observed\nvariables without latent parents. ParceLiNGAM [11] or RCD [12, 13] can\nidentify the ancestral relationship between two observed variables if at least\none of them does not have a latent parent, and remove the influence of the\nobserved variable without a latent parent.\nModels 1–3 in Figure E.1 contain observed variables that do not have a\nlatent parent. According to Definition 1.1, X1 and X2 in Models 1 and 3\nbelong to distinct clusters whose latent parents are L1 and L2, respectively,\nwhereas in Model 2, X1 and X2 share the same latent parent L1. Model 3\ncontains a directed path between clusters, whereas Models 1 and 2 do not.\nWe consider the model reduction procedure for Models 1–3 individually.\nExample E.1 (Model 1). By using ParceLiNGAM or RCD, we can identify\nX4 →X1, X4 →X2, X1 →X5, and X3 →X5. Since X5 is a sink node,\nthe induced subgraph obtained by removing X5 represents the marginal model\nover the remaining variables. Since X4 is a source node, if we replace X1\nand X2 with the residuals r(4)\n1\nand r(4)\n2\nobtained by regressing them on X4,\nthen the induced subgraph obtained by removing X4 represents the conditional\ndistribution given X4. As a result, Model 1 is reduced to the model shown in\nFigure E.1 (d). This model satisfies Assumptions A1–A3.\nExample E.2 (Model 2). X1 and X2 are confounded by L1, and they are\nmediated through X3. By using ParceLiNGAM or RCD, the ancestral rela-\ntionship among X1, X2, X3 can be identified. Let r(1)\n3\nbe the residual obtained\nby regressing X3 on X1. Let ˜r(3)\n2\nbe the residual obtained by regressing X2\non r(1)\n3 . According to [11] and [12, 13], the model for L1, X1, and r(3)\n2\ncorre-\nsponds to the one shown in Figure E.1 (e). This model satisfies Assumptions\nA1–A3.\nExample E.3 (Model 3). In Model 3, X1 ∈Anc(X3), and they are mediated\nby X5. By using ParceLiNGAM or RCD, the ancestral relationship among\nX1, X3, X5 can be identified. Let r(1)\n5\nbe the residual obtained by regressing X5\non X1. Let ˜r(5)\n3\nbe the residual obtained by regressing X3 on r(1)\n5 . According\nto [11] and [12, 13], by reasoning in the same way as for Models 1 and 2,\nModel 3 is reduced to the model shown in Figure E.1 (f). This model doesn’t\nsatisfy Assumptions A1–A3.\nUsing [11] and [12, 13], ancestral relations between pairs of observed vari-\nables that include at least one variable without a latent parent can be iden-\n57\n\nL1\nX1\nL2\nX2\nX3\nX4\nX5\n(a) Model 1\nL1\nX1\nX2\nX3\n(b) Model 2\nL1\nX1\nL2\nX2\nX3\nX4\nX5\n(c) Model 3\nL1\nr(4)\n1\nL2\nX3\nr(4)\n2\n(d) Reduced model of Model 1\nL1\nX1\n˜r(3)\n2\n(e) Reduced model of Model 2\nL1\nr(4)\n1\nL2\nr(4)\n2\n˜r(5)\n3\n(f) Reduced model of Model 3\nFigure E.1: Three models that can be reduced.\ntified. The graph obtained by the model reduction procedure is constructed\nby iteratively applying the following steps:\n(i) iteratively remove observed variables without latent parents that ap-\npear as source or sink nodes, updating the induced subgraph at each\nstep so that any new source or sink nodes are subsequently removed;\n(ii) when an observed variable without a latent parent serves as a mediator,\nremove the variable and connect its parent and child with a directed\nedge.\nIf no directed path exists between any two observed variables with distinct\nlatent parents, the model obtained through the model reduction procedure\nsatisfies Assumptions A1–A3. Conversely, if there exist two observed vari-\nables with distinct latent parents that are connected by a directed path,\nthe model obtained through the model reduction procedure does not satisfy\nAssumption A3. In summary, Assumption A1 can be generalized to\nA1′. Each observed variable has at most one latent parent.\nProposition E.4. Given observed data generated from an LvLiNGAM MG\nthat satisfies the assumptions A1′ and A2–A5, the latent causal structure\n58\n\namong the latent variables, the directed edges from the latent variables to\nthe observed variables, and the ancestral relationships among the observed\nvariables can be identified by using the proposed method in combination with\nParceLiNGAM and RCD.\n59\n"},"cleaned":{"kind":"string","value":"Causal Discovery for Linear DAGs with Dependent Latent Variables via Higher-order Cumulants Ming Caia, Penggang Gaoa, Hisayuki Haraa,b aGraduate -cho, Kyoto, 606-8501, Japan bInstitute for Liberal Arts and Sciences, Kyoto University, Yoshida Nihonmatsu-cho, Kyoto, 606-8501, Japan Abstract This paper addresses the problem of estimating causal directed acyclic graphs in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM). Existing methods assume mutually independent latent confounders or cannot properly handle models with causal relationships among observed variables. We propose a novel algorithm that identifies causal DAGs in LvLiNGAM, allowing causal structures among latent variables, among observed variables, and between the two. The proposed method leverages higher-order cumulants of observed data to identify the causal structure. Extensive simulations and experiments with real-world data demonstrate the validity and practical utility of the proposed algorithm. Keywords: canonical model, causal discovery, cumulants, DAG, latent confounder, Triad constraints 1. Introduction Estimating causal directed acyclic graphs (DAGs) in the presence of latent confounders has been a major challenge in causal analysis. Conventional causal discovery methods, such as the Peter-Clark (PC) algorithm [1], Greedy Equivalence Search (GES) [2], and the Linear Non-Gaussian Acyclic Model (LiNGAM) [3, 4], focus solely on the causal model without latent confounders. Fast Causal Inference (FCI) [1] extends the PC algorithm to handle latent variables, recovering a partial ancestral graph (PAG) under the faithfulness assumption. However, FCI is computationally intensive and, moreover, often 16 Oct 2025 fails to determine the causal directions. Really Fast Causal Inference (RFCI) [5] trades off some independence tests for speed, at the loss of estimating accuracy. Greedy Fast Causal Inference (GFCI) [6] hybridizes GES and FCI but inherits the limitation of FCI. The assumption of linearity and non-Gaussian disturbances in the causal model enables the identification of causal structures beyond the PAG. The linear non-Gaussian acyclic model with latent confounders (LvLiNGAM) is an extension of LiNGAM that incorporates latent confounders. Hoyer et al. [7] demonstrated that LvLiNGAM can be transformed into a canonical model in which all latent variables are mutually independent and causally precede the observed variables. They proposed estimating the canonical models using overcomplete ICA [8], assuming that the number of latent variables is known. Overcomplete ICA can identify the causal DAG only up to permutations and scaling of the variables. Thus, substantial computational effort is required to identify the true causal DAG from the many candidate models. Another limitation of overcomplete ICA is its tendency to converge to local optima. Salehkaleybar et al. [9] improved the algorithm by reducing the candidate models. Other methods for estimating LvLiNGAM, based on linear regression analysis and independence testing, have also been developed.[10, 11, 12, 13]. Furthermore, Multiple Latent Confounders LiNGAM (ML-CLiNGAM) [14] and FRITL [15] initially identify the causal skeleton using a constraint-based method, and then estimate the causal directions of the undirected edges in the skeleton using linear regression and independence tests. While these methods can identify structures among observed variables that are not confounded by latent variables, they cannot necessarily determine the causal direction between two variables confounded by latent variables. More recently, methods using higher-order cumulants have led to new developments in the identification of canonical LvLiNGAMs. Cai et al. [16] assume that each latent variable has at least three observed children, and that there exists a subset of these children that are not connected by any other observed or latent variables. Then, cumulants are employed to identify onelatent-component structures and latent influences are recursively removed to recover the underlying causal relationships. Chen et al. [17] show that if two observed variables share one latent confounder, the causal direction between them can be identified by leveraging higher-order cumulants. Schkoda et al. [18] introduced ReLVLiNGAM, a recursive approach that leverages higher-order cumulants to estimate canonical LvLiNGAM with multiple la2 tent parents. One strength of ReLVLiNGAM is that it does not require prior knowledge of the number of latent variables. The methods reviewed so far are estimation methods for the canonical LvLiNGAM. A few methods, however, have been proposed to estimate the causal DAG of LvLiNGAM when latent variables exhibit causal relationships. A variable is said to be pure if it is conditionally independent of other observed variables given its latent parents; otherwise, it is called impure. Silva et al. [19] showed that the latent DAG is identifiable under the assumption that each latent variable has at least three pure children, by employing tetrad conditions on the covariance of the observed variables. Cai et al. [20] proposed a two-phase algorithm, LSTC (learning the structure of latent variables based on Triad Constraints), to identify the causal DAG where each latent variable has at least two children, all of which are pure, and each observed variable has a single latent parent. Xie et al. [21] generalized LSTC and defined the linear non-Gaussian latent variable model (LiNGLaM), where observed variables may have multiple latent parents but no causal edges among them, and proved its identifiability. In [20] and [21], causal clusters are defined as follows: Definition 1.1 (Causal cluster [20, 21]). A set of observed variables that share the same latent parents is called a causal cluster. Their methods consist of two main steps: identifying causal clusters and then recovering the causal order of latent variables. LSTC and the algorithm for LiNGLaM estimate clusters of observed variables by leveraging the Triad constraints or the generalized independence noise (GIN) conditions. It is also possible to define clusters in the same manner as Definition 1.1 for models where causal edges exist among observed variables. However, when impure observed variables exist, their method might fail to identify the clusters, resulting in an incorrect estimation of both the number of latent variables and latent DAGs. Several recent studies have shown that LvLiNGAM remains identifiable even when some observed variables are impure [22, 23, 24]. However, these methods still rely on the existence of at least some pure observed variables in each cluster. 1.1. Contributions In this paper, we relax the pure observed children assumption of Cai et al. [20] and investigate the identifiability of the causal DAG for an extended model that allows causal structures both among latent variables and among 3 observed variables. Using higher-order cumulants of the observed data, we show the identifiability of the causal DAG of a class of LvLiNGAM and propose a practical algorithm for estimating the class. The proposed method first estimates clusters using the approaches of [20, 21]. When causal edges exist among observed variables, the clusters estimated by using Triad constraints or GIN conditions may be over-segmented compared to the true clusters. The proposed method leverages higher-order cumulants of observed variables to refine these clusters, estimates causal edges within clusters, determines the causal order among latent variables, and finally estimates the exact causal structure among latent variables. In summary, our main contributions are as follows: 1. Demonstrate identifiability of causal DAGs in a class of LvLiNGAM, allowing causal relationships among latent and observed variables. 2. Extend the causal cluster estimation methods of [20] and [21] to handle cases where directed edges exist among observed variables within clusters. 3. Propose a top-down algorithm using higher-order cumulants to infer the causal order of latent variables. 4. Develop a bottom-up recursive procedure to reconstruct the latent causal DAG from latent causal orders. The rest of this paper is organized as follows. Section 2 defines the class of LvLiNGAM considered in this study. In Section 2, we also summarize some basic facts on higher-order cumulants. Section 3 describes the proposed method in detail. Section 4 presents numerical simulations to demonstrate the effectiveness of the proposed method. Section 5 evaluates the usefulness of the proposed method by applying it to the Political Democracy dataset [25]. Finally, Section 6 concludes the paper. All proofs of theorems, corollaries, and lemmas in the main text are provided in the Appendices. 2. Preliminaries 2.1. LvLiNGAM Let X = (X1, . . . , Xp)⊤and L = (L1, . . . , Lq)⊤be vectors of observed and latent variables, respectively. In this paper, we identify these vectors with the corresponding set of variables. Define V = X ∪L = {V1, . . . , Vp+q}. Let G = (V , E) be a causal DAG. Vi →Vj denotes a directed edge from Vi 4 to Vj. Anc(Vi), Pa(Vi), and Ch(Vi) are the sets of ancestors, parents, and children of Vi, respectively. We use Vi ≺Vj to indicate that Vi precedes Vj in a causal order. The LvLiNGAM considered in this paper is formulated as L X = A 0 Λ B L X + ε e , (2.1) where A = {aji}, B = {bji}, and Λ = {λji} are matrices of causal coefficients, while ε and e denote vectors of independent non-Gaussian disturbances associated with L and X, respectively. Let aji, λji, and bji be the causal coefficients from Li to Lj, from Li to Xj, and from Xi to Xj, respectively. Due to the arbitrariness of the scale of latent variables, we may, without loss of generality, set one of the coefficients λji to 1 for some Xj ∈Ch(Li). Hereafter, such a normalization will often be used. A and B can be transformed into lower triangular matrices by row and column permutations. We assume that the elements of ε and e are mutually independent and follow non-Gaussian continuous distributions. Let MG denote the LvLiNGAM defined by G. As shown in (2.1), we assume in this paper that all observed variables are not ancestors of any latent variables. Consider the following reduced form of (2.1), L X = (Iq -A)-1 0 (Ip -B)-1Λ(Iq -A)-1 (Ip -B)-1 ε e . Let αll ji, αol ji, and αoo ji represent the total effects from Li to Lj, Li to Xj, and Xi to Xj, respectively. Thus, (Iq -A)-1 = {αll ji}, (Ip -B)-1Λ(Iq -A)-1 = {αol ji}, and (Ip -B)-1 = {αoo ji }. The total effect from Vi to Vj is denoted by αji, with the superscript omitted. M := [(Ip -B)-1Λ(Iq -A)-1, (Ip -B)-1] is called a mixing matrix of the model (2.1). Denote u = (ε⊤, e⊤)⊤. Then, X is written as X = Mu, (2.2) which conforms to the formulation of the overcomplete ICA problem [8, 26, 7]. M is said to be irreducible if every pair of columns is linearly independent. G is said to be minimal if and only if M is irreducible. If G is not minimal, some latent variables can be absorbed into other latent variables, resulting in a minimal graph [9]. MG is called the canonical model when A = 0 5 and M is irreducible. Hoyer et al. [7] showed that any LvLiNGAM can be transformed into an observationally equivalent canonical model. For example, the LvLiNGAM defined by the DAG in Figure 2.1 (a) is the canonical model of the LvLiNGAM defined by the DAG in Figure 2.1 (b). Hoyer et al. [7] also demonstrated that, when the number of latent variables is known, the canonical model can be identified up to observational equivalent models using overcomplete ICA. Salehkaleybar et al. [9] showed that, even when A ̸= 0, the irreducibility of M is a necessary and sufficient condition for the identifiability of the number of latent variables. However, they did not provide an algorithm for estimating the number of latent variables. Schkoda et al. [18] proposed ReLVLiNGAM to estimate the canonical model with generic coefficients even when the number of latent variables is unknown. However, the canonical model derived from an LvLiNGAM with A ̸= 0 lies in a measure-zero subset of the parameter space, which prevents ReLVLiNGAM from accurately identifying the number of latent confounders between two observed variables in such cases. For example, ReLVLiNGAM may not identify the canonical model in Figure 2.1 (a) from data generated by the LvLiNGAM in Figure 2.1 (b). Cai et al. [20] and Xie et al. [21] demonstrated that within LvLiNGAMs where all the observed children of latent variables are pure, there exists a class, such as the models shown in Figure 2.1 (b), in which the causal order among latent variables is identifiable. They proposed algorithms for estimating the causal order. However, the complete causal structure cannot be identified solely from the causal order, and their algorithm cannot be generL1 L2 L3 X1 X2 X3 X4 X5 X6 (a) An example of canonical LvLiNGAM L1 L2 L3 X1 X4 X2 X5 X3 X6 (b) An LvLiNGAM that can be identified by [20, 21] Figure 2.1: Examples of LvLiNGAMs 6 alized to cases where causal edges exist among observed variables or where latent variables do not have sufficient pure children. In this paper, we introduce the following class of models, which generalizes the class of models in Cai et al. [20] by allowing causal edges among the observed variables, and consider the problem of identifying the causal order among observed variables within each cluster as well as the causal structure among the latent variables. A1. Each observed variable has only one latent parent. A2. Each latent variable has at least two children, at least one of which is observed. A3. There are no direct causal paths between causal clusters. A4. The model satisfies the faithfulness assumption. A5. The higher-order cumulant of each component of the disturbance u is nonzero. In Section 3, we demonstrate that the causal structure of latent variables and the causal order of observed variables for the LvLiNGAM that satisfies Assumption A1-A5 are identifiable, and we provide an algorithm for estimating the causal DAG for this class. The proposed method enables the identification not only of the causal order among latent variables but also of their complete causal structure. Under Assumption A1, every observed variable is assumed to have one latent parent. However, even if there exist observed variables without latent parents, the estimation problem can sometimes be reduced to a model satisfying Assumption A1 by applying ParceLiNGAM [11] or repetitive causal discovery (RCD) [12, 13] as a preprocessing step of the proposed method. Details are provided in Appendix E. 2.2. Cumulants The proposed method leverages higher-order cumulants of observed data to identify the causal structure among latent variables. In this subsection, we summarize some facts on higher-order cumulants. First, we introduce the definition of a higher-order cumulant. Definition 2.1 (Cumulants [27]). Let i1, . . . , ik ∈{1, . . . , p}. The k-th order cumulant of random vector (Xi1, . . . , Xik) is c(k) i1,...,ik = cum(k)(Xi1, . . . , Xik) 7 = X (I1,...,Ih) (-1)h-1(h -1)!E \"Y j∈I1 Xj # · · · E \"Y j∈Ih Xj # , where the sum is taken over all partitions (I1, . . . , Ih) of (i1, . . . , ik). If i1 = · · · = ik = i, we write cum(k)(Xi) to denote cum(k)(Xi, . . . , Xi). The k-th order cumulants of the observed variables of LvLiNGAM satisfy c(k) i1,i2,...,ik = cum(k)(Xi1, . . . , Xik) = q X j=1 αol i1j · · · αol ikjcum(k)(εj) + p X j=1 αoo i1j · · · αoo ikjcum(k)(ej) We consider an LvLiNGAM in which all variables except Xi and Xj are regarded as latent variables. We refer to the canonical model that is observationally equivalent to this model as the canonical model over Xi and Xj. Let Conf(Xi, Xj) = {L′ 1, L′ 2, · · · , L′ l} be the set of latent confounders in the canonical model over Xi and Xj, where all L′ h ∈Conf(Xi, Xj) are mutually independent. Without loss of generality, we assume that Xj /∈ Anc(Xi). Then, Xi and Xj are expressed as Xi = l X h=1 α′ ihL′ h + vi, Xj = l X h=1 α′ jhL′ h + αoo ji vi + vj, (2.3) where vi and vj are disturbances, and α′ ih and α′ jh are total effects from L′ h to Xi and Xj, respectively, in the canonical model over them. We note that the model (2.3) is a canonical model with generic parameters, and that lis equal to the number of confounders in the original model MG. Schkoda et al. [18] proposed an algorithm for estimating the canonical model with generic parameters by leveraging higher-order cumulants. Several of their theorems concerning higher-order cumulants are also applicable to the canonical model over Xi and Xj. They define a Pk2-k1+1 i=0 i ×k1 matrix A(k1,k2) (Xi→Xj) as follows: 8 A(k1,k2) (Xi→Xj) = c(k1) i,i,...,i c(k1) i,i,...,j . . . c(k1) i,i,...,j c(k1+1) i,i,i,...,i c(k1+1) i,i,i,...,j . . . c(k1+1) i,i,j,...,j c(k1+1) j,i,i,...,i c(k1+1) j,i,i,...,j . . . c(k1+1) j,i,j,...,j ... ... ... ... c(k2) i,...,i,i,i,...,i,i c(k2) i,...,i,i,i,...,i,j . . . c(k2) i,...,i,i,j,...,j,j ... ... ... ... c(k2) j,...,j,i,i,...,i,i c(k2) j,...,j,i,i,...,i,j . . . c(k2) j,...,j,i,j,...,j,j , (2.4) where k1 q, some clusters must be merged to recover the true clustering. X can be partitioned into maximal subsets of mutually dependent variables. Each observed variable in these subsets has a corresponding latent parent. If the causal order of the latent parents within each subset is determined, then the causal order of the entire latent variable set ˆL is uniquely determined. Henceforth, we assume, without loss of generality, that X itself forms one such maximal subset. 3.2.1. Determining the Source Latent Variable Since we assume that X consists of mutually dependent variables, G contains only one source node among the latent variables. Theorem 3.5 provides the necessary and sufficient condition for a latent variable to be a source node. Theorem 3.5. Let Xi denote the observed variable with the highest causal order among ˆCi. Then, Li is generically a latent source in G if and only if Conf(Xi, Xj) are identical across all Xj ∈X \\ {Xi} such that Xi ̸⊥⊥Xj in the canonical model over Xi and Xj, with their common value being {Li}. Note that in Stage I, the ancestral relationships among the observed variables are determined. Hence, the causal order within each cluster can also be determined. Let Xj be the observed variable with the highest causal order among ˆCj for j = 1, . . . , K and define Xoc = {X1, . . . , XK}. When | ˆCi| ≥2, let Xi′ be any element in ˆCi \\ {Xi}. Define Xi by Xi = {Xi′}, if | ˆCi| ≥2 ∅, if | ˆCi| = 1. (3.2) Let L(i,j) denote a latent confounder of Xi and Xj in the canonical model over them. In the implementation, we verify whether the conditions of Theorem 3.5 are satisfied by using Corollary 3.6. Corollary 3.6. Assume k ≥3. Li is generically a latent source in G if and only if one of the following two cases holds: 1. Xi = ∅and |Xoc \\ {Xi}| = 1 13 L1 X1 X2 X3 (a) L1 L2 X1 X2 X3 (b) Figure 3.2: An example of merging clusters in Stage II 2. |(Xoc ∪Xi) \\ {Xi}| ≥2 and the following all hold: (a) In the canonical model over Xi and Xj, |Conf(Xi, Xj)| = 1 for Xj ∈(Xoc ∪Xi) \\ {Xi} such that Xi ̸⊥⊥Xj. (b) c(k) (Xi→Xj)(L(i,j)) are identical for Xj ∈(Xoc ∪Xi) \\ {Xi}. When Xi = ∅and |Xoc \\ {Xi}| = 1, it is trivial by Assumption A2 that Li is a latent source. Otherwise, for Li to be a latent source, it is necessary that |Conf(Xi, Xj)| = 1 for all Xj ∈(Xoc ∪Xi) \\ {Xi}. This can be verified by using Condition 1 of Proposition 2.2. In addition, if c(k) (Xi→Xj)(L(i,j)) for Xj ∈(Xoc ∪Xi) \\ {Xi} are identical, Li can be regarded as a latent source. When Xi ∈Anc(Xi′), the equation (2.5) yields two distinct solutions, c(k) (Xi→Xi′)(L(i,i′)) = c(k) (Xi→Xi′)(Li) and c(k) (Xi→Xi′)(ei), that are identifiable only up to a permutation of the two. If either of these two solutions equals c(k) (Xi→Xj)(L(i,j)) for all Xj ∈Xoc \\ {Xi}, then Li can be identified as the latent source. Example 3.7. Consider the models in Figure 3.2. For both models (a) and (b), the clusters estimated in Stage I are ˆC1 = {X1} and ˆC2 = {X2, X3}, and let L1 and L2 be the latent parents assigned to ˆC1 and ˆC2, respectively. Then, Xoc = {X1, X2}. In the model (a), we can assume λ11 = 1 without loss of generality. Then, the model (a) is expressed as X1 = ε1 + e1, X2 = λ21ε1 + e2, X3 = (λ21b32 + λ31)ε1 + b32e2 + e3. By Proposition 2.4 and assuming k ≥2, we can obtain c(k) (X1→X2)(L(1,2)) = cum(k)(ε1), c(k) (X2→X1)(L(2,1)) = c(k) (X2→X3)(L(2,3)) = cum(k)(λ21ε1). 14 Algorithm 2 Finding latent sources Input: Mutually dependent Xoc, ˆC, and AL Output: Xoc, ˆC, and a set of ancestral relationships between latent variables AL 1: Each cluster is assigned one latent parent and let ˆL be the set of latent parents 2: Apply Corollary 3.6 to find the latent sources Ls 3: Assume Ls ∈Ls and ˆCs ∈ˆC 4: if |Ls| ≥2 then 5: Merge the corresponding clusters into ˆCs and update ˆC and Xoc 6: Identify all latent parents in Ls with Ls 7: end if 8: for all Li ∈ˆL \\ Ls do 9: Anc(Li) ←{Ls} 10: end for 11: Xoc ←Xoc \\ {Xs} 12: AL ←AL ∪{Anc(Li) | Li ∈ˆL \\ Ls} ∪{Anc(Ls) = ∅} 13: return Xoc, ˆC, and AL Since |Xoc \\{X1}| = |{X2}| = 1 and c(k) (X2→X1)(L(2,1)) = c(k) (X2→X3)(L(2,3)), both L1 and L2 are determined as latent sources. The dependence between X1 and X2 leads to L1 and L2 being regarded as a single latent source, resulting in the merging of ˆC1 and ˆC2. In the model (b), we can assume λ11 = λ22 = 1 without loss of generality. Then, the model (b) is described as X1 = ε1 + e1, X2 = (a21ε1 + ε2) + e2, X3 = (b32 + λ31)(a21ε1 + ε2) + b32e2 + e3. Then, c(k) (X1→X2)(L(1,2)) = cum(k)(ε1), c(k) (X2→X1)(L(2,1)) = cum(k)(a21ε1) ̸= c(k) (X2→X3)(L(2,3)) = cum(k)(a21ε1 + ε2). Therefore, L1 is a latent source, while L2 is not. As in model (a), multiple latent variables may also be identified as latent sources. In such cases, their observed children are merged into a single cluster. Once Li is established as a latent source, it implies that Li is an ancestor of the other elements in ˆL. The procedure of Section 3.2.1 is summarized in Algorithm 2. 15 3.2.2. Determining the Causal Order of Latent Variables Next, we address the identification of subsequent latent sources after finding L1 in the preceding procedure. If the influence of the latent source can be removed from the observed descendant, the subsequent latent source may be identified through a procedure analogous to the one previously applied. The statistic ̃e(Xi,Xh), defined below, serves as a key quantity for removing such influence. Definition 3.8. Let Xi and Xh be two observed variables. Define ̃e(Xi,Xh) as ̃e(Xi,Xh) = Xi -ρ(Xi,Xh)Xh, where ρ(Xi,Xh) = cum(Xi, Xi, Xh, Xh) cum(Xi, Xh, Xh, Xh) Xi ̸⊥⊥Xh, 0 Xi ⊥⊥Xh. Under Assumption 5, when Xi ̸⊥⊥Xh, ρ(Xi,Xh) is shown to be generically finite and non-zero. See Lemma A.2 in the Appendix for details. Let Lh be the latent source, and let Xh be its observed child with the highest causal order. When there is no directed path between Xi and Xh, ̃e(Xi,Xh) can be regarded as Xi after removing the influence of Lh. Example 3.9. Consider the model in Figure 3.3 (a). We can assume λ22 = λ33 = 1 without loss of generality. Then, X1, X2 and X3 are described as X1 = ε1 + e1, X2 = a21ε1 + ε2 + e2, X3 = a31ε1 + ε3 + e3. We can easily show that ρ(X2,X1) = a21. Hence, we have ̃e(X2,X1) = -a21e1 + ε2 + e2. It can be seen that ̃e(X2,X1) does not depend on L1 = ε1, and that ̃e(X2,X1) and X3 are mutually independent. Example 3.10. Consider the model in Figure 3.3 (b). We can assume that λ11 = λ22 = λ33 = 1 without loss of generality. Then, the model is described as X1 = ε1 + e1, X2 = a21ε1 + ε2 + e2, 16 L1 L2 L3 X1 X3 X2 (a) L1 L2 L3 X1 X2 X3 X4 X5 (b) Figure 3.3: Examples of LvLiNGAMs X3 = a32a21ε1 + a32ε2 + ε3 + e3, X4 = λ42(a21ε1 + ε2) + e4, X5 = (λ53 + b53)(a32a21ε1 + a32ε2 + ε3) + b53e3 + e5. We can easily show that ρ(X2,X1) = a21 and ρ(X3,X1) = a32a21. Hence, we have ̃e(X2,X1) = -a21e1 + ε2 + e2, ̃e(X3,X1) = -a32a21e1 + a32ε2 + ε3 + e3. It can be seen that ̃e(X2,X1) and ̃e(X3,X1) are obtained by replacing L1 = ε1 with -e1. The model for ( ̃e(X2,X1), X3) and ( ̃e(X2,X1), X5) are described by canonical models with Conf( ̃e(X2,X1), X3) = Conf( ̃e(X2,X1), X5) = {ε2}, respectively. The model for ( ̃e(X3,X1), X2) and ( ̃e(X3,X1), X5) are described by canonical models with Conf( ̃e(X3,X1), X2) = {ε2} and Conf( ̃e(X3,X1), X5) = {a32ε2 + ε3, e3}, respectively. X5 contains {ε1, ε2, ε3, e3, e5}, and ̃e(X3,X1) contains {ε2, ε3, e1, e3}. Since these sets are not in an inclusion relationship, it follows from Lemma 5 of Salehkaleybar et al. [9] that there is no ancestral relationship between ̃e(X3,X1) and X5. It is noteworthy that ̃e(X3,X1) and X5 share two latent confounders, and that no ancestral relationship exists between them even though X3 ∈Anc(X5) in the original graph. Let L1 be the current latent source identified by the preceding procedure. Let G-({L1}) be the subgraph of G induced by V \\ ({L1} ∪ˆC1). By generalizing the discussions in Examples 3.9 and 3.10, we obtain the following theorems. Theorem 3.11. For Xi, Xj ∈Xoc \\ {X1} and their respective latent parent Li and Lj, Li ⊥⊥Lj | L1 if and only if ̃e(Xi,X1) ⊥⊥Xj. 17 Theorem 3.12. Let Li denote the latent parent of Xi ∈Xoc \\ {X1}. If | ˆCi| ≥2, let Xi′ be an element of ˆCi \\ {Xi}. Xi is defined in the same manner as (3.2). Then, Li is generically a source in G-({L1}) if and only if the following two conditions hold: 1. Conf( ̃e(Xi,X1), Xj) are identical for all Xj ∈Xoc \\ {X1, Xi} such that ̃e(Xi,X1) ̸⊥⊥Xj, with their common value being {εi}. 2. If Xi ̸= ∅, Conf( ̃e(Xi,X1), Xi′) ∩Conf( ̃e(Xi,X1), Xj) are identical for all Xj ∈Xoc \\ {X1, Xi}, with their common value being {εi}. By applying Theorem 3.11, we can obtain the family of maximal dependent subsets of Xoc \\ {X1} in the conditional distribution given L1. Theorem 3.12 allows us to verify whether Li is a latent source in G-({L1}). By recursively iterating such a procedure, the ancestral relationships among the latent variables can be identified. To achieve this, it is necessary to generalize ̃e(Xi,X1) defined as in Definition 3.13. Let G-({L1, . . . , Ls-1}) denote the subgraph of G induced by V except for {L1, . . . , Ls-1} and their observed children and L1, . . . , Ls-1 be latent sources in G, G-({L1}), G-({L1, L2}), . . . , G-({L1, . . . , Ls-2}), respectively. Then, {L1, . . . , Ls-1} has a causal order L1 ≺· · · ≺Ls-1. Definition 3.13. For i ≥s, ̃e(Xi, ̃es) is defined as follows. ̃e(Xi, ̃es) = Xi s = 1, Xi -Ps-1 h=1 ρ(Xi, ̃e(Xh, ̃eh)) ̃e(Xh, ̃eh) s > 1, where ̃es = ( ̃e(X1, ̃e1), . . . , ̃e(Xs-1, ̃es-1)). ̃e(Xi, ̃es) can be regarded as a statistic with the information of L1, . . . , Ls-1 eliminated from Xi. The following lemma shows that ̃e(Xi, ̃es) is obtained by replacing the information of ε1, . . . , εs-1 with that of e1, . . . , es-1. Lemma 3.14. Let X1, . . . , Xs-1, and Xi be the observed children with the highest causal order of L1, . . . , Ls-1, and Li, respectively. ̃e(Xi, ̃es) can be expressed as ̃e(Xi, ̃es) = εi + U[i] i = s εi + Pi-1 h=s αll ihεh + ei i > s and s = 1 εi + Pi-1 h=s αll ihεh + U[s-1] + ei i > s and s > 1, 18 where U[i] and U[s-1] are linear combinations of {e1, . . . , ei} and {e1, . . . , es-1}, respectively. By using ̃e(Xi, ̃es) in Definition 3.13, we obtain Theorems 3.15 and 3.16, which generalize Theorems 3.11 and 3.12, respectively. Theorem 3.15. For Xi, Xj ∈Xoc \\ {X1, . . . , Xs-1} and their respective latent parent Li and Lj, Li ⊥⊥Lj | {L1, . . . , Ls-1} if and only if ̃e(Xi, ̃es) ⊥⊥Xj. Theorem 3.16. Let Li be the latent parent of Xi ∈Xoc \\ {X1, . . . , Xs-1}. If | ˆCi| ≥2, let Xi′ be an element of ˆCi \\ {Xi}. Xi is defined in the same manner as (3.2). Then, Li is generically a latent source in G-({L1, . . . , Ls-1}) if and only if the following two conditions hold: 1. Conf( ̃e(Xi, ̃es), Xj) are identical for all Xj ∈Xoc \\ {X1, . . . , Xs-1, Xi} such that ̃e(Xi, ̃es) ̸⊥⊥Xj, with their common value being {εi}. 2. When Xi ̸= ∅, Conf( ̃e(Xi, ̃es), Xi′) ∩Conf( ̃e(Xi, ̃es), Xj) are identical for all Xj ∈Xoc \\ {X1, . . . , Xs-1, Xi} such that ̃e(Xi, ̃es) ̸⊥⊥Xj, with their common value being {εi}. As in Theorem 3.11, by applying Theorem 3.15, we can identify the family of maximal dependent subsets of Xoc \\ {X1, . . . , Xs-1} in the conditional distribution given {L1, . . . , Ls-1}. For each maximal dependent subset, we can apply Theorem 3.16 to identify the next latent source. In the implementation, we verify whether the conditions of Theorem 3.16 are satisfied using Corollary 3.17, which generalizes Corollary 3.6. Corollary 3.17. Assume k ≥3. Li is generically a latent source in G-({L1, . . . , Ls-1}) if and only if one of the following two cases holds: 1. Xi = ∅and |Xoc \\ {X1, . . . , Xs-1, Xi}| = 1. 2. |(Xoc ∪Xi) \\ {X1, . . . , Xs-1, Xi}| ≥2, and the following all hold: (a) In the canonical model over ̃e(Xi, ̃es) and Xj, |Conf( ̃e(Xi, ̃es), Xj)| = 1 for all Xj ∈Xoc \\ {X1, . . . , Xs-1, Xi} such that ̃e(Xi, ̃es) ̸⊥⊥Xj. (b) c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) are identical for all Xj ∈Xoc\\{X1, . . . , Xs-1, Xi} such that ̃e(Xi, ̃es) ̸⊥⊥Xj, where L(i,j) is the unique latent confounder in the canonical model over ̃e(Xi, ̃es) and Xj. 19 (c) ̃e(Xi, ̃es) and Xi′ has a latent confounder L(i,i′) in the canonical model over them that satisfies c(k) ( ̃e(Xi, ̃es)→Xi′)(L(i,i′)) = c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) for all Xj ∈Xoc\\{X1, . . . , Xs-1, Xi} such that ̃e(Xi, ̃es) ̸⊥⊥Xj, when Xi ̸= ∅. To determine whether Li is a latent source of G-({L1, . . . , Ls-1}), we first examine, using Condition 1 of Proposition 2.2, whether |Conf( ̃e(Xi, ̃es), Xj)| = 1, as in Section 3.2.1. If c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) are identical for Xj ∈(Xoc ∪ Xi)\\{X1, . . . , Xs-1, Xi}, Li is identified as a latent source. As in the previous case, when Xi ̸= ∅and Xi ∈Anc(Xi′), the equation (2.5) yields two distinct solutions for the higher-order cumulants of latent confounders. Here, we determine that Li is a latent source in G-({L1, . . . , Ls-1}) if either of two solutions of (2.5) equals to c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) for Xj ∈Xoc \\{X1, . . . , Xs-1, Xi}. If multiple latent sources are identified for any element in a mutually dependent maximal subset of Xoc \\ {X1, . . . , Xs-1}, the corresponding clusters must be merged. As latent sources are successively identified, the correct set of latent variables L, the ancestral relationships among L, and the correct clusters are also successively identified. The procedure of Section 3.2.2 is presented in Algorithm 3. Algorithm 4 combines Algorithms 2 and 3 to provide the complete procedure for Stage II. Example 3.18. For the model in Figure 3.1 (a), the estimated clusters obtained in Stage I are {X1}, {X2}, {X3, X5}, and {X4}, with their corresponding latent parents denoted as L1, L2, L3, and L4, respectively. Set Xoc = {X1, X2, X3, X4}. Only X1 satisfies Corollary 3.6, and thus L1 is identified as the initial latent source. Then, we remove X1 from Xoc and update it to Xoc = {X2, X3, X4}. Next, since it can be shown that only L2 satisfies Corollary 3.17, i.e., c(3) ( ̃e(X2,X1)→X3)(L(2,3)) = c(3) ( ̃e(X2,X1)→X4)(L(2,4)), it follows that L2 is the latent source of G-({L1}). Similarly, we remove X2 from the current Xoc and update it to Xoc = {X3, X4}. Let X3′ = X5. In G-({L1, L2}), we compute ̃e(X3, ̃e3) and ̃e(X4, ̃e3), and find that c(3) ( ̃e(X3, ̃e3)→X4)(L(3,4)) = c(3) ( ̃e(X3, ̃e3)→X5)(L(3,5)), |Xoc ∪∅\\ {X4}| = 1, 20 Algorithm 3 Finding subsequent latent sources Input: Xoc, ˆC, and AL Output: ˆC and AL 1: Apply Corollary 3.17 to find the set of latent sources L0 in G-(Ls) 2: if L0 = ∅then 3: return ˆC and AL 4: end if 5: if |L0| ≥2 then 6: for all pairs Xi, Xj ∈ S k:Lk∈L0 ˆCk ∩Xoc do 7: if ̃e(Xi, ̃es) ̸⊥⊥Xj then 8: Merge ˆCj into ˆCi 9: ˆC ←ˆC \\ { ˆCj}, ˆL ←ˆL \\ {Lj}, Xoc ←Xoc \\ {Xj}, AL ←AL \\ {Anc(Lj)} 10: end if 11: end for 12: end if 13: for all Xi ∈ S k:Lk∈L0 ˆCk ∩Xoc do 14: X(i) oc ←∅ 15: for all Xj ∈Xoc \\ {Xi} do 16: if Xj ̸⊥⊥ ̃e(Xi, ̃es) then 17: Anc(Lj) ←Anc(Lj) ∪{Li}, X(i) oc ←X(i) oc ∪{Xj} 18: end if 19: end for 20: ˆC, AL ←Algorithm 3 (X(i) oc , ˆC, AL) 21: end for 22: return ˆC and AL indicating both L3 and L4 are latent sources by Corollary 3.17. Furhtermore, we conclude that {X3, X5} and {X4} should be merged into one cluster confounded by L3. 3.3. Stage III: Identifying Causal Structure among Latent Variables By the end of Stage II, the clusters of observed variables have been identified, as well as the ancestral relationships among latent variables and among observed variables. The ancestral relationships among L alone do not uniquely determine the complete causal structure of L. Here, we propose a bottom-up algorithm to estimate the causal structure of the latent variables. Note that if the ancestral relationships among L are known, a causal order of L can also be obtained. Theorem 3.19 provides an estimator of the causal coefficients between latent variables. 21 Algorithm 4 Finding the ancestral relationships between latent variables Input: X, AO, and ˆC Output: ˆC and AL 1: AL →∅ 2: for all mutually dependent Xoc do 3: Xoc, ˆC, AL ←Algorithm 2 (Xoc, ˆC, AL) 4: ˆC, AL ←Algorithm 3 (Xoc, ˆC, AL) 5: end for 6: return ˆC and AL Theorem 3.19. Assume that Anc(Li) = {L1, . . . , Li-1} with the causal order L1 ≺· · · ≺Li-1. Let X1, . . . , Xi be the observed children of L1, . . . , Li with the highest causal order, respectively. Define ̃ri,k-1 as ̃ri,k-1 = Xi, k = 1 Xi -Pi-1 h=i-(k-1) aihXh, k ≥2 When we set λ11 = · · · = λii = 1, ai,i-k = ρ( ̃ri,k-1, ̃e(Xi-k, ̃ei-k)) generically holds. In addition, under Assumption A4, it holds generically that ai,i-k = 0 if and only if ̃ri,k-1 ⊥⊥ ̃e(Xi-k, ̃ei-k). If the only information available is the ancestral relationships among {L1, . . . , Li}, we cannot determine whether there is an edge Li-k →Li in G. However, according to Theorem 3.19, if ̃ri,k-1 ⊥⊥ ̃e(Xi-k, ̃ei-k), then ai,i-k = 0, and thus it follows that Li-k →Li does not exist. Algorithm 5 describes how Theorem 3.19 is applied to estimate the causal structure among L. Example 3.20. For the model in Figure 3.1 (a), the estimated causal order of latent variables is L1 ≺L2 ≺L3 with Xoc = {X1, X2, X3}. Assume initially that L1, L2, and L3 form a complete graph. Then X1, X2, X3, and ̃e(X2, ̃e2) are X1 = ε1 + e1, X2 = a21ε1 + ε2 + e2, X3 = (a21a32 + a31)ε1 + a32ε2 + ε3 + e3, ̃e(X2, ̃e2) = ̃e(X2,X1) = ε2 + e2 -a21e1. We estimate a32 and a31 using Theorem 3.19 as follows: a32 = ρ(X3, ̃e(X2, ̃e2)), 22 ̃r31 = X3 -a32X2 = a31ε1 + ε3 -a32e2 + e3, a31 = ρ( ̃r31,X1). Thus, if ̃r31 ⊥⊥X1, then a31 = 0. In this case, we can conclude that L1 →L3 does not exist. Algorithm 5 Finding causal structure among latent variables Input: Xoc, L, AL Output: An adjacency matrix Aadj of L 1: function Adjacency(Xoc, Li, Lopen, Aadj, ̃ri) 2: if |Lopen| = 0 then 3: return Aadj 4: end if 5: Initialize Lnext ←∅ 6: for all Lj ∈Lopen do 7: ˆaij ←0, Lnext ←Lnext ∪Pa(Lj) 8: if ∃{Lk, Lh} ⊂Lnext s.t. Lk ∈Anc(Lh) then 9: Lnext ←Lnext \\ {Lk} 10: end if 11: if ̃ri ̸⊥⊥ ̃e(Xj, ̃ej) then 12: ˆaij ←an empirical counterpart of aij 13: end if 14: ̃ri ← ̃ri -ˆaijXj 15: if ˆaji ̸= 0 then 16: Aadj[i, j] ←1 17: end if 18: end for 19: Lopen ←Lnext 20: Aadj ←Adjacency(Xoc, Li, Lopen, Aadj, ̃ri) 21: end function 22: function Main(Xoc, L, AL) 23: Initialize Aadj ←{0}|L|×|L| 24: for all Li ∈L do 25: ̃ri ←Xi, Lopen ←Pa(Li) 26: if ∃{Lk, Lh} ⊂Lopen s.t. Lk ∈Anc(Lh) then 27: Lopen ←Lopen \\ {Lk} 28: end if 29: Aadj ←Adjacency(Xoc, Li, Lopen, Aadj, ̃ri) 30: end for 31: return Aadj 32: end function 23 3.4. Summary This section integrates Algorithms 1, 4, and 5 into Algorithm 6, which identifies the clusters of observed variables, the causal structure between latent variables, and the ancestral relationships between observed variables under the assumptions A1-A5. Since the causal clusters ˆC have been correctly identified, the directed edges from L to X are also identified. Although the ancestral relationships among observed variables can be identified, their exact causal structure remains undetermined. In conclusion, we obtain the following result: Theorem 3.21. Given observed data generated from an LvLiNGAM MG in (2.1) that satisfies the assumptions A1-A5, the proposed method can identify the latent causal structure among L, causal edges from L to X, and ancestral relationships among X. Algorithm 6 Identify the Causal Structure among Latent Variables Input: X = (X1, . . . , Xp)⊤ Output: AO, ˆC, and Aadj 1: ˆC, AO ←Algorithm 1 (X) ▷Estimate over-segmented clusters 2: AL, ˆC ←Algorithm 4 (X, AO, ˆC) ▷Identify the causal order among latent variables 3: Aadj ←Algorithm 5 (Xoc, L, AL) ▷Find the causal structure among latent variables 4: return AO, ˆC, and Aadj 4. Simulations In this section, we assess the effectiveness of the proposed method by comparing it with the algorithms proposed by Xie et al. [21] for estimating LiNGLaM and by Xie et al. [23] for estimating LiNGLaH, as well as with ReLVLiNGAM [18], which serves as the estimation method for the canonical model with generic parameters. For convenience, we hereafter refer to both the model class introduced by Xie et al. [21] and its estimation algorithm as LiNGLaM, and likewise use LiNGLaH to denote both the model class and the estimation algorithm proposed by Xie et al. [23]. 4.1. Settings In the simulation, the true models are set to six LvLiNGAMs defined by the DAGs shown in Figures 4.1 (a)-(f). We refer to these models as Models (a)-(f), respectively. All these models satisfy Assumptions A1-A3. 24 All disturbances are assumed to follow a log-normal distribution, ui ∼ Lognormal(-1.1, 0.8), shifted to have zero mean by subtracting its expected value. The coefficient λii from Li to Xi is fixed at 1. Other coefficients in Λ and A are drawn from Uniform(1.1, 1.5), while those in B are drawn from Uniform(0.5, 0.9). When all causal coefficients are positive, the faithfulness condition is satisfied. The higher-order cumulant of a log-normal distribution is non-zero. None of the models (a)-(f) is LiNGLaM or LiNGLaH. The models (a) and (b) are generic canonical models, whereas the canonical models derived from Figures 4.1 (c)-(f) do not satisfy the genericity assumption of Schkoda et al. [18]. The sample sizes N are set to 1000, 2000, 4000, 8000, and 16000. The number of iterations is set to 100. We evaluate the performance of the proposed method and other methods using the following metrics. • Ncl, Nls, Nos, and Ncs: The counts of iterations in which the resulting clusters, the latent structures, the ancestral relationships among X, and the latent structure and the ancestral relationships among X are correctly estimated, respectively. • PREll, RECll, and F1ll: Averages of Precision, Recall, and F1-score of the estimated edges among latent variables, respectively, when clusters are correctly estimated. • PREoo, RECoo, and F1oo: Averages of Precision, Recall, and F1-score of the estimated causal ancestral relationships among observed variables, respectively, when clusters are correctly estimated. LiNGLaM and LiNGLaH assume that each cluster contains at least two observed variables. When a cluster includes only a single observed variable, these methods may fail to assign it to any cluster, resulting in it being left without an associated latent parent. Here, we treat such variables as individual clusters and assign each a latent parent. 4.2. Implementation Hilbert-Schmidt independence criterion (HSIC) [28] is employed for the independence tests in the proposed method. As HSIC becomes computationally expensive for large sample sizes, we randomly select 2,000 samples for HSIC when N ≥2000. The significance level of HSIC is set to αind = 0.05. When estimating the number of latent variables and the ancestral relationships among the observed variables, we apply Proposition 2.2. Following 25 L1 X1 X2 X3 (a) L1 X1 X2 X3 (b) L1 L2 X1 X2 X3 (c) L1 L2 X1 X2 X3 X4 (d) L1 L2 L3 X1 X2 X3 X4 (e) L1 L2 L3 X1 X2 X3 X4 (f) Figure 4.1: Six models for simulations Schkoda et al. [18], the rank of A(k1,k2) (Xi→Xj) is determined from its singular values. Let σr denote the r-th largest singular value of A(k1,k2) (Xi→Xj) and let τs be a predefined threshold. If σr/σ1 ≤τs, we set σr to zero. To ensure termination in the estimation of the number of confounders between two observed variables, we impose an upper bound on the number of latent variables, following Schkoda et al. [18]. In this experiment, we set the upper bound on the number of latent variables to two in both our proposed method and ReLVLiNGAM. When estimating latent sources, we use Corollaries 3.6 and 3.17. To check whether |Conf(Xi, Xj)| = 1 in the canonical model over Xi and Xj, one possible approach is to apply Proposition 2.2. Theorem A.7 in the Appendix shows that |Conf(Xi, Xj)| = 1 is equivalent to (c(6) i,i,i,j,j,j)2 = c(6) i,i,i,i,j,jc(6) i,i,j,j,j,j. Based on this fact, one can alternatively check whether Conf(Xi, Xj) = 1 by using the criterion |(c(6) i,i,i,j,j,j)2 -c(6) i,i,i,i,j,jc(6) i,i,j,j,j,j| max (c(6) i,i,i,j,j,j)2, |c(6) i,i,i,i,j,jc(6) i,i,j,j,j,j| s and s = 1, we only discuss the remaining two cases. We first prove the case where s = i by induction on i. When i = 1, ̃e(X1, ̃e1) = X1 = ε1 + e1, 52 where U[1] = ε1. Assume that the inductive assumption holds up to i. Then, ̃e(Xh, ̃eh) = εh + U[h], 1 ≤h ≤i. Since Xi+1 is expressed as Xi+1 = εi+1 + ei+1 + i X h=1 αll i+1,hεh, we have ρ(Xi+1, ̃e(Xh, ̃eh)) = αll i+1,h for h = 1, . . . , i, according to Definition 3.13. Hence, we have ̃e(Xi+1, ̃ei+1) = Xi+1 - i X h=1 ρ(Xi+1, ̃e(Xh, ̃eh)) ̃e(Xh, ̃eh) = εi+1 + ei+1 + i X h=1 αll i+1,hεh - i X h=1 αll i+1,h(εh + U[h]) = εi+1 + ei+1 - i X h=1 αll i+1,hU[h] = εi+1 + U[i+1], where U[i+1] = ei+1 -Pi h=1 αll i+1,hU[h]. Thus, the claim holds for all i by induction. Next, we discuss the case where i > s and s > 1. According to Definition 3.13, ̃e(Xi, ̃es) = Xi - s-1 X h=1 ρ(Xi, ̃e(Xh, ̃eh)) ̃e(Xh, ̃eh), ρ(Xi, ̃e(Xh, ̃eh)) = αll ih. Using the conclusion of the case where i = s, we obtain ̃e(Xi, ̃es) = εi + i-1 X h=1 αll ihεh + ei - s-1 X h=1 αll ih εh + U[h] = εi + i-1 X h=s αll ihεh + U[s-1] + ei. 53 C.6. The proof of Theorem 3.15 Proof. The proof of this theorem follows similarly to that of Theorem 3.11. C.7. The proof of Theorem 3.16 Proof. The proof of this theorem follows similarly to that of Theorem 3.12. C.8. The proof of Corollary 3.17 According to Theorem 3.16, sufficiency is immediate. We therefore prove only necessity by showing that if Li is not a latent source in G-({L1, . . . , Ls-1}), then neither case 1 nor case 2 holds. If Li is not a latent source, Xi ̸= ∅or |Xoc \\ {X1, . . . , Xs-1, Xi}| ≥2, and therefore case 1 is not satisfied. We will show that one of the conditions (a), (b), and (c) is not satisfied. First, note that condition (a) does not hold whenever there exists Xj ∈Xoc \\ {X1, . . . , Xs-1, Xi} with |Conf( ̃e(Xi, ̃es), Xj)| ̸= 1. Hence, assume that condition (a) holds. Assume that Ls is the latent source of G-({L1, . . . , Ls-1}), and that Xs is its observed child with the highest causal order. Let Lj be the latent parent of Xj ∈Xoc \\ {X1, . . . , Xs-1, Xi}, respectively. Since the condition (a) holds, |Conf( ̃e(Xi, ̃es), Xs)| = |Conf( ̃e(Xi, ̃es), Xj)| = 1, where ̃es = ( ̃e(X1, ̃e1), . . . , ̃e(Xs-1, ̃es-1)). Then, Xs and ̃e(Xi, ̃es) are written as Xs = s-1 X h=1 αll shεh + εs + es, ̃e(Xi, ̃es) = εi + i-1 X h=s αll ihεh + U[s-1] + ei, according to Lemma 3.14. Hence, we have Conf( ̃e(Xi, ̃es), Xs) = {εs}. Assume that Li has a latent child Lj and that none of the descendants of Li are parents of Lj. Xj is expressed by Xj = j-1 X h=1 αll jhεh + εj + ej. 54 Both ̃e(Xi, ̃es) and Xj involve linear combinations of εs, . . . , εi. Since {εs, . . . , εi} are mutually independent and |Conf( ̃e(Xi, ̃es), Xj)| = 1, αll jh = αll jiαll ih according to Hoyer et al. [7], and then Xj can be rewritten as Xj = s-1 X i=1 αll jhεh + αll ji εi + i-1 X h=s αll ihεh ! + εj + ej, j = i + 1, s-1 X i=1 αll jhεh + αll ji εi + i-1 X h=s αll ihεh ! + j-1 X h=i+1 αll jhεh + εj + ej, j > i + 1. Therefore, c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) = cum(k) εi + i-1 X h=s αll ihεh ! , c(k) ( ̃e(Xi, ̃es)→Xs)(L(i,s)) = cum(k)(αll isεs), according to Lemma A.8. Therefore, we conclude that c(k) ( ̃e(Xi, ̃es)→Xj)(L(i,j)) ̸= c(k) ( ̃e(Xi, ̃es)→Xs)(L(i,s)) generically, and condition (b) is not satisfied. Next, we assume that Li does not have latent children, so that Xi ̸= ∅. Assume that Xi′ ∈Xi and it can be expressed as Xi′ = i-1 X h=1 αll ihεh + εi + bi′iei + ei′. Then, Conf( ̃e(Xi, ̃es), Xi′) = n εi + Pi-1 h=s αll ihεh o , bi′i = 0, n εi + Pi-1 h=s αll ihεh, ei o , bi′i ̸= 0. In either case, there is one latent L(i,i′) satisfies that c(k) ( ̃e(Xi, ̃es)→Xi′)(L(i,i′)) = cum(k) εi + i-1 X h=s αll ihεh ! , which generically implies c(k) ( ̃e(Xi, ̃es)→Xi′)(L(i,i′)) ̸= c(k) ( ̃e(Xi, ̃es)→Xs)(L(i,s)). Thus, the condition (c) is not satisfied. 55 D. Proofs of Theorems in Section 3.3 D.1. Theorem 3.19 Proof. From Lemma 3.14, ̃e(Xi-k, ̃ei-k) = εi-k + U[i-k]. (D.1) By definition, ̃ri,k-1 is written as ̃ri,k-1 = Xi - i-1 X h=i-(k-1) aihXh = i-1 X h=1 aihLh + εi + ei - i-1 X h=i-(k-1) aih(Lh + eh) = i-(k+1) X h=1 aihLh + ai,i-kεi-k + εi + ei - i-1 X h=i-(k-1) aiheh = V[i-(k+1)] + ai,i-kεi-k + εi + U[i]\\[i-k], (D.2) where V[i-(k+1)] is a linear combination of {ε1, . . . , εi-(k+1)} and U[i]\\[i-k] is a linear combination of {ei-(k-1), . . . , ei}. From (D.1) and (D.2), we can show that ̃ri,k-1 ⊥⊥ ̃e(Xi-k, ̃ei-k) ⇔ai,i-k = 0, and otherwise ρ(Xi, ̃e(Xi-1, ̃ei-1)) = a2 i,i-1 cum(4)(εi-1) ai,i-1 cum(4)(εi-1) = ai,i-1 generically holds. E. Reducing an LvLiNGAM In this paper, we have discussed the identifiability of LvLiNGAM under the assumption that each observed variable has exactly one latent parent. However, even when some observed variables do not have latent parents, by iteratively marginalizing out sink nodes and conditioning on source nodes to remove such variables one by one, the model can be progressively reduced to one in which each observed variable has a single latent parent. This can 56 be achieved by first estimating the causal structure involving the observed variables without latent parents. ParceLiNGAM [11] or RCD [12, 13] can identify the ancestral relationship between two observed variables if at least one of them does not have a latent parent, and remove the influence of the observed variable without a latent parent. Models 1-3 in Figure E.1 contain observed variables that do not have a latent parent. According to Definition 1.1, X1 and X2 in Models 1 and 3 belong to distinct clusters whose latent parents are L1 and L2, respectively, whereas in Model 2, X1 and X2 share the same latent parent L1. Model 3 contains a directed path between clusters, whereas Models 1 and 2 do not. We consider the model reduction procedure for Models 1-3 individually. Example E.1 (Model 1). By using ParceLiNGAM or RCD, we can identify X4 →X1, X4 →X2, X1 →X5, and X3 →X5. Since X5 is a sink node, the induced subgraph obtained by removing X5 represents the marginal model over the remaining variables. Since X4 is a source node, if we replace X1 and X2 with the residuals r(4) 1 and r(4) 2 obtained by regressing them on X4, then the induced subgraph obtained by removing X4 represents the conditional distribution given X4. As a result, Model 1 is reduced to the model shown in Figure E.1 (d). This model satisfies Assumptions A1-A3. Example E.2 (Model 2). X1 and X2 are confounded by L1, and they are mediated through X3. By using ParceLiNGAM or RCD, the ancestral relationship among X1, X2, X3 can be identified. Let r(1) 3 be the residual obtained by regressing X3 on X1. Let ̃r(3) 2 be the residual obtained by regressing X2 on r(1) 3 . According to [11] and [12, 13], the model for L1, X1, and r(3) 2 corresponds to the one shown in Figure E.1 (e). This model satisfies Assumptions A1-A3. Example E.3 (Model 3). In Model 3, X1 ∈Anc(X3), and they are mediated by X5. By using ParceLiNGAM or RCD, the ancestral relationship among X1, X3, X5 can be identified. Let r(1) 5 be the residual obtained by regressing X5 on X1. Let ̃r(5) 3 be the residual obtained by regressing X3 on r(1) 5 . According to [11] and [12, 13], by reasoning in the same way as for Models 1 and 2, Model 3 is reduced to the model shown in Figure E.1 (f). This model doesn't satisfy Assumptions A1-A3. Using [11] and [12, 13], ancestral relations between pairs of observed variables that include at least one variable without a latent parent can be iden57 L1 X1 L2 X2 X3 X4 X5 (a) Model 1 L1 X1 X2 X3 (b) Model 2 L1 X1 L2 X2 X3 X4 X5 (c) Model 3 L1 r(4) 1 L2 X3 r(4) 2 (d) Reduced model of Model 1 L1 X1 ̃r(3) 2 (e) Reduced model of Model 2 L1 r(4) 1 L2 r(4) 2 ̃r(5) 3 (f) Reduced model of Model 3 Figure E.1: Three models that can be reduced. tified. The graph obtained by the model reduction procedure is constructed by iteratively applying the following steps: (i) iteratively remove observed variables without latent parents that appear as source or sink nodes, updating the induced subgraph at each step so that any new source or sink nodes are subsequently removed; (ii) when an observed variable without a latent parent serves as a mediator, remove the variable and connect its parent and child with a directed edge. If no directed path exists between any two observed variables with distinct latent parents, the model obtained through the model reduction procedure satisfies Assumptions A1-A3. Conversely, if there exist two observed variables with distinct latent parents that are connected by a directed path, the model obtained through the model reduction procedure does not satisfy Assumption A3. In summary, Assumption A1 can be generalized to A1′. Each observed variable has at most one latent parent. Proposition E.4. Given observed data generated from an LvLiNGAM MG that satisfies the assumptions A1′ and A2-A5, the latent causal structure 58 among the latent variables, the directed edges from the latent variables to the observed variables, and the ancestral relationships among the observed variables can be identified by using the proposed method in combination with ParceLiNGAM and RCD. 59"}}},{"rowIdx":199,"cells":{"id":{"kind":"string","value":"2510.14781"},"raw":{"kind":"string","value":"SciPost Physics Codebases\nSubmission\nParaToric 1.0-beta: Continuous-time quantum Monte Carlo\nfor the toric code in a parallel field\nSimon M. Linsel⋆and Lode Pollet†\nDepartment of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),\nLudwig-Maximilians-Universität München, Theresienstr. 37, München D-80333, Germany\nMunich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799\nMünchen, Germany\n⋆simon.linsel@lmu.de ,\n† lode.pollet@lmu.de\nAbstract\nWe introduce ParaToric, a C++ package for simulating the toric code in a parallel field\n(i.e., X- and Z-fields) at finite temperature. We implement and extend the continuous-\ntime quantum Monte Carlo algorithm of Wu, Deng, and Prokof’ev on the square, triangu-\nlar, honeycomb, and cubic lattices with open and periodic boundaries, respectively. The\npackage is expandable to arbitrary lattice geometries and custom observables diagonal in\neither the X- or Z-basis. ParaToric also supports snapshot extraction in both bases, mak-\ning it ideal for generating training/benchmarking data for other methods, such as lattice\ngauge theories, cold atom or other quantum simulators, quantum spin liquids, artificial\nintelligence, and quantum error correction. The software provides bindings to C/C++\nand Python, and is thus almost universally integrable into other software projects.\nCopyright attribution to authors.\nThis work is a submission to SciPost Physics Codebases.\nLicense information to appear upon publication.\nPublication information to appear upon publication.\nReceived Date\nAccepted Date\nPublished Date\nContents\n1\nIntroduction\n3\n2\nThe toric code in a parallel field\n3\n2.1\nHamiltonian\n3\n2.2\nLattice geometries\n3\n2.3\nObservables\n4\n3\nInstallation & interfaces\n6\n3.1\nC++ interface\n6\n3.1.1\nBuild & Installation\n7\n3.1.2\nPublic class ExtendedToricCode\n8\n3.1.3\nConfiguration type\n8\n3.1.4\nReturn type\n11\n3.1.5\nC++ usage examples\n11\n3.2\nC++ command-line interface\n13\n3.2.1\nBuild & Installation\n13\n1\narXiv:2510.14781v1 [quant-ph] 16 Oct 2025\n\nSciPost Physics Codebases\nSubmission\n3.2.2\netc_sample\n15\n3.2.3\netc_hysteresis\n16\n3.2.4\netc_thermalization\n16\n3.2.5\nHDF5 structure\n16\n3.3\nC interface\n16\n3.3.1\nBuild & Installation\n16\n3.3.2\nStatus & error handling\n17\n3.3.3\nOpaque handle\n17\n3.3.4\nConfiguration type\n17\n3.3.5\nReturn type\n18\n3.3.6\nProcedures (mirror the C++ API)\n19\n3.3.7\nC usage example\n19\n3.4\nPython bindings\n21\n3.4.1\nBuild & Installation\n21\n3.4.2\nModule layout\n22\n3.4.3\nUsage example\n24\n3.5\nPython command-line interface\n24\n3.5.1\nBuild & Installation\n25\n3.5.2\nT-sweep\n26\n3.5.3\nh-sweep\n27\n3.5.4\nλ-sweep\n27\n3.5.5\n◦-sweep\n28\n3.5.6\nHysteresis-sweep\n28\n3.5.7\nThermalization\n29\n4\nUsing ParaToric\n29\n4.1\nMonte Carlo Updates\n29\n4.2\nMonte Carlo Diagnostics\n30\n4.2.1\nThermalization mode\n30\n4.2.2\nIntegrated autocorrelation time\n30\n4.2.3\nError bars\n31\n4.3\nTips & tricks\n31\n4.3.1\nProbing ground state physics\n31\n4.3.2\nProbing first-order transitions\n31\n4.3.3\nChoosing the basis\n31\n4.3.4\nChoosing N_thermalization\n31\n4.3.5\nChoosing N_samples\n31\n4.3.6\nChoosing N_between_samples\n32\n4.3.7\nChoosing N_resamples\n32\n4.3.8\nExtracting snapshots\n32\n4.3.9\nAdding new observables/lattices/updates\n32\n4.4\nBenchmarks\n32\n4.4.1\nThermalization\n32\n4.4.2\nIntegrated autocorrelation time\n33\n4.4.3\nRun-time\n33\n4.4.4\nTopological phase transition\n35\n5\nConclusion & Outlook\n36\nReferences\n37\n2\n\nSciPost Physics Codebases\nSubmission\n1\nIntroduction\nThe toric code is one of the most fundamental and most-studied models in modern condensed\nmatter physics. It was first written down by Kitaev [1] and is the simplest example of a model\nhosting a topological phase (a gapped Z2 quantum spin liquid) and anyonic excitations. The\ntoric code is also the foundational model for error-correcting codes [2, 3] and has deep con-\nnections to the Ising gauge theory [4].\nThe toric code can be extended with fields which, when strong enough, destroy the topo-\nlogical order. This model is sign-problem-free, thus making quantum Monte Carlo the method\nof choice. Wu, Deng, and Prokof’ev developed a continuous-time quantum Monte Carlo algo-\nrithm [5]. ParaToric implements and extends this algorithm with new updates which enable\nergodicity at large temperatures and at zero off-diagonal field, thus significantly improving the\napplicability of the algorithm.\nParaToric implements a wide range of lattices, boundary conditions, and observables. It is\nalso possible to extend ParaToric with new interactions, observables, and lattices. We provide\ndocumented interfaces in C, C++, and Python as well as command-line interfaces, making\nthe integration of ParaToric into other projects and programming languages straightforward.\nParaToric will save simulation results to HDF5 files and snapshots to GraphML files (XML-\nbased), with a focus on interoperability with other packages. ParaToric comes with an MIT\nlicense.\n2\nThe toric code in a parallel field\n2.1\nHamiltonian\nParaToric implements and extends the continuous-time quantum Monte Carlo (QMC) algo-\nrithm by Wu, Deng, and Prokof’ev [5] to simulate the toric code in a parallel field (also called\nperturbed toric code or extended toric code)\nˆH = −µ\nX\nv\nˆAv −J\nX\np\nˆBp −h\nX\nl\nˆσx\nl −λ\nX\nl\nˆσz\nl ,\n(1)\nwhere J,λ > 0 in the ˆ\nσx-basis and µ,h > 0 in the ˆ\nσz-basis (otherwise the model has a sign-\nproblem). ˆσx\nl and ˆσz\nl are Pauli matrices defined on the links of the underlying lattice. The\nstar term ˆAv contains all links adjacent to lattice site v, the plaquette term ˆBp contains all links\nthat belong to the same elementary plaquette p of the underlying lattice. The temperature\nT = 1/β is finite. For readers interested in extending the code, we note that it is relatively\nstraightforward to add interactions that are diagonal in the chosen basis, such as (long-range)\nIsing interactions. Off-diagonal interactions require a more careful review and extension of\nthe Monte Carlo updates to ensure ergodicity. However, diagonal interactions can also lead to\nsampling problems, especially when they introduce frustration.\n2.2\nLattice geometries\nWe implement the square, honeycomb, triangular, and cubic lattices, see Fig. 1. On the cubic\nlattice, the plaquettes contain the four links of cube faces, not the twelve links of the cube\n3\n\nSciPost Physics Codebases\nSubmission\n(a)\n(b)\n̂Av\n̂Bp\n̂σx\n̂σz\n̂Av\n̂Bp\n̂σz\n̂σx\n(c)\n̂σz\n̂σx\n̂Av\n̂Bp\n(d)\n̂Av\n̂Bp\nFigure 1: Implemented lattices. We implement the extended toric code (1) on the\nsquare (a), honeycomb (b), triangular (c), and cubic lattices (d). For each lattice, we\nshow the star (ˆAv) and plaquette (ˆBp) terms. The cubic lattice importantly features\nstar interactions of six links and plaquette interactions of four links on the faces of\ncubes.\n(that model has a different m-anyon structure). We implement open and periodic boundaries,\nrespectively. New lattices can be added in src/lattice/lattice.cpp.\n2.3\nObservables\nHere we list all observables that ParaToric implements for the extended toric code. Custom\nobservables can be added in src/mcmc/extended_toric_code_qmc.hpp. For each ob-\nservable ˆO, we calculate the expectation value 〈ˆO〉and the Binder ratio UO = 〈ˆO4〉\n〈ˆO2〉2 with error\nbars obtained from bootstrapping (see below), respectively.\nanyon_count\nThe number of e-anyons ( ˆ\nσx-basis) or m-anyons ( ˆ\nσz-basis) in the system.\nanyon_density\nˆ\nσx-basis: The number of e-anyons divided by the number of lattice sites.\nˆ\nσz-basis: The number of m-anyons divided by the number of plaquettes.\ndelta\nThe difference of the star and plaquette expectation values: ∆= 〈ˆAv〉−〈ˆBp〉.\nenergy\nThe total energy E = 〈ˆH〉.\nenergy_h\nThe electric field term Eh = 〈−h\nP\nl ˆσx\nl 〉.\nenergy_lmbda\nThe gauge field term Eλ = 〈−λ\nP\nl ˆσz\nl 〉. We write λ as lmbda because some\nprogramming languages feature lambda as a keyword.\n4\n\nSciPost Physics Codebases\nSubmission\nenergy_J\nThe plaquette term EJ = 〈−J\nP\np ˆBp〉.\nenergy_mu\nThe star term Eµ = 〈−µ\nP\nv ˆAv〉.\nfredenhagen_marcu\nThe equal-time Fredenhagen-Marcu loop operator [6–8]:\nOx/z\nFM = lim\nL→∞\n〈\nQ\nl∈Cx/z\n1/2 ˆσx/z\nl\n〉\nÇ\n|〈\nQ\nl∈Cx/z ˆσx/z\nl\n〉|\n,\n(2)\nC x/z\n1/2 is half, C x/z is a full Wilson loop in the ˆ\nσx-basis (’t Hooft loop in the ˆ\nσz-basis). The loop is\nautomatically constructed for all supported lattices, and the perimeter scales with O(L), where\nL is the linear system size. When probing perimeter/area laws, the user should change L. We\ncurrently do not support off-diagonal loop operators, e.g., measuring products of ˆσx-operators\nin the ˆσz-basis.\nlargest_cluster\nThe largest connected cluster of neighboring bonds with ˆσx = −1\n(ˆσz = −1) in the ˆσx-basis (ˆσz-basis). This observable is used to calculate the percolation\nstrength, see [9].\npercolation_probability\nMeasures the bond percolation probability, i.e. if we can\nwind around the system while only traversing bonds with ˆσx = −1 (ˆσz = −1) in the ˆσx-basis\n(ˆσz-basis). Formally, it is the expectation value 〈ˆΠx/z〉of the projector\nˆΠx/z =\nX\nW(j)̸=0\n|{ˆσx/z}j〉〈{ˆσx/z}j|,\n(3)\nover all possible configurations {ˆσx/z}j with non-zero winding number W(j) of connected link\nclusters of neighboring ˆσx/z = −1. These clusters are called percolating clusters. For details,\nsee [9–11].\npercolation_strength\nIf a snapshot does not have a percolating cluster, the percolation\nstrength is 0. If a snapshot has a percolating cluster, the percolation strength is defined as the\nresult of largest_cluster divided by the total number of links in the system. For details,\nsee [9,11].\nplaquette_percolation_probability\nSimilar to the percolation probability of bonds.\nTwo plaquettes are in the same cluster if they share a link l with ˆ\nτx\nl = −1. For details, see [11].\nplaquette_z\nThe plaquette expectation value 〈ˆBp〉.\nsigma_x\nThe electric field expectation value 〈ˆσx〉.\nsigma_x_susceptibility\nThe static susceptibility\nχ x = 1\nN\nZ β\n0\n〈ˆσx(0) ˆσx(τ)〉c dτ,\n(4)\nwhere N is the total number of links (qubits) and the integral is over the imaginary time τ1.\nImportantly, χ x can be calculated both in the ˆσx- and the ˆσz-basis.\n1The dynamical (fidelity) susceptibility is a trivial extension and contains an extra τ dependency in the integral.\n5\n\nSciPost Physics Codebases\nSubmission\nsigma_z\nThe gauge field expectation value 〈ˆσz〉.\nsigma_z_susceptibility\nThe static susceptibility\nχz = 1\nN\nZ β\n0\n〈ˆσz(0) ˆσz(τ)〉c dτ,\n(5)\nwhere N is the total number of links (qubits) and the integral is over the imaginary time τ.\nImportantly, χz can be calculated both in the ˆσx- and the ˆσz-basis.\nstaggered_imaginary_times\nOrder parameter from [5]. It is defined as\nOx/z\nSIT = 1\nβ\n\u0002\n(τk\n1 −0) −(τk\n2 −τk\n1) + ... + (−1)N(k)−1\n(τk\nN(k) −τk\nN(k)−1) + (−1)N(k)(β −τk\nN(k))\n\u0003\n,\n(6)\nwhere τk\nn is the imaginary time of the n-th tuple spin flip of type k. k is a plaquette p (star\ns) of links in the ˆσx-basis (ˆσz-basis). This order parameter can neither be evaluated from\nsnapshots, nor from any other method that does not have access to imaginary time.\nstar_x\nThe star expectation value 〈ˆAv〉.\nstring_number\nThe total number of links with ˆσx = −1 in the ˆσx-basis (ˆσz = −1 in the\nˆσz-basis).\n3\nInstallation & interfaces\nThere are five ways to use paratoric, directly from within code (C, C++, Python) or via the\ncommand-line (C++, Python). All interfaces require compiling C++ code. We tested the\ncompilation with GCC 15 and Clang 20.\nAll interfaces implement three functionalities.\nThermalization simulations are used to\nbenchmark the thermalization process of a Markov chain and are primarily a diagnostic tool.\nRegular sampling routines are used for generating snapshots and measuring observables, e.g.,\nin the context of continuous phase transitions. Hysteresis routines are a variant of the regu-\nlar sampling routines where not one but an array of Hamiltonian parameters is provided and\nonly one Markov chain is used for all parameters. The order of the Hamiltonian parameters\nin the input array matters: The last state of the previous parameter is used as an initial state\nfor the thermalization phase of the next parameters. This simulation type should primarily be\nused when mapping out hysteresis curves in the vicinity of first-order phase transitions, hence\nthe name. Since the hysteresis simulation returns the values of not one but many parameter\nsets, the output types are generally different from the regular sampling. It is also much slower\nthan regular sampling, because the simulation for different parameters can in general not be\nparallelized.\n3.1\nC++ interface\nThe C++ interface enables users to use a ParaToric public header from within another C++\nproject.\n6\n\nSciPost Physics Codebases\nSubmission\n3.1.1\nBuild & Installation\nThe core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions may work,\nbut were not tested). To compile it, run:\ncmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\\n-DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\\n-DPARATORIC_BUILD_TESTS=ON\ncmake --build build -jN\nctest --test-dir build -jN --output-on-failure\ncmake --install build\nReplace N with the number of cores to use, e.g. -j4 for 4 cores.\n• -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you’re a developer.\n• -DCMAKE_INSTALL_PREFIX. By default, executables install to ${CMAKE_SOURCE_DI\nR}/${CMAKE_INSTALL_BINDIR}/, headers to ${CMAKE_INSTALL_INCLUDEDIR}/p\naratoric, and static libraries to ${CMAKE_SOURCE_DIR}/${CMAKE_INSTALL_LIBD\nIR}/. The Python scripts expect ${CMAKE_SOURCE_DIR}/bin/; this directory always\ncontains the paratoric executable. To install into a custom directory, pass it via -DCM\nAKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo\nry/.\n• -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json\nfor tooling.\n• -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The\ncore itself does not need MPI.\n• -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and\nClang.\n• -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires\na CPU which supports AVX2.\n• -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recom-\nmended).\nCMake usage (installed package)\ncmake_minimum_required(VERSION 3.23)\nproject(my_qmc_app CXX)\nfind_package(paratoric CONFIG REQUIRED)\n# provides paratoric::core\nadd_executable(myapp main.cpp)\ntarget_link_libraries(myapp PRIVATE paratoric::core)\nCMake usage (as subdirectory)\nIf the core lives in deps/paratoric, add it and link to the same target:\nadd_subdirectory(deps/paratoric)\nadd_executable(myapp main.cpp)\ntarget_link_libraries(myapp PRIVATE paratoric::core)\n7\n\nSciPost Physics Codebases\nSubmission\n3.1.2\nPublic class ExtendedToricCode\nThe interface class ExtendedToricCode lives in the public header #include<paratoric\n/mcmc/extended_toric_code.hpp>. All symbols are in the paratoric namespace. All\nmethods are static, take a single Config object and return a Result object. The required\nfields in config are documented for each method within the docstrings.\nResult ExtendedToricCode::get_thermalization(Config config)\nRun ther-\nmalization only. Required fields: lat_spec.{basis,lattice_type,system_size,beta\n,boundaries,default_spin}, param_spec.{mu,h,J,lmbda,h_therm,lmbda_therm\n}, sim_spec.{N_thermalization,N_resamples,custom_therm,observables,seed\n}, out_spec.{path_out,save_snapshots}.\nResult ExtendedToricCode::get_sample(Config config)\nRun a production mea-\nsurement pass. Returns the observables selected in config. Required fields: lat_spec.{ba\nsis,lattice_type,system_size,beta,boundaries,default_spin}, param_spec.\n{mu,h,J,lmbda}, sim_spec.{N_samples,N_thermalization,N_between_samples,\nN_resamples,observables,seed}, out_spec.{path_out,save_snapshots}.\nResult ExtendedToricCode::get_hysteresis(Config config)\nPerform a hys-\nteresis sweep, where the last state of the previous parameter is used as the initial state of the\nfollowing parameter in h_hys & lmbda_hys. Required fields: lat_spec.{basis,lattice\n_type,system_size,beta,boundaries,default_spin}, param_spec.{mu,J,h_hys\n,lmbda_hys}, sim_spec.{N_samples,N_thermalization,N_between_samples,N_r\nesamples,observables,seed}, out_spec.{paths_out,save_snapshots}.\n3.1.3\nConfiguration type\nThe struct Config (declared in <paratoric/types/types.hpp>) contains multiple nested\nspecifications.\nTop-level configuration: Config\nField\nType\nPurpose\nsim_spec\nSimSpec\nSimulation / MC controls (backend-consumed).\nparam_spec ParamSpec Model couplings / parameters (backend-consumed).\nlat_spec\nLatSpec\nLattice geometry and basis.\nout_spec\nOutSpec\nOutput folders and snapshot toggles.\n8\n\nSciPost Physics Codebases\nSubmission\nSimulation specification (config.sim_spec)\nField\nType\nMeaning / Defaults\nN_samples\nint\nNumber of recorded snapshots. Default 1000.\nN_thermalization\nint\nNumber of warmup steps before sampling. Typ-\nically O(Ld), where L is the system size and d is\nthe dimensionality. Default 10000.\nN_between_samples int\nSteps between consecutive snapshots.\nHigher\nvalue decreases autocorrelation and improves er-\nror bars. Typically O(Ld), where L is the system\nsize and d is the dimensionality. Default 1000.\nN_resamples\nint\nBootstrap resamples for errors. Default 1000.\ncustom_therm\nbool\nUse custom thermalization schedule. Default fa\nlse.\nseed\nint\nPRNG seed. 0 means “random seed.” Default 0.\nobservables\nvector<string> Names of observables to record each snapshot.\nFor options, see Sec. 2.3.\nParameter specification (config.param_spec)\nField\nType\nMeaning / Defaults\nmu\ndouble\nStar term coefficient. Default 1.0.\nh\ndouble\nElectric field term. Default 0.0.\nJ\ndouble\nPlaquette term. Default 1.0.\nlmbda\ndouble\nGauge-field term. Default 0.0.\nh_therm\ndouble\nThermalization value for h when using custom\nschedules. Default NaN (unused).\nlmbda_therm double\nThermalization value for lmbda when using cus-\ntom schedules. Default NaN (unused).\nh_hys\nvector<double> Sweep values of h for hysteresis runs. Default\nempty. Length must match lmbda_hys.\nlmbda_hys\nvector<double> Sweep values of lmbda for hysteresis runs. De-\nfault empty. Length must match h_hys.\nLattice specification (config.lat_spec)\nField\nType\nMeaning / Valid values\nbasis\nchar\nSpin eigenbasis for the simulation. Must be ’x’\nor ’z’.\nlattice_type string The lattice (\"square\", \"triangular\", \"hone\nycomb\" or \"cubic\").\nsystem_size\nint\nLinear system size (per dimension).\nbeta\ndouble Inverse temperature β > 0.\nboundaries\nstring Boundary condition: \"periodic\" or \"open\".\ndefault_spin int\nInitial link spin, must be +1 or −1.\n9\n\nSciPost Physics Codebases\nSubmission\nOutput specification (config.out_spec)\nField\nType\nMeaning\npath_out\nstring\nPrimary output folder name.\npaths_out\nvector<string> Hysteresis subfolder names. Length must match h_hys.\nsave_snapshots bool\nSave snapshots toggle. Default false.\n10\n\nSciPost Physics Codebases\nSubmission\n3.1.4\nReturn type\nField\nC++ Type\nMeaning\nseries\nvector<vector<variant<comple\nx<double>,double>>>\nTime series of all requested\nobservables, thermalization is\nexcluded (except for\nthermalization simulation). Outer\nindex = observable, inner index =\ntime point.\nacc_ratio\nvector<double>\nTime series of Monte Carlo\nacceptance ratios.\nmean\nvector<double>\nBootstrap observable means.\nmean_std\nvector<double>\nBootstrap standard errors of the\nmean.\nbinder\nvector<double>\nBootstrap binder ratios.\nbinder_std\nvector<double>\nBootstrap standard errors of the\nbinder ratios.\ntau_int\nvector<double>\nEstimated integrated\nautocorrelation times.\nseries_hys\nvector<vector<vector<variant\n<complex<double>,double>>>>\nHysteresis time series of all\nrequested observables,\nthermalization is excluded (except\nfor thermalization simulation).\nOuter vector = hysteresis\nparameters (order as in h_hys,lm\nbda_hys), middle vector =\nobservables (order as in observa\nbles), inner vector = time series.\nmean_hys\nvector<vector<double>>\nHysteresis bootstrap observable\nmeans. Outer vector = hysteresis\nparameters (order as in h_hys,lm\nbda_hys), inner vector =\nobservables (order as in observa\nbles).\nmean_std_hys\nvector<vector<double>>\nHysteresis bootstrap standard\nerrors of the mean. Indices as\nabove.\nbinder_hys\nvector<vector<double>>\nHysteresis bootstrap binder ratios.\nIndices as above.\nbinder_std_hys vector<vector<double>>\nHysteresis bootstrap standard\nerrors of the binder ratios. Indices\nas above.\ntau_int_hys\nvector<vector<double>>\nHysteresis estimated integrated\nautocorrelation times. Indices as\nabove.\n3.1.5\nC++ usage examples\nListing 1: C++ API - Minimal call\n11\n\nSciPost Physics Codebases\nSubmission\n// C++23\n#include <iostream>\n#include <print>\n#include <vector>\n#include <string>\n#include <paratoric/mcmc/extended_toric_code.hpp>\n#include <paratoric/types/types.hpp>\nint main() {\nusing namespace paratoric;\nConfig cfg{};\n// ---- lattice sub-config (required) ----\ncfg.lat_spec.basis = 'z'; // or 'x'\ncfg.lat_spec.lattice_type = \"square\"; // or \"cubic\", \"honeycomb\", ...\ncfg.lat_spec.system_size = 16;\ncfg.lat_spec.beta = 8.0;\ncfg.lat_spec.boundaries = \"periodic\"; // or \"open\"\ncfg.lat_spec.default_spin = 1;\n// ---- Hamiltonian parameters ----\ncfg.param_spec.mu = 1.0; // star term\ncfg.param_spec.J = 1.0; // plaquette term\ncfg.param_spec.h = 0.20; // electric field term\ncfg.param_spec.lmbda = 0.00; // gauge-field term\n// Optional thermalization schedule values (used if custom_therm = true)\ncfg.param_spec.h_therm = std::numeric_limits<double>::quiet_NaN();\ncfg.param_spec.lmbda_therm = std::numeric_limits<double>::quiet_NaN();\n// (Optional) Hysteresis sweep grids - only read by get_hysteresis(...)\ncfg.param_spec.h_hys = {}; // e.g. {0.0, 0.1, 0.2, 0.3, 0.2, 0.1, 0.0}\ncfg.param_spec.lmbda_hys = {}; // e.g. {0.0, 0.1, 0.2, 0.3, 0.2, 0.1, 0.0}\n// ---- Simulation (MC) controls ----\ncfg.sim_spec.N_samples = 0; // 0 => thermalization-only\ncfg.sim_spec.N_thermalization = 5000; // warmup steps\ncfg.sim_spec.N_between_samples = 10; // thinning between snapshots\ncfg.sim_spec.N_resamples = 1000; // bootstrap\ncfg.sim_spec.custom_therm = false; // set true to use *_therm values\ncfg.sim_spec.seed = 12345; // 0 => random seed\n// Observables to record each snapshot (backend-recognized names)\ncfg.sim_spec.observables = {\n\"energy\", // total energy\n\"plaquette_z\", // plaquette energy\n\"anyon_count\", // number of anyons (x-basis: e-anyons, z-basis: m-anyons\n)\n\"fredenhagen_marcu\" // example: Wilson/'t Hooft loop proxy\n};\n// ---- Output / I/O policy ----\ncfg.out_spec.path_out = \"runs/sample\"; // single-run output dir\ncfg.out_spec.paths_out = {}; // filled only for hysteresis\ncfg.out_spec.save_snapshots = false; // set true to dump every snapshot\n12\n\nSciPost Physics Codebases\nSubmission\ncfg.out_spec.full_time_series = true; // save full time series (FCS)\n// 1) Check thermalization\nResult warmup = ExtendedToricCode::get_thermalization(cfg);\nstd::print(\"Thermalization␣series:␣{}\", warmup.series);\n// 2) Production sample (set N_samples > 0 and call get_sample)\ncfg.sim_spec.N_samples = 2000;\nResult out = ExtendedToricCode::get_sample(cfg);\nstd::print(\"Production␣autocorrelations:␣{}\", out.tau_int);\nreturn 0;\n}\n3.2\nC++ command-line interface\nParaToric ships a C++ command-line interface ${CMAKE_INSTALL_PREFIX}/${CMAKE_INS\nTALL_BINDIR}/paratoric that orchestrates C++ backends, runs sweeps, and writes HDF5\n(observables) and XML (snapshots) outputs.\n3.2.1\nBuild & Installation\nThe command-line interface requires HDF5 ≥1.14.3 (older HDF5 versions may work, but were\nnot tested). The core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions\nmay work, but were not tested). To compile it, run:\ncmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\\n-DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\\n-DPARATORIC_BUILD_TESTS=ON -DPARATORIC_BUILD_CLI=ON\ncmake --build build -jN\nctest --test-dir build -jN --output-on-failure\ncmake --install build\nReplace N with the number of cores to use, e.g. -j4 for 4 cores.\n• -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you’re a developer.\n• -DCMAKE_INSTALL_PREFIX. By default, executables install to ${CMAKE_SOURCE_DI\nR}/${CMAKE_INSTALL_BINDIR}/, headers to ${CMAKE_INSTALL_INCLUDEDIR}/p\naratoric, and static libraries to ${CMAKE_SOURCE_DIR}/${CMAKE_INSTALL_LIBD\nIR}/. The Python scripts expect ${CMAKE_SOURCE_DIR}/bin/; this directory always\ncontains the paratoric executable. To install into a custom directory, pass it via -DCM\nAKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo\nry/.\n• -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json\nfor tooling.\n• -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The\ncore itself does not need MPI.\n• -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and\nClang.\n13\n\nSciPost Physics Codebases\nSubmission\n• -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires\na CPU which supports AVX2.\n• -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recom-\nmended).\n• -DPARATORIC_BUILD_CLI=PROJECT_IS_TOP_LEVEL Required for both the C++ and\nPython command line interface.\n14\n\nSciPost Physics Codebases\nSubmission\nGlobal options\nLong flag\nShort\nType\nDescription\n--simulation\n-sim\nstring\nSimulation mode: etc_sample,\netc_hysteresis, etc_therm\nalization.\n--N_samples\n-Ns\nint\nNumber of recorded samples.\n--N_thermalization\n-Nth\nint\nThermalization (warmup) steps.\n--N_between_samples\n-Nbs\nint\nSteps\nbetween\nsamples\n(thin-\nning).\n--beta\n-bet\ndouble\nInverse temperature β = 1/T.\n--mu_constant\n-muc\ndouble\nStar-term coupling µ.\n--J_constant\n-Jc\ndouble\nPlaquette coupling J.\n--h_constant\n-hc\ndouble\nField h.\n--lmbda_constant\n-lmbdac\ndouble\nField λ.\n--h_constant_therm\n-hct\ndouble\nThermalization value for h (used\nif custom therm).\n--lmbda_constant_therm\n-lmbdact\ndouble\nThermalization value for λ.\n--h_hysteresis\n-hhys\nlist<double>\nHysteresis schedule for h (space-\nseparated). Length must match l\nmbdahys.\n--lmbda_hysteresis\n-lmbdahys\nlist<double>\nHysteresis schedule for λ. Length\nmust match hhys.\n--N_resamples\n-Nr\nint\nBootstrap resamples (error bars).\n--custom_therm\n-cth\nbool\nUse thermalization values (0/1).\n--observables\n-obs\nlist<string>\nMeasured\nobservables\n(space-\nseparated).\n--seed\n-s\nint\nPRNG seed; 0 means random\nseed.\n--basis\n-bas\nchar\nSpin basis: ’x’ or ’z’.\n--lattice_type\n-lat\nstring\nLattice type (e.g. square, cubi\nc, ...).\n--system_size\n-L\nint\nLinear lattice size (per dimen-\nsion).\n--boundaries\n-bound\nstring\nperiodic or open.\n--default_spin\n-dsp\nint\nInitial link spin (+1 or -1).\n--output_directory\n-outdir\npath\nOutput directory path.\n--folder_name\n-fn\nstring\nSubfolder (of output directory)\nname for single run.\n--folder_names\n-fns\nlist<string>\nSubfolders (of output directory)\nfor hysteresis steps. Length must\nmatch lmbdahys.\n--snapshots\n-snap\nbool\nSave snapshots into specified sub-\nfolders of output directory.\n--full_time_series\n-fts\nbool\nSave full time series toggle.\n--process_index\n-procid\nint\nProcess\nidentifier\n(logging/de-\nbug).\n3.2.2\netc_sample\nRuns a production measurement pass with the supplied configuration.\nListing 2: Example usage\n15\n\nSciPost Physics Codebases\nSubmission\n./paratoric -sim etc_sample -Ns 2000 -Nth 5000 -Nbs 10 -Nr 1000 -bet 16.0 -\nmuc 1 -Jc 1 -hc 0.2 -lmbdac 0.0 -obs energy plaquette_z anyon_count -bas\nz -lat square -L 16 -bound periodic -dsp 1 -outdir ./runs/sample -snap\n=0 -fcs=1\n3.2.3\netc_hysteresis\nRuns a parameter sweep where the last state of step i initializes step i+1. Provide --h_hy\nsteresis and --lmbda_hysteresis as space-separated lists, and --folder_names for\nper-step outputs.\nListing 3: Example usage\n./paratoric -sim etc_hysteresis -Ns 1000 -Nth 2000 -Nbs 50 -Nr 500 -bet\n12.0 -muc 1 -Jc 1 -lmbdahys 0.2 0.2 0.2 0.2 0.2 0.2 0.2 -hhys 0.0 0.1\n0.2 0.3 0.2 0.1 0.0 -obs energy fredenhagen_marcu -bas x -lat square -L\n12 -bound periodic -dsp 1 -outdir ./runs/hys -fns step0 step1 step2\nstep3 step4 step5 step6\n3.2.4\netc_thermalization\nPerforms thermalization only (no production sampling).\nListing 4: Example usage\n./paratoric -sim etc_thermalization -Ns 0 -Nth 5000 -Nbs 10 -Nr 500 -bet\n10.0 -muc 1 -Jc 1 -hc 0.3 -lmbdac 0.1 -hct 0.4 -lmbdact 0.2 -cth 1 -obs\nenergy anyon_density -bas z -lat square -L 10 -bound open -dsp 1 -outdir\n./runs/therm -snap=1\n3.2.5\nHDF5 structure\nThe output HDF5 has the structure simulation/results/acc_ratio for an array of the\nacceptance weights (only for thermalization), simulation/results/observable_name/\nseries for the time series (if it was enabled), and simulation/results/observable_\nname/{mean,mean_error,binder,binder_error,autocorrelation_time}. For the\nregular sampling, mean, mean_error, binder, binder_error and autocorrelation_\ntime contain doubles. For the hysteresis, they contain an array of values for the hysteresis\nparameters (in the order of h_hysteresis and lmbda_hysteresis).\n3.3\nC interface\nThe C interface enables users to use a ParaToric public header from within another C project\nor another programming language that supports a C-style interface. The C interface exposes\na stable ABI. It mirrors the C++ interface. Include the public header #include<paratoric/\nmcmc/extended_toric_code_c.h>. All functions return ptc_status_t.\n3.3.1\nBuild & Installation\nThe code can be compiled in exactly the same fashion as for the C++ interface.\n16\n\nSciPost Physics Codebases\nSubmission\nCMake usage (installed package).\ncmake_minimum_required(VERSION 3.23)\nproject(my_qmc_c C)\nfind_package(paratoric CONFIG REQUIRED)\n# provides paratoric::core\nadd_executable(cdemo main.c)\ntarget_link_libraries(cdemo PRIVATE paratoric::core)\nCMake usage (as subdirectory).\nadd_subdirectory(deps/paratoric)\nadd_executable(cdemo main.c)\ntarget_link_libraries(cdemo PRIVATE paratoric::core)\n3.3.2\nStatus & error handling\nName\nC type/values\nMeaning\nptc_status_t\n{PTC_STATUS_OK=0, PTC_STATU\nS_INVALID_ARGUMENT=1, PTC_S\nTATUS_RUNTIME_ERROR=2, PTC_\nSTATUS_NO_MEMORY=3, PTC_STA\nTUS_INTERNAL_ERROR=4}\nReturn code of every API call.\nptc_last_error() const char*\nThread–local error string.\nValid\nuntil next call.\n3.3.3\nOpaque handle\nCreate and destroy the interface instance. Use ptc_create(ptc_handle_t **out) and p\ntc_destroy(ptc_handle_t *h).\n3.3.4\nConfiguration type\nTop-level ptc_config_t aggregates four nested specs. Field names mirror the C++ Config.\nTop-level configuration: ptc_config_t\nField\nType\nPurpose\nsim\nptc_sim_spec_t\nMonte Carlo parameters.\nparams ptc_param_spec_t Hamiltonian parameters.\nlat\nptc_lat_spec_t\nLattice parameters.\nout\nptc_out_spec_t\nOutput paths and snapshot toggle.\n17\n\nSciPost Physics Codebases\nSubmission\nSimulation specification (config.sim)\nField\nType\nMeaning\nN_samples\nint\nNumber of snapshots.\nN_thermalization\nint\nThermalization steps.\nN_between_samples int\nThinning between snapshots.\nN_resamples\nint\nBootstrap resamples.\ncustom_therm\nbool\nCustom thermalization schedule.\nseed\nint\nPRNG seed (0 = random).\nobservables\nconst char* const* Array of observable names (nullable).\nN_observables\nsize_t\nLength of observables.\nParameter specification (config.params)\nField\nType\nMeaning\nmu,h,J,lmbda\ndouble\nCouplings (star, electric, plaquette, gauge).\nh_therm\ndouble\nThermalization value for h if custom_therm=true.\nlmbda_therm\ndouble\nThermalization value for lmbda if custom_therm=t\nrue.\nh_hys\nconst double* Hysteresis schedule for h (nullable).\nh_hys_len\nsize_t\nLength of h_hys. Must match lmbda_hys_len.\nlmbda_hys\nconst double* Hysteresis schedule for lmbda (nullable).\nlmbda_hys_len size_t\nLength of lmbda_hys. Must match h_hys_len.\nLattice specification (config.lat)\nField\nType\nMeaning / Valid values\nbasis\nchar\nSpin basis: ’x’ or ’z’.\nlattice_type const char* E.g. \"triangular\", \"square\", ...\nsystem_size\nint\nLinear system size per dimension.\nbeta\ndouble\nInverse temperature.\nboundaries\nconst char* \"periodic\" or \"open\".\ndefault_spin int\nInitial link spin: +1 or -1.\nOutput specification (config.out)\nField\nType\nMeaning\npath_out\nconst char*\nSingle output directory (nullable).\npaths_out\nconst char* const* Output directories for hysteresis steps (nul-\nlable).\nN_paths_out\nsize_t\nLength of paths_out. Must match h_hys_l\nen and lmbda_hys_len.\nsave_snapshots bool\nToggle snapshot dumping.\n3.3.5\nReturn type\nAll outputs are owned by the caller. Call ptc_result_destroy(&r) to free and zero.\n18\n\nSciPost Physics Codebases\nSubmission\nField\nC type\nMeaning\nseries\nptc_series_t\nTime series (real/complex) of all\nrequested observables, thermaliza-\ntion is excluded (except for ther-\nmalization simulation).\nOuter in-\ndex = observable, inner index =\ntime point.\nacc_ratio\nptc_dvec_t\nMC acceptance ratios for each up-\ndate (thermalization).\nmean, mean_std\nptc_dvec_t\nBootstrap mean and standard error\n(order as in observables).\nbinder, binder_std\nptc_dvec_t\nBinder ratios and standard error\n(order as in observables).\ntau_int\nptc_dvec_t\nIntegrated\nautocorrelation\ntime\n(order as in observables).\nseries_hys\nptc_series_block\ns_t\nHysteresis time series. Outer index\n= hysteresis parameters (order as\nin h_hys,lmbda_hys), middle in-\ndex = observables (order as in ob\nservables), inner vector = time\nseries.\nmean_hys, mean_std_hys, binder\n_hys, binder_std_hys, tau_int_\nhys\nptc_dmat_t\nOuter index = hysteresis parame-\nters (order as in h_hys,lmbda_h\nys), inner index = observables (or-\nder as in observables).\n3.3.6\nProcedures (mirror the C++ API)\nAll fill a ptc_result_t *out on success. Return PTC_STATUS_OK on success.\nptc_get_thermalization(ptc_handle_t *h, const ptc_config_t *cfg, ptc_r\nesult_t *out)\nRun thermalization only. Required fields: cfg->lat.{basis,lattice\n_type,system_size,beta,boundaries,default_spin}, cfg->params.{mu,h,J,lm\nbda}, cfg->sim.{N_thermalization,N_resamples,observables,N_observables,\nseed}, cfg->out.{path_out,save_snapshots}.\nptc_get_sample(ptc_handle_t *h, const ptc_config_t *cfg, ptc_result_t\n*out)\nRun a production measurement pass. Required fields: cfg->lat.{basis,latti\nce_type,system_size,beta,boundaries,default_spin}, cfg->params.{mu,h,J,\nlmbda,h_therm,lmbda_therm}, cfg->sim.{N_samples,N_thermalization,N_betw\neen_samples,N_resamples,custom_therm,observables,N_observables,seed}, c\nfg->out.{path_out,save_snapshots}.\nptc_get_hysteresis(ptc_handle_t *h, const ptc_config_t *cfg, ptc_resul\nt_t *out)\nRun a hysteresis sweep over h_hys and/or lmbda_hys. The last state of step\ni initializes step i+1. Required fields: cfg->lat.{basis,lattice_type,system_size\n,beta,boundaries,default_spin}, cfg->params.{mu,h_hys,h_hys_len,J,lmbda\n_hys,lmbda_hys_len}, cfg->sim.{N_samples,N_thermalization,N_between_sam\nples,N_resamples,observables,N_observables,seed}, cfg->out.{paths_out,N\n_paths_out,save_snapshots}.\n3.3.7\nC usage example\n19\n\nSciPost Physics Codebases\nSubmission\nListing 5: C API - Minimal call\n#include <stdio.h>\n#include <math.h>\n#include <paratoric/mcmc/extended_toric_code_c.h>\nint main(void) {\nptc_handle_t* h = NULL;\nif (ptc_create(&h) != PTC_STATUS_OK) { puts(\"create␣failed\"); return 1; }\nptc_lat_spec_t lat = {\n.basis = 'z',\n.lattice_type = \"square\",\n.system_size = 16,\n.beta = 8.0,\n.boundaries = \"periodic\",\n.default_spin = 1\n};\nptc_param_spec_t ps = {\n.mu = 1.0, .h = 0.2, .J = 1.0, .lmbda = 0.0,\n.h_therm = NAN, .lmbda_therm = NAN,\n.h_hys = NULL, .h_hys_len = 0,\n.lmbda_hys = NULL, .lmbda_hys_len = 0\n};\nconst char* obs[] = {\"energy\",\"plaquette_z\",\"anyon_count\"};\nptc_sim_spec_t sim = {\n.N_samples = 0, /* thermalization-only initially */\n.N_thermalization = 5000,\n.N_between_samples = 10,\n.N_resamples = 1000,\n.custom_therm = false,\n.seed = 12345,\n.observables = obs,\n.N_observables = sizeof(obs)/sizeof(obs[0])\n};\nptc_out_spec_t outspec = {\n.path_out = \"runs/sample\",\n.paths_out = NULL, .N_paths_out = 0,\n.save_snapshots = false\n};\nptc_config_t cfg = { .sim = sim, .params = ps, .lat = lat, .out = outspec\n};\nptc_result_t warm = {0};\nptc_status_t st = ptc_get_thermalization(h, &cfg, &warm);\nif (st != PTC_STATUS_OK) { puts(ptc_last_error()); ptc_destroy(h); return\n2; }\nptc_result_destroy(&warm);\ncfg.sim.N_samples = 2000;\nptc_result_t res = {0};\nst = ptc_get_sample(h, &cfg, &res);\nif (st != PTC_STATUS_OK) { puts(ptc_last_error()); ptc_destroy(h); return\n3; }\n20\n\nSciPost Physics Codebases\nSubmission\n// use res.mean, res.tau_int, ...\nptc_result_destroy(&res);\nptc_destroy(h);\nreturn 0;\n}\nMemory rules.\nYou own all buffers in ptc_result_t. Call ptc_result_destroy once\nper successful call.\n3.4\nPython bindings\nParaToric exposes a compiled Python extension module _paratoric with a submodule ext\nended_toric_code. The bindings convert C++ vectors into NumPy arrays and release the\nglobal interpreter lock (GIL) while running the C++ kernels.\n3.4.1\nBuild & Installation\nThe core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions may work,\nbut were not tested). The Python bindings require a Python installation with Numpy and\nPyBind11 (tested with version 3.0.1). Pybind11 is included as a git submodule (you need to\npull it!). To compile the Python bindings, run:\ncmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\\n-DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\\n-DPARATORIC_BUILD_TESTS=ON -DPARATORIC_BUILD_PYBIND=ON \\\n-DPython3_EXECUTABLE=\"$(which python)\" -DPYBIND11_FINDPYTHON=ON \\\n-DPARATORIC_INSTALL_TO_SITE=ON -DPARATORIC_PIP_EDITABLE_INSTALL=ON\ncmake --build build -jN\nctest --test-dir build -jN --output-on-failure\ncmake --install build\nReplace N with the number of cores to use, e.g. -j4 for 4 cores.\n• -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you’re a developer.\n• -DCMAKE_INSTALL_PREFIX. By default, executables install to ${CMAKE_SOURCE_DI\nR}/${CMAKE_INSTALL_BINDIR}/, headers to ${CMAKE_INSTALL_INCLUDEDIR}/p\naratoric, and static libraries to ${CMAKE_SOURCE_DIR}/${CMAKE_INSTALL_LIBD\nIR}/. The Python scripts expect ${CMAKE_SOURCE_DIR}/bin/; this directory always\ncontains the paratoric executable. To install into a custom directory, pass it via -DCM\nAKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo\nry/.\n• -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json\nfor tooling.\n• -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The\ncore itself does not need MPI.\n• -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and\nClang.\n21\n\nSciPost Physics Codebases\nSubmission\n• -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires\na CPU which supports AVX2.\n• -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recom-\nmended).\n• -DPARATORIC_BUILD_PYBIND=OFF. Compile Python bindings.\n• -DDPARATORIC_INSTALL_TO_SITE=OFF. Install ParaToric Python module to site pack-\nages.\n• -DPARATORIC_PIP_EDITABLE_INSTALL=OFF. Install ParaToric Python module via\npip as an editable module.\n• -DPARATORIC_PIP_OFFLINE_INSTALL=OFF. Turn on when installing to pip without\ninternet access. Requires NumPy and setuptools.\n3.4.2\nModule layout\n• paratoric._paratoric: compiled extension (PyBind11). Submodule: extended_t\noric_code.\n• paratoric.extended_toric_code: convenient alias\n• Running python -m paratoric enters the package entry point (__main__.py).\nNumPy return formats\nAll time series with potentially complex values are returned as complex128. Real observables\nappear with zero imaginary part. Shapes are documented in the function references below.\nAPI reference (paratoric.extended_toric_code)\nget_thermalization(...)\nRun only the warmup and return per-snapshot observables\nand MC acceptance ratios. Internally converts std::variant<complex<double>,double\n> to complex128 and std::vector<double> to float64 arrays. The GIL is released while\nthe C++ routine executes.\nParameter\nType / default\nMeaning\nN_thermalization\nint\nWarmup steps.\nN_resamples\nint=1000\nBootstrap resamples.\nobservables\nlist[str]\nNames per snapshot.\nseed\nint=0\nPRNG seed (0 ⇒random).\nmu,h,J,lmbda\nfloat\nHamiltonian parameters.\nbasis\n{’x’,’z’}=’x’\nSpin eigenbasis.\nlattice_type\nstr\nE.g. \"triangular\", \"square\", ...\nsystem_size\nint\nLinear size per dimension.\nbeta\nfloat\nInverse temperature.\nboundaries\nstr=\"periodic\"\nBoundary condition.\ndefault_spin\nint=1\nInitial link spin (+1/-1).\nsave_snapshots\nbool=false\nEnable snapshot files.\npath_out\npath|None=None\nOutput directory (if saving).\nReturns: (series, acc_ratio)\nseries: ndarray(complex128) of shape (n_ob\ns, N_thermalization); acc_ratio: ndarray(float64) of shape (N_thermalizati\non,).\n22\n\nSciPost Physics Codebases\nSubmission\nget_sample(...)\nRun thermalization and production sampling; return series and boot-\nstrap statistics. Converts nested C++ containers to NumPy arrays and releases the GIL during\ncomputation.\nParameter\nType / default\nMeaning\nN_samples\nint\nStored samples per observable.\nN_thermalization\nint\nWarmup steps before sampling.\nN_between_samples\nint\nThinning between samples.\nN_resamples\nint=1000\nBootstrap resamples.\ncustom_therm\nbool=false\nUse h_therm, lmbda_therm dur-\ning warmup.\nobservables\nlist[str]\nNames per snapshot.\nseed\nint=0\nPRNG seed (0 ⇒random).\nmu,h,J,lmbda\nfloat\nHamiltonian parameters.\nh_therm,lmbda_therm\nfloat=0\nWarmup\nparameters\nif\ncustom\ntherm.\nbasis\n{’x’,’z’}=’x’\nSpin eigenbasis.\nlattice_type,system_size,bet\na\nstr,int,float\nLattice and temperature.\nboundaries,default_spin\nstr,int=(\"period\nic\",1)\nBC and initial spin.\nsave_snapshots,path_out\nbool=False,path|\nNone=None\nOptional I/O.\nReturns: tuple of six arrays\nseries (complex128): (n_obs, N_samples); mean,\nmean_std, binder, binder_std, tau_int (float64): each (n_obs,).\nget_hysteresis(...)\nRun a sweep where each step uses the previous state as its ini-\ntial condition. Returns stacked arrays across steps; path handling validates per-step output\ndirectories when saving snapshots.\nParameter\nType / default\nMeaning\nN_samples,N_thermalization,N_b\netween_samples\nint,int,int\nCadence per step.\nN_resamples\nint=1000\nBootstrap resamples.\nobservables\nlist[str]\nNames per snapshot.\nseed\nint=0\nPRNG seed.\nmu,J\nfloat\nStar and plaquette couplings.\nh_hys,lmbda_hys\nlist[float]\nHysteresis values, length must\nmatch.\nbasis\n{’x’,’z’}=’x’\nSpin asis.\nlattice_type,system_size,beta\nstr,int,float\nLattice and temperature.\nboundaries,default_spin\nstr,int=(\"periodic\n\",1)\nBC and initial spin.\nsave_snapshots\nbool=false\nEnable stepwise I/O.\npaths_out\nlist[path]|None=No\nne\nOutput path per step (size must\nmatch h_hys if saving).\nReturns: tuple of six arrays\nseries3d (complex128): (n_steps, n_obs, N_sam\nples); mean2d, std2d, binder2d, binder_std2d, tau2d (float64): each (n_steps,\nn_obs). The number of steps equals len(h_hys) (and len(lmbda_hys)).\n23\n\nSciPost Physics Codebases\nSubmission\nArray dtypes and shapes (summary)\nFunction\nName / dtype\nShape\nget_thermalization\nseries (complex128)\n(n_obs, N_thermalization)\nacc_ratio (float64)\n(N_thermalization,)\nget_sample\nseries (complex128)\n(n_obs, N_samples)\nmean, mean_std, bind\ner, binder_std, tau_\nint (float64)\neach (n_obs,)\nget_hysteresis\nseries3d (complex12\n8)\n(n_steps, n_obs, N_samples)\nmean2d, std2d, binde\nr2d, binder_std2d, t\nau2d (float64)\neach (n_steps, n_obs)\nNotes on performance\nThe bindings release the global interpreter lock (GIL) during heavy compute (py::gil_sco\nped_release), enabling multi-threaded C++ execution if the backend uses threads or when\ncalling from multiprocessing workers. Conversions handle 1D/2D/3D containers and enforce\nconsistent inner lengths before copying to NumPy.\n3.4.3\nUsage example\nListing 6: Importing and calling from Python\n>>> import numpy as np\n>>> from paratoric import extended_toric_code as etc\n>>> series, acc_ratio = etc.get_thermalization(\n... N_thermalization=2000, N_resamples=500,\n... observables=[\"energy\",\"plaquette_z\",\"anyon_count\"],\n... seed=0, mu=1.0, h=0.2, J=1.0, lmbda=0.0,\n... basis='z', lattice_type=\"square\", system_size=16, beta=8.0,\n... boundaries=\"periodic\", default_spin=1,\n... save_snapshots=False, path_out=None)\n>>> series.shape, series.dtype\n((3, 2000), dtype('complex128'))\n>>> out = etc.get_sample(\n... N_samples=1000, N_thermalization=5000, N_between_samples=10,\n... N_resamples=1000, custom_therm=False,\n... observables=[\"energy\",\"plaquette_z\"],\n... seed=0, mu=1.0, h=0.2, h_therm=0.0,\n... J=1.0, lmbda=0.0, lmbda_therm=0.0,\n... basis='z', lattice_type=\"square\", system_size=16, beta=8.0,\n... boundaries=\"periodic\", default_spin=1,\n... save_snapshots=False, path_out=None)\n>>> (series_s, mean, mean_std, binder, binder_std, tau_int) = out\n3.5\nPython command-line interface\nParaToric ships a Python command-line interface /python/cli/paratoric.py that orches-\ntrates C++ backends, runs sweeps, and writes HDF5/XML outputs. It requires NumPy, mat-\nplotlib, and h5py.\n24\n\nSciPost Physics Codebases\nSubmission\n3.5.1\nBuild & Installation\nThe command-line interface requires HDF5 ≥1.14.3 (older HDF5 versions may work, but were\nnot tested). The core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions\nmay work, but were not tested). To compile it, run:\ncmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\\n-DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\\n-DPARATORIC_BUILD_TESTS=ON -DPARATORIC_BUILD_CLI=ON\ncmake --build build -jN\nctest --test-dir build -jN --output-on-failure\ncmake --install build\nReplace N with the number of cores to use, e.g. -j4 for 4 cores.\n• -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you’re a developer.\n• -DCMAKE_INSTALL_PREFIX. By default, executables install to ${CMAKE_SOURCE_DI\nR}/${CMAKE_INSTALL_BINDIR}/, headers to ${CMAKE_INSTALL_INCLUDEDIR}/p\naratoric, and static libraries to ${CMAKE_SOURCE_DIR}/${CMAKE_INSTALL_LIBD\nIR}/. The Python scripts expect ${CMAKE_SOURCE_DIR}/bin/; this directory always\ncontains the paratoric executable. To install into a custom directory, pass it via -DCM\nAKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo\nry/.\n• -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json\nfor tooling.\n• -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The\ncore itself does not need MPI.\n• -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and\nClang.\n• -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires\na CPU which supports AVX2.\n• -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recom-\nmended).\n• -DPARATORIC_BUILD_CLI=PROJECT_IS_TOP_LEVEL Required for both the C++ and\nPython command line interface.\n25\n\nSciPost Physics Codebases\nSubmission\nGeneral options\nLong flag\nShort\nDescription\n--help\n-h\nShow help and exit.\n--simulation\n-sim\nSimulation type selector.\n--N_thermalization\n-Nth\nThermalization steps (proposed updates).\n--N_samples\n-Ns\nNumber of samples/snapshots.\n--N_between_steps\n-Nbs\nSteps between successive samples (thinning).\n--N_resamples\n-Nr\nBootstrap resamples.\n--custom_therm\n-cth\nUse thermalization values for h,λ (0 or 1).\n--observables\n-obs\nSpace-separated list, e.g. fredenhagen_marcu\npercolation_probability energy.\n--seed\n-seed\nPRNG seed; 0 means random seed.\n--mu_constant\n-muc\nValue of µ.\n--J_constant\n-Jc\nValue of J.\n--h_constant\n-hc\nValue of h.\n--h_constant_therm\n-hct\nThermalization value of h.\n--lmbda_constant\n-lmbdac\nValue of λ.\n--lmbda_constant_therm\n-lmbdact\nThermalization value of λ.\n--output_directory\n-outdir\nOutput directory.\n--snapshots\n-snap\nSave snapshots toggle (0/1).\n--full_time_series\n-fts\nSave full time series toggle (0/1).\n--processes\n-proc\nLogical CPU count for Python multiprocessing. 0\nmeans all available cores. Negative numbers −x\nmean use all cores minus x. Default is -4.\nLattice-specific options\nLong flag\nShort\nDescription\n--help\n-h\nShow help and exit.\n--basis\n-bas\nSpin basis: x or z.\n--lattice_type\n-lat\nsquare, cubic, triangular, honeycomb, ...\n--system_size\n-L\nLinear size; in 2D, 30 yields a 30×30 lattice (unit\ncells).\n--temperature\n-T\nTemperature T = 1/β > 0.\n--boundaries\n-bound\nperiodic or open.\n--default_spin\n-dsp\nInitial edge spin: 1 or -1.\nThe command line interface offers several sweep modes. All are embarrassingly parallel;\nset --processes close to the number of steps when possible.\n3.5.2\nT-sweep\nRuns T_steps independent Markov chains for evenly spaced temperatures in [T_lower, T_upper]\nand plots all requested observables.\nListing 7: Example usage\n26\n\nSciPost Physics Codebases\nSubmission\npython3 ./python/cli/paratoric.py -sim etc_T_sweep -Ns 1000 -muc 1 -Nth\n2000 -Nbs 100 -Tl 0.5 -Tu 5 -Ts 30 -hc 0.1 -Jc 1 -lmbdac 0.1 -Nr 1000 -\nobs percolation_strength percolation_probability\nplaquette_percolation_probability largest_cluster string_number energy\nenergy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x\nplaquette_z staggered_imaginary_times delta anyon_count anyon_density\nfredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -\nbas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_T_sweep.\n--T_lower\n-Tl\nLower bound of T.\n--T_upper\n-Tu\nUpper bound of T.\n--T_steps\n-Ts\nNumber of temperatures between bounds.\n3.5.3\nh-sweep\nRuns h_steps independent chains in parallel for evenly spaced h in [h_lower, h_upper].\nListing 8: Example usage\npython3 ./python/cli/paratoric.py -sim etc_h_sweep -Ns 1000 -muc 1 -Nth\n2000 -Nbs 100 -hl 0.1 -hu 0.5 -hs 8 -T 0.03 -Jc 1 -lmbdac 0.2 -Nr 1000 -\nobs percolation_strength percolation_probability\nplaquette_percolation_probability largest_cluster string_number energy\nenergy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x\nplaquette_z staggered_imaginary_times delta anyon_count anyon_density\nfredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -\nbas x -lat square -L 6 -bound periodic -dsp 1 -outdir /path/to/out\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_h_sweep.\n--h_lower\n-hl\nLower bound of h.\n--h_upper\n-hu\nUpper bound of h.\n--h_steps\n-hs\nNumber of field steps between bounds.\n3.5.4\nλ-sweep\nRuns lmbda_steps independent chains in parallel for evenly spaced λ in [λ_lower, λ_upper].\nListing 9: Example usage\npython3 ./python/cli/paratoric.py -sim etc_lmbda_sweep -Ns 1000 -muc 1 -Nth\n2000 -Nbs 100 -lmbdal 0.01 -lmbdau 1.0 -lmbdas 15 -T 0.1 -hc 0.3 -Jc 1\n-Nr 1000 -obs percolation_strength percolation_probability\nplaquette_percolation_probability largest_cluster string_number energy\nenergy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x\nplaquette_z staggered_imaginary_times delta anyon_count anyon_density\nfredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -\nbas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out\n27\n\nSciPost Physics Codebases\nSubmission\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_lmbda_sweep.\n--lmbda_lower\n-lmbdal\nLower bound of λ.\n--lmbda_upper\n-lmbdau\nUpper bound of λ.\n--lmbda_steps\n-lmbdas\nNumber of field steps between bounds.\n3.5.5\n◦-sweep\nRuns Theta_steps independent chains in parallel along a circle in (λ,h) centered at (lmbda\n_constant,h_constant) with radius radius, for angles Θ ∈[Θ_lower, Θ_upper] (angles\nmeasured anti-clockwise from the λ-axis).\nListing 10: Example usage\npython3 ./python/cli/paratoric.py -sim etc_circle_sweep -Ns 1000 -muc 1 -\nNth 2000 -Nbs 100 -lmbdac 0.4 -rad 0.3 -Thl 0 -Thu 3.141 -Ths 15 -T 0.1\n-hc 0.4 -Jc 1 -Nr 1000 -obs percolation_strength percolation_probability\nplaquette_percolation_probability largest_cluster string_number energy\nenergy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x\nplaquette_z staggered_imaginary_times delta anyon_count anyon_density\nfredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -\nbas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_circle_sweep.\n--lmbda_constant\n-lmbdac\nCircle center in λ.\n--h_constant\n-hc\nCircle center in h.\n--radius\n-rad\nCircle radius.\n--Theta_lower\n-Thl\nLower bound of Θ.\n--Theta_upper\n-Thu\nUpper bound of Θ.\n--Theta_steps\n-Ths\nNumber of angles between bounds.\n3.5.6\nHysteresis-sweep\nUse the hysteresis schedule specified in hhys and lmbdahys. This mode will run two Markov\nchains, one in the original parameter order specified in hhys and lmbdahys, and one with a\nreversed parameter order, i.e., it calculates both branches of the hysteresis loop.\nListing 11: Example usage\npython3 ./python/cli/paratoric.py -sim etc_hysteresis -Nbs 5000 -Ns 10000 -\nmuc 1 -Nth 20000 -hhys 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -T\n0.1 -Jc -1 -lmbdahys 0.5 0.525 0.55 0.575 0.6 0.625 0.65 0.675 0.7 0.725\n0.75 -Nr 1000 -obs plaquette_percolation_probability\npercolation_strength percolation_probability largest_cluster\nstring_number energy energy_h energy_mu energy_J energy_lmbda sigma_x\nsigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count\nanyon_density fredenhagen_marcu sigma_x_susceptibility\nsigma_z_susceptibility -s 0 -bas z -lat square -L 4 -bound periodic -dsp\n1 -outdir /path/to/out\n28\n\nSciPost Physics Codebases\nSubmission\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_hysteresis.\n--lmbda_hysteresis\n-lmbdahys\nHysteresis schedule for λ. Length must match hh\nys.\n--h_hysteresis\n-hhys\nHysteresis schedule for h. Length must match lm\nbdahys.\n3.5.7\nThermalization\nRuns repetitions independent chains in parallel and reports observables and MC accep-\ntance ratios every step, averaged over chains.\nListing 12: Example usage\npython3 ./python/cli/paratoric.py -sim etc_thermalization -muc 1 -Nth 2000\n-reps 10 -lmbdac 2 -T 0.1 -hc 0.3 -Jc 1 -Nr 1000 -obs\npercolation_strength percolation_probability\nplaquette_percolation_probability largest_cluster string_number energy\nenergy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x\nplaquette_z staggered_imaginary_times delta anyon_count anyon_density\nfredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -\nbas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out\nSweep-specific flags.\nLong flag\nShort\nDescription\n--simulation\n-sim\nUse etc_thermalization.\n--repetitions\n-reps\nNumber of Markov chains to average.\n4\nUsing ParaToric\n4.1\nMonte Carlo Updates\nThere is no need for the user to explicitly call specific updates or interact with internal C++\nclasses when using the documented interfaces. Internally, we use all five updates described\nin the original algorithm by Wu, Deng, and Prokof’ev [5]. These must furthermore be sup-\nplemented with the following two updates: Because for high temperatures and for zero off-\ndiagonal fields the spin at imaginary time 0 = β cannot be flipped, we allow for flipping the\nspin on the entire imaginary axis on one bond or on a plaquette (star) in the ˆσx-basis (ˆσz-\nbasis).\nThese updates only change the energy terms diagonal in the given basis and is trivial when\ncaching the total integrated diagonal energy (the update locally flips the sign of the total in-\ntegrated potential energy). Another advantage is that integrated autocorrelation times for\nobservables diagonal in the given basis improve even in regimes that were previously accessi-\nble. All seven updates are equally likely to be proposed, and we use a 64-bit Mersenne-Twister\nfor pseudorandom numbers [12] with the ability to externally set the seed. Some updates\nhave early exits for input parameters for which they will always be rejected.\n29\n\nSciPost Physics Codebases\nSubmission\n4.2\nMonte Carlo Diagnostics\nThere are two compilation modes, Release and Debug. In production runs, one should\nalways use the Release mode; however, it still gives the user enough information to diagnose\nsampling problems without severe performance impacts.\n4.2.1\nThermalization mode\nWe provide thermalization routines which should be used before production runs to ensure\nproper thermalization (also known as burn-in). Thermalization times can vary drastically\nbetween different observables and initial conditions. We provide an example of sufficient and\ninsufficient thermalization in Fig. 2. We recommend using the provided Python command-line\ninterface, which will also plot the thermalization of all measured observables for the user.\nIn thermalization runs, we also return the Monte Carlo acceptance ratio of every update.\nThis can also be used to diagnose freezing (in the measurement phase, use the integrated\nautocorrelation time instead), e.g., when the acceptance ratio is always identical and/or very\nlow.\nIn case one suspects experiencing a serious sampling problem, we recommend recompiling\nthe project in the Debug mode, which provides a wide array of runtime debug information\nabout the proposed steps, acceptance ratios, and intermediate results. However, do not use\nthe Debug mode in production runs, as it negatively impacts performance.\n4.2.2\nIntegrated autocorrelation time\nWhen measuring observables, we first thermalize the system with N_thermalization steps,\nthen measure N_samples times with N_between_samples steps between measurements.\nThe normalized autocorrelation function ρO(k) of an observable Ok (observable O measured\nat time k) applied to a discrete time series of length N is given by:\nρO(k) = C(k)\nC(0),\nC(k) =\n1\nN −k\nN−k−1\nX\ni=0\n(Oi −¯O)(Oi+k −¯O),\n¯O = 1\nN\nN−1\nX\ni=0\nOi.\n(7)\nIt is a statistical measure of the correlations between measurements of observable O at times\ni and i + k.2 We define the integrated autocorrelation time\nτO\nint = 1\n2 +\nX\nk≥1\nρO(k).\n(8)\nLarge τint are generally undesirable since they increase error bars and can lead to bias. In case\nof perfect sampling, we would have ρO(0) = 1 and ρO(k) = 0 ∀k ≥1, i.e., each measurement\nis only correlated with itself but not with other measurements and τint = 1/2. In practice, this\nis usually not feasible, and we have to work with a finite autocorrelation time τint > 1/2.\nWhen using ParaToric, we strongly recommend monitoring τint for all simulations and all\nobservables. It is automatically calculated for every observable based on the full time series. As\na rule of thumb, the autocorrelation is fine as long as τint ≪Nsamples, otherwise it leads to bias\nand seriously underestimated error bars. In the vicinity of phase transitions, τint dramatically\nincreases (“critical slowing down”) [13]. Importantly, τint can differ vastly between different\nobservables! If the autocorrelation is too high, increase the number of steps between samples.\nIn more complicated cases, one may need to adapt the update proposal distributions and/or\nthe updates themselves as a last resort.\n2In ParaToric, the autocorrelation function is calculated efficiently using fast Fourier transforms.\n30\n\nSciPost Physics Codebases\nSubmission\nIt is also important to mention that ParaToric only computes a statistical estimate of τint.\nMany factors determine how accurate this estimate is, and crucially, the system needs to be\nproperly thermalized. In principle, one can use τint in the way it is computed above directly\nfor calculating error bars of correlated time series; however, ParaToric uses a more robust\nbootstrapping approach.\n4.2.3\nError bars\nParaToric applies the stationary bootstrap [14–16] for all error bars, thus capturing autocor-\nrelation effects. Large τint will lead to worse error bars. The only parameter that the user\ncan change is the number of bootstrap resamples N_between_steps. The default is 1000,\nwhich is enough in most cases. Note that a too low value of N_between_steps increases\nthe relative computational cost of performing the measurements, which may negatively affect\nthe code efficiency at no statistical gain. If the error bars are too large, either the number of\nsamples is too low (in which case one should increase N_samples) or the autocorrelation is\ntoo large (in which case one could additionally increase N_between_samples).\n4.3\nTips & tricks\n4.3.1\nProbing ground state physics\nThe algorithm implemented by ParaToric fundamentally requires a finite temperature T > 0.\nHowever, in QMC simulations, there is always a finite-size energy gap (the difference between\nthe energy of the ground state and the first excited state). Additionally, some phases like the\ntopological ground state of the toric code have a physical bulk gap (even at L →∞). As long\nas the temperature is well below the total gap, we are exponentially close to the ground state.\nUsually, a temperature T ∼1/L suffices for the toric code although other situations may arise.\n4.3.2\nProbing first-order transitions\nParaToric provides functionalities to probe weak and strong first-order phase transitions. The\nhysteresis mode can be used to probe hysteresis loops in the vicinity of strong first-order phase\ntransitions, by repeating the simulation two times and mirroring the order of the parameters\nin h_hysteresis and lmbda_hysteresis. Weak first-order transitions can be detected by\nplotting a time series histogram of an observable (it exhibits a double-peak structure). Both\napproaches have been used in the context of the toric code [11].\n4.3.3\nChoosing the basis\nSometimes, one can work in both the ˆσx and the ˆσz-basis. The performance can vary drasti-\ncally! Generally, the ˆσx-basis is more efficient for h/J > λ/µ and vice versa.\n4.3.4\nChoosing N_thermalization\nBased on our experience, we can say that Nthermalization = 500Ld/T is a sensible choice for\nsmall fields, where d is the dimensionality of the system. Nevertheless, one should make use\nof the provided tools to benchmark thermalization, see Sec. 4.2, and rather err on the side of\nsafety.\n4.3.5\nChoosing N_samples\nNeglecting\nautocorrelation\neffects,\nthe\nerror\nof\nan\nobservable\n∆O\nscales\nas\n∆O ∼1/\nÆ\nNsamples. More samples are, in principle, always better and lead to lower error\n31\n\nSciPost Physics Codebases\nSubmission\nbars. Smoothness of a curve of statistical results also requires that error bars be small in rela-\ntion to the parameter grid size. If one increases the parameter resolution (e.g., in the field h),\nthen one typically also increases N_samples.\n4.3.6\nChoosing N_between_samples\nThe optimal choice for N_samples is the integrated autocorrelation time. A good guess of\nthe autocorrelation time based on previous simulations for smaller system sizes or nearby\nparameter points can result in substantial computational saving costs in production runs for\nlarge system sizes. Near continuous phase transitions, the integrated autocorrelation time has\nan additional dependence τint ∼Lz, where z is the dynamical exponent of the universality\nclass of the transition. For a 2D system in the vicinity of a continuous phase transition, a\nsensible scaling for N_between_samples could be O(L2 × β × Lz) (O(L2) links, each has\noff-diagonal spin flips O(β)).\n4.3.7\nChoosing N_resamples\nAs with N_samples, more is better (but also more costly). Usually N_resamples ≈1000 is\na sensible choice.\n4.3.8\nExtracting snapshots\nWhen the option save_snapshots is enabled, ParaToric will write the snapshots into the\ndirectory specified in path_out (or in the paths paths_out for hysteresis sweeps). The\nsnapshots are saved in the GraphML format (XML-based), which is supported by many major\ngraph libraries. One snapshot will be saved for every measurement of observables, i.e., N_s\namples snapshots in total. All snapshots are written into a single file to save disk space and\nsimultaneously offer a structured, self-documenting format. Every edge stores a list of spins.\nThe first spin belongs to the first snapshot, the second one to the second snapshot, and so on.\nThere are no special requirements for disks or memory bandwidth; the snapshots are kept in\nRAM and are only written to disk after the simulation has finished.\n4.3.9\nAdding new observables/lattices/updates\nAfter adding features to the code, always benchmark them using analytical results, other nu-\nmerical methods (exact diagonalization, tensor networks, ...), and unit tests. We advise using\na fixed seed during development, e.g., when checking whether two methods produce the exact\nsame result. The code has some built-in features to check self-consistency, e.g., at the end of\neach simulation, the code checks whether the cached total energy is numerically close to the\ntotal energy calculated from scratch. Do not turn off these features, as they will point you\ntoward bugs!\n4.4\nBenchmarks\n4.4.1\nThermalization\nIn Fig. 2 (which we already discussed before and repeat here for completeness) we plot the\ngauge field energy ∝λ for two systems: one is sufficiently thermalized, the other one is not.\nThe plots are a direct output of the Python command-line interface. Always make sure that\nthe system is well thermalized.\n32\n\nSciPost Physics Codebases\nSubmission\n(b)\n(a)\nFigure 2: Good and bad thermalization plots produced by the Python command-\nline interface. We show the gauge field energy ∝λ. (a) The system is well thermal-\nized; after its initial drop, the energy fluctuates around the expectation value. (b)\nThe system is not yet thermalized; the floating average is still decreasing.\n4.4.2\nIntegrated autocorrelation time\nHere, we demonstrate how the integrated autocorrelation time τint grows with decreasing N_\nbetween_samples. We use the following setup:\nListing 13: N_between_samples benchmarking setup\n>>> import numpy as np\n>>> from paratoric import extended_toric_code as etc\n>>> series, mean, std, binder, binder_std, tau_int = extended_toric_code.\nget_sample(N_samples=100000, N_thermalization=10000, N_between_samples\n=1, N_resamples=1000, custom_therm=False, observables=[\"energy\"], seed\n=0, mu=1.0, h=0.0, h_therm=0.0, J=1.0, lmbda=0, lmbda_therm=0.0, basis='\nx', lattice_type=\"square\", system_size=4, beta=10, boundaries=\"periodic\"\n, default_spin=1, save_snapshots=False)\nWe only run the simulation once per N_between_samples. The results are:\nN_between_samples\n1\n10\n100\n500\n1000\nτint (energy)\n1895\n141.4\n19.7\n3.24\n1.64\nFor very small N_between_samples, τint is very high: in cases where the update is re-\njected, the configuration is identical to the one measured before! A choice of N_between_s\namples between 500 and 1000 would be a good tradeoff between τint and runtime for this\nexample. Increasing N_between_samples to well over 1000 would be a waste of CPU time.\n4.4.3\nRun-time\nWe benchmark the run-time for two realistic parameter sets on the square lattice and varying\nsystem size. The first setup simulates the toric code without fields:3\nListing 14: L benchmarking setup 1\n>>> import numpy as np\n>>> from paratoric import extended_toric_code as etc\n3All tests were run on a laptop; some conditions, like the CPU temperature, were not identical for all simulations.\nThe benchmarks are therefore only an approximation.\n33\n\nSciPost Physics Codebases\nSubmission\n>>> L=20\n>>> series, mean, std, binder, binder_std, tau_int = extended_toric_code.\nget_sample(N_samples=10000, N_thermalization=500*L*L*L,\nN_between_samples=8*L*L*L, N_resamples=1000, custom_therm=False,\nobservables=[\"energy\", \"sigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.0,\nh_therm=0.0, J=1.0, lmbda=0, lmbda_therm=0.0, basis='x', lattice_type=\"\nsquare\", system_size=L, beta=L, boundaries=\"periodic\", default_spin=1,\nsave_snapshots=False)\nWe only run one test per system size. The results are:\nL\n4\n8\n12\n16\n20\nRuntime (s)\n3.1\n21.3\n75\n197\n379\nFrom our experience, for large L the update complexity is approximately O(L3 logβ), owing\nto the chosen cubic dependency of N_thermalization and N_between_samples and a\nO(logβ) dependence of operations on the imaginary time axis, see β benchmark below. The\nsystem size itself does not impact the performance, as the interactions are local. On computing\nclusters, we have realized system sizes of up to L = 80 for the square lattice; this number will\nonly increase in the future as CPUs get faster.\nThe second setup simulates the toric code with fields in both ˆσx and ˆσz-direction:\nListing 15: L benchmarking setup 2\n>>> import numpy as np\n>>> from paratoric import extended_toric_code as etc\n>>> L=20\n>>> series, mean, std, binder, binder_std, tau_int = extended_toric_code.\nget_sample(N_samples=10000, N_thermalization=500*L*L*L,\nN_between_samples=8*L*L*L, N_resamples=1000, custom_therm=False,\nobservables=[\"energy\", \"sigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.2, J\n=1.0, lmbda=0.2, basis='x', lattice_type=\"square\", system_size=L, beta=L\n, boundaries=\"periodic\", default_spin=1, save_snapshots=False)\nWe only run one test per system size. The results are:\nL\n4\n8\n12\n16\n20\nRuntime (s)\n3.9\n34.1\n133\n323\n689\nWe also test the run-time dependence of the inverse temperature β, with the following\nsetup:\nListing 16: β benchmarking setup\n>>> import numpy as np\n>>> from paratoric import extended_toric_code as etc\n>>> series, mean, std, binder, binder_std, tau_int = extended_toric_code.\nget_sample(N_samples=10000, N_thermalization=20000, N_between_samples\n=2000, N_resamples=1000, custom_therm=False, observables=[\"energy\", \"\nsigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.2, J=1.0, lmbda=0.2, basis='x'\n, lattice_type=\"square\", system_size=10, beta=20, boundaries=\"periodic\",\ndefault_spin=1, save_snapshots=False)\nWe only run one test per β. The results are:\nβ\n4\n8\n12\n16\n20\nRuntime (s)\n14.9\n17.2\n19.1\n20.0\n22.1\n34\n\nSciPost Physics Codebases\nSubmission\n0.20\n0.25\n0.30\n0.35\n0.40\nh\n0\n1\nOx\nFM\nsquare, ⁄ = 0.2\nL = 10\nL = 20\nL = 30\nL = 40\n0.20\n0.25\n0.30\n0.35\n0.40\nh\n0.0\n0.5\nOx\nSI\nsquare, ⁄ = 0.2\nL = 10\nL = 20\nL = 30\nL = 40\n0.31\n0.32\n0.33\n0.34\nh\n100\n101\n102\nUΠx\nsquare, ⁄ = 0.2\nL = 10\nL = 20\nL = 30\nL = 40\n0.20\n0.25\n0.30\n0.35\n0.40\nh\n0\n1\nΠx\nsquare, ⁄ = 0.2\nL = 10\nL = 20\nL = 30\nL = 40\n(b)\n(a)\n(c)\n(d)\nFigure 3: Topological phase transition of the extended toric code (1) on the square\nlattice. The critical field is known and located at hc(λ = 0.2) ≈0.33 [5,9]. Our re-\nsults agree with the value published in the literature, within error bars. (a) The per-\ncolation probability Binder ratio UΠx [9] features a crossing point around h = 0.33.\n(b) The percolation probability Πx is non-zero in the topological phase and zero in\nthe trivial phase. The transition gets sharper with increasing system size. (c) The\nstaggered imaginary time order parameter Ox\nSI is zero in the topological phase and\nnon-zero in the trivial phase. (d) The Fredenhagen-Marcu order parameter Ox\nFM is\nzero in the topological phase and non-zero in the trivial phase. The loop length grows\nwith O(L). Compared to the other order parameters, it is very noisy because it is a\nmulti-body correlator and on top of that a division of two exponentially small num-\nbers.\nThis benchmark illustrates an appealing feature of our implementation: there is almost no\nslowing down when increasing β implying that very low temperatures are within reach with\nParaToric. This paradoxical result (because the number n of off-diagonal star/plaquette and\nmagnetic field operators must physically scale linearly in β) is explained by the fact that most\nsearches within the imaginary time axis scale as O(log n) by making use of binary searches.\n4.4.4\nTopological phase transition\nWe probe the well-known topological phase transition in the ground state of the extended toric\ncode (1) on the square lattice, where we have a gapped Z2 quantum spin liquid for small fields\nh,λ and a topologically trivial phase for high fields. We set J = µ = 1, λ = 0.2 and sweep h\nover the known critical value hc(λ = 0.2) ≈0.33 [5,9] for L ∈{10,20,30,40} in the ˆσx-basis.\nThe temperature is set to T = 1/L to capture ground state physics. We take 30000 snapshots,\nwith 8L3 steps in between snapshots and 500L3 thermalization steps.4. We confirm that the\nsystem is well thermalized and that all integrated autocorrelation times are below 10, i.e.,\nthe produced snapshots can safely be considered identically and independently distributed.\nWe show the percolation probability, the Fredenhagen-Marcu string order parameter, and the\nstaggered imaginary time order parameter in Fig. 3. All of them reproduce the known phase\nboundary.\n4If we were interested in quantities like critical exponents and we need to go very close to the critical field,\nwe should take into account the dynamical exponent z (τint ∼Lz) in the number of steps between snapshots to\naccount for critical slowing down. Now the error bars are just larger near the critical field.\n35\n\nSciPost Physics Codebases\nSubmission\n5\nConclusion & Outlook\nWe have presented ParaToric, a continuous-time quantum Monte Carlo solver for the toric code\nin a parallel field. ParaToric builds on the existing work by Wu, Deng, and Prokof’ev [5] and\nis also applicable to high temperature and low off-diagonal couplings.\nParaToric can store snapshots, which makes it ideally suited to generate training/bench-\nmarking data for applications in other fields, such as lattice gauge theories, cold atom or other\nquantum simulators, quantum spin liquids, artificial intelligence, and quantum error correc-\ntion. We believe it also serves a pedagogical purpose. Another strength of ParaToric is its in-\nteroperability with other programming languages. The C interface is compatible with virtually\nall programming languages, thus ParaToric can be seamlessly integrated into other projects.\nParaToric comes with an MIT license. For future release of ParaToric we plan extensions\nalong the following lines:\n• Additional lattices such as the kagome and the ruby lattice. Given the underlying graph\nstructure used in ParaToric, such extensions are straightforward.\n• Additional observables: we think here of, for instance, the finite temperature extension\nof the fidelity susceptibility to diagnose the phase transitions in the absence of a local\norder parameter. It would be worthwhile to have additional off-diagonal observables\nsuch as the off-diagonal Fredenhagen-Marcu string operators, or correlation functions\nbetween off-diagonal operators in space and or time. Measurements of the Renyi entropy\nare also high on the to-do list. The latter two classes require however major changes to\nthe code, and testing.\n• Additional interaction types. There are many classes of models in which topological\norder may be emergent instead of explicit as in the toric code. Such models typically\nhave additional interactions than the ones covered in ParaToric, such as longer-range\nIsing interactions, and miss some others (typically the plaquette type interactions, and\nsometimes even the star terms). It is in general an open problem how to efficiently\nsimulate such models at the lowest temperatures (even for sign-free models). Extending\nParaToric to dealing with other types of interactions can thus serve as an additional tool\nfor benchmarking purposes and algorithmic exploration.\nAcknowledgements\nThe authors acknowledge fruitful discussions with A. Bohrdt, G. De Paciani, G. Dünnweber, F.\nGrusdt, L. Homeier, and N. V. Prokof’ev.\nAuthor contributions\nSML did the main coding and planning work with input from LP. All\nauthors contributed to the writing of the manuscript.\nFunding information\nThis research was funded by the European Research Council (ERC)\nunder the European Union’s Horizon 2020 research and innovation program – ERC Starting\nGrant SimUcQuam (Grant Agreement No. 948141), and by the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111\n– project number 390814868.\n36\n\nSciPost Physics Codebases\nSubmission\nReferences\n[1] A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303(1), 2\n(2003), doi:https://doi.org/10.1016/S0003-4916(02)00018-0.\n[2] E. Dennis, A. Kitaev, A. Landahl and J. Preskill, Topological quantum memory, Journal of\nMathematical Physics 43(9), 4452 (2002), doi:10.1063/1.1499754, https://pubs.aip.\norg/aip/jmp/article-pdf/43/9/4452/19183135/4452_1_online.pdf.\n[3] A. G. Fowler, M. Mariantoni, J. M. Martinis and A. N. Cleland,\nSurface codes: To-\nwards practical large-scale quantum computation,\nPhys. Rev. A 86, 032324 (2012),\ndoi:10.1103/PhysRevA.86.032324.\n[4] J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51,\n659 (1979), doi:10.1103/RevModPhys.51.659.\n[5] F. Wu, Y. Deng and N. Prokof’ev,\nPhase diagram of the toric code model in a parallel\nmagnetic field, Phys. Rev. B 85, 195104 (2012), doi:10.1103/PhysRevB.85.195104.\n[6] K. Fredenhagen and M. Marcu, Charged states in z2 gauge theories, Communications in\nMathematical Physics 92(1), 81 (1983), doi:10.1007/BF01206315.\n[7] K. Fredenhagen and M. Marcu,\nConfinement criterion for qcd with dynamical quarks,\nPhys. Rev. Lett. 56, 223 (1986), doi:10.1103/PhysRevLett.56.223.\n[8] K. Fredenhagen and M. Marcu, Dual interpretation of order parameters for lattice gauge\ntheories with matter fields, Nuclear Physics B - Proceedings Supplements 4, 352 (1988),\ndoi:https://doi.org/10.1016/0920-5632(88)90124-7.\n[9] S. M. Linsel, A. Bohrdt, L. Homeier, L. Pollet and F. Grusdt, Percolation as a confine-\nment order parameter in z2 lattice gauge theories, Phys. Rev. B 110, L241101 (2024),\ndoi:10.1103/PhysRevB.110.L241101.\n[10] G. Dünnweber, S. M. Linsel, A. Bohrdt and F. Grusdt, Percolation renormalization group\nanalysis of confinement in z2 lattice gauge theories, Phys. Rev. B 111, 024314 (2025),\ndoi:10.1103/PhysRevB.111.024314.\n[11] S. M. Linsel, L. Pollet and F. Grusdt,\nIndependent e- and m-anyon confinement in the\nparallel field toric code on non-square lattices, doi:10.48550/arXiv.2504.03512 (2025).\n[12] M. Matsumoto and T. Nishimura, Mersenne twister: a 623-dimensionally equidistributed\nuniform pseudo-random number generator,\nACM Trans. Model. Comput. Simul. 8, 3\n(1998).\n[13] U. Wolff, Critical slowing down, Nuclear Physics B - Proceedings Supplements 17, 93\n(1990), doi:https://doi.org/10.1016/0920-5632(90)90224-I.\n[14] D. N. Politis and J. P. Romano, The stationary bootstrap, Journal of the American Statis-\ntical Association 89(428), 1303 (1994), doi:10.1080/01621459.1994.10476870.\n[15] D. N. Politis and H. White, Automatic block-length selection for the dependent bootstrap,\nEconometric Reviews 23(1), 53 (2004), doi:10.1081/ETC-120028836.\n[16] A. Patton, D. N. Politis and H. White, Correction to “automatic block-length selection for the\ndependent bootstrap” by d. politis and h. white, Econometric Reviews 28(4), 372 (2009),\ndoi:10.1080/07474930802459016.\n37\n"},"cleaned":{"kind":"string","value":"SciPost Physics Codebases Submission ParaToric 1.0-beta: Continuous-time quantum Monte Carlo for the toric code in a parallel field Simon M. Linsel⋆and Lode Pollet† (ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37, München D-80333, Germany Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany ⋆ , † Abstract We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., X- and Z-fields) at finite temperature. We implement and extend the continuoustime quantum Monte Carlo algorithm of Wu, Deng, and Prokof'ev on the square, triangular, honeycomb, and cubic lattices with open and periodic boundaries, respectively. The package is expandable to arbitrary lattice geometries and custom observables diagonal in either the X- or Z-basis. ParaToric also supports snapshot extraction in both bases, making it ideal for generating training/benchmarking data for other methods, such as lattice gauge theories, cold atom or other quantum simulators, quantum spin liquids, artificial intelligence, and quantum error correction. The software provides bindings to C/C++ and Python, and is thus almost universally integrable into other software projects. Copyright attribution to authors. This work is a submission to SciPost Physics Codebases. License information to appear upon publication. Publication information to appear upon publication. Received Date Accepted Date Published Date Contents 1 Introduction 3 2 The toric code in a parallel field 3 2.1 Hamiltonian 3 2.2 Lattice geometries 3 2.3 Observables 4 3 Installation & interfaces 6 3.1 C++ interface 6 3.1.1 Build & Installation 7 3.1.2 Public class ExtendedToricCode 8 3.1.3 Configuration type 8 3.1.4 Return type 11 3.1.5 C++ usage examples 11 3.2 C++ command-line interface 13 3.2.1 Build & Installation 13 1 16 Oct 2025 SciPost Physics Codebases Submission 3.2.2 etc_sample 15 3.2.3 etc_hysteresis 16 3.2.4 etc_thermalization 16 3.2.5 HDF5 structure 16 3.3 C interface 16 3.3.1 Build & Installation 16 3.3.2 Status & error handling 17 3.3.3 Opaque handle 17 3.3.4 Configuration type 17 3.3.5 Return type 18 3.3.6 Procedures (mirror the C++ API) 19 3.3.7 C usage example 19 3.4 Python bindings 21 3.4.1 Build & Installation 21 3.4.2 Module layout 22 3.4.3 Usage example 24 3.5 Python command-line interface 24 3.5.1 Build & Installation 25 3.5.2 T-sweep 26 3.5.3 h-sweep 27 3.5.4 λ-sweep 27 3.5.5 ◦-sweep 28 3.5.6 Hysteresis-sweep 28 3.5.7 Thermalization 29 4 Using ParaToric 29 4.1 Monte Carlo Updates 29 4.2 Monte Carlo Diagnostics 30 4.2.1 Thermalization mode 30 4.2.2 Integrated autocorrelation time 30 4.2.3 Error bars 31 4.3 Tips & tricks 31 4.3.1 Probing ground state physics 31 4.3.2 Probing first-order transitions 31 4.3.3 Choosing the basis 31 4.3.4 Choosing N_thermalization 31 4.3.5 Choosing N_samples 31 4.3.6 Choosing N_between_samples 32 4.3.7 Choosing N_resamples 32 4.3.8 Extracting snapshots 32 4.3.9 Adding new observables/lattices/updates 32 4.4 Benchmarks 32 4.4.1 Thermalization 32 4.4.2 Integrated autocorrelation time 33 4.4.3 Run-time 33 4.4.4 Topological phase transition 35 5 Conclusion & Outlook 36 References 37 2 SciPost Physics Codebases Submission 1 Introduction The toric code is one of the most fundamental and most-studied models in modern condensed matter physics. It was first written down by Kitaev [1] and is the simplest example of a model hosting a topological phase (a gapped Z2 quantum spin liquid) and anyonic excitations. The toric code is also the foundational model for error-correcting codes [2, 3] and has deep connections to the Ising gauge theory [4]. The toric code can be extended with fields which, when strong enough, destroy the topological order. This model is sign-problem-free, thus making quantum Monte Carlo the method of choice. Wu, Deng, and Prokof'ev developed a continuous-time quantum Monte Carlo algorithm [5]. ParaToric implements and extends this algorithm with new updates which enable ergodicity at large temperatures and at zero off-diagonal field, thus significantly improving the applicability of the algorithm. ParaToric implements a wide range of lattices, boundary conditions, and observables. It is also possible to extend ParaToric with new interactions, observables, and lattices. We provide documented interfaces in C, C++, and Python as well as command-line interfaces, making the integration of ParaToric into other projects and programming languages straightforward. ParaToric will save simulation results to HDF5 files and snapshots to GraphML files (XMLbased), with a focus on interoperability with other packages. ParaToric comes with an MIT license. 2 The toric code in a parallel field 2.1 Hamiltonian ParaToric implements and extends the continuous-time quantum Monte Carlo (QMC) algorithm by Wu, Deng, and Prokof'ev [5] to simulate the toric code in a parallel field (also called perturbed toric code or extended toric code) ˆH = -μ X v ˆAv -J X p ˆBp -h X l ˆσx l -λ X l ˆσz l , (1) where J,λ > 0 in the ˆ σx-basis and μ,h > 0 in the ˆ σz-basis (otherwise the model has a signproblem). ˆσx l and ˆσz l are Pauli matrices defined on the links of the underlying lattice. The star term ˆAv contains all links adjacent to lattice site v, the plaquette term ˆBp contains all links that belong to the same elementary plaquette p of the underlying lattice. The temperature T = 1/β is finite. For readers interested in extending the code, we note that it is relatively straightforward to add interactions that are diagonal in the chosen basis, such as (long-range) Ising interactions. Off-diagonal interactions require a more careful review and extension of the Monte Carlo updates to ensure ergodicity. However, diagonal interactions can also lead to sampling problems, especially when they introduce frustration. 2.2 Lattice geometries We implement the square, honeycomb, triangular, and cubic lattices, see Fig. 1. On the cubic lattice, the plaquettes contain the four links of cube faces, not the twelve links of the cube 3 SciPost Physics Codebases Submission (a) (b) ̂Av ̂Bp ̂σx ̂σz ̂Av ̂Bp ̂σz ̂σx (c) ̂σz ̂σx ̂Av ̂Bp (d) ̂Av ̂Bp Figure 1: Implemented lattices. We implement the extended toric code (1) on the square (a), honeycomb (b), triangular (c), and cubic lattices (d). For each lattice, we show the star (ˆAv) and plaquette (ˆBp) terms. The cubic lattice importantly features star interactions of six links and plaquette interactions of four links on the faces of cubes. (that model has a different m-anyon structure). We implement open and periodic boundaries, respectively. New lattices can be added in src/lattice/lattice.cpp. 2.3 Observables Here we list all observables that ParaToric implements for the extended toric code. Custom observables can be added in src/mcmc/extended_toric_code_qmc.hpp. For each observable ˆO, we calculate the expectation value 〈ˆO〉and the Binder ratio UO = 〈ˆO4〉 〈ˆO2〉2 with error bars obtained from bootstrapping (see below), respectively. anyon_count The number of e-anyons ( ˆ σx-basis) or m-anyons ( ˆ σz-basis) in the system. anyon_density ˆ σx-basis: The number of e-anyons divided by the number of lattice sites. ˆ σz-basis: The number of m-anyons divided by the number of plaquettes. delta The difference of the star and plaquette expectation values: ∆= 〈ˆAv〉-〈ˆBp〉. energy The total energy E = 〈ˆH〉. energy_h The electric field term Eh = 〈-h P l ˆσx l 〉. energy_lmbda The gauge field term Eλ = 〈-λ P l ˆσz l 〉. We write λ as lmbda because some programming languages feature lambda as a keyword. 4 SciPost Physics Codebases Submission energy_J The plaquette term EJ = 〈-J P p ˆBp〉. energy_mu The star term Eμ = 〈-μ P v ˆAv〉. fredenhagen_marcu The equal-time Fredenhagen-Marcu loop operator [6-8]: Ox/z FM = lim L→∞ 〈 Q l∈Cx/z 1/2 ˆσx/z l 〉 Ç |〈 Q l∈Cx/z ˆσx/z l 〉| , (2) C x/z 1/2 is half, C x/z is a full Wilson loop in the ˆ σx-basis ('t Hooft loop in the ˆ σz-basis). The loop is automatically constructed for all supported lattices, and the perimeter scales with O(L), where L is the linear system size. When probing perimeter/area laws, the user should change L. We currently do not support off-diagonal loop operators, e.g., measuring products of ˆσx-operators in the ˆσz-basis. largest_cluster The largest connected cluster of neighboring bonds with ˆσx = -1 (ˆσz = -1) in the ˆσx-basis (ˆσz-basis). This observable is used to calculate the percolation strength, see [9]. percolation_probability Measures the bond percolation probability, i.e. if we can wind around the system while only traversing bonds with ˆσx = -1 (ˆσz = -1) in the ˆσx-basis (ˆσz-basis). Formally, it is the expectation value 〈ˆΠx/z〉of the projector ˆΠx/z = X W(j)̸=0 |{ˆσx/z}j〉〈{ˆσx/z}j|, (3) over all possible configurations {ˆσx/z}j with non-zero winding number W(j) of connected link clusters of neighboring ˆσx/z = -1. These clusters are called percolating clusters. For details, see [9-11]. percolation_strength If a snapshot does not have a percolating cluster, the percolation strength is 0. If a snapshot has a percolating cluster, the percolation strength is defined as the result of largest_cluster divided by the total number of links in the system. For details, see [9,11]. plaquette_percolation_probability Similar to the percolation probability of bonds. Two plaquettes are in the same cluster if they share a link l with ˆ τx l = -1. For details, see [11]. plaquette_z The plaquette expectation value 〈ˆBp〉. sigma_x The electric field expectation value 〈ˆσx〉. sigma_x_susceptibility The static susceptibility χ x = 1 N Z β 0 〈ˆσx(0) ˆσx(τ)〉c dτ, (4) where N is the total number of links (qubits) and the integral is over the imaginary time τ1. Importantly, χ x can be calculated both in the ˆσx- and the ˆσz-basis. 1The dynamical (fidelity) susceptibility is a trivial extension and contains an extra τ dependency in the integral. 5 SciPost Physics Codebases Submission sigma_z The gauge field expectation value 〈ˆσz〉. sigma_z_susceptibility The static susceptibility χz = 1 N Z β 0 〈ˆσz(0) ˆσz(τ)〉c dτ, (5) where N is the total number of links (qubits) and the integral is over the imaginary time τ. Importantly, χz can be calculated both in the ˆσx- and the ˆσz-basis. staggered_imaginary_times Order parameter from [5]. It is defined as Ox/z SIT = 1 β (τk 1 -0) -(τk 2 -τk 1) + ... + (-1)N(k)-1 (τk N(k) -τk N(k)-1) + (-1)N(k)(β -τk N(k)) , (6) where τk n is the imaginary time of the n-th tuple spin flip of type k. k is a plaquette p (star s) of links in the ˆσx-basis (ˆσz-basis). This order parameter can neither be evaluated from snapshots, nor from any other method that does not have access to imaginary time. star_x The star expectation value 〈ˆAv〉. string_number The total number of links with ˆσx = -1 in the ˆσx-basis (ˆσz = -1 in the ˆσz-basis). 3 Installation & interfaces There are five ways to use paratoric, directly from within code (C, C++, Python) or via the command-line (C++, Python). All interfaces require compiling C++ code. We tested the compilation with GCC 15 and Clang 20. All interfaces implement three functionalities. Thermalization simulations are used to benchmark the thermalization process of a Markov chain and are primarily a diagnostic tool. Regular sampling routines are used for generating snapshots and measuring observables, e.g., in the context of continuous phase transitions. Hysteresis routines are a variant of the regular sampling routines where not one but an array of Hamiltonian parameters is provided and only one Markov chain is used for all parameters. The order of the Hamiltonian parameters in the input array matters: The last state of the previous parameter is used as an initial state for the thermalization phase of the next parameters. This simulation type should primarily be used when mapping out hysteresis curves in the vicinity of first-order phase transitions, hence the name. Since the hysteresis simulation returns the values of not one but many parameter sets, the output types are generally different from the regular sampling. It is also much slower than regular sampling, because the simulation for different parameters can in general not be parallelized. 3.1 C++ interface The C++ interface enables users to use a ParaToric public header from within another C++ project. 6 SciPost Physics Codebases Submission 3.1.1 Build & Installation The core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions may work, but were not tested). To compile it, run: cmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\ -DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\ -DPARATORIC_BUILD_TESTS=ON cmake --build build -jN ctest --test-dir build -jN --output-on-failure cmake --install build Replace N with the number of cores to use, e.g. -j4 for 4 cores. • -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you're a developer. • -DCMAKE_INSTALL_PREFIX. By default, executables install to {CMAKE_INSTALL_BINDIR}/, headers to {CMAKE_SOURCE_DIR}/ {CMAKE_SOURCE_DIR}/bin/; this directory always contains the paratoric executable. To install into a custom directory, pass it via -DCM AKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo ry/. • -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json for tooling. • -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The core itself does not need MPI. • -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and Clang. • -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires a CPU which supports AVX2. • -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recommended). CMake usage (installed package) cmake_minimum_required(VERSION 3.23) project(my_qmc_app CXX) find_package(paratoric CONFIG REQUIRED) # provides paratoric::core add_executable(myapp main.cpp) target_link_libraries(myapp PRIVATE paratoric::core) CMake usage (as subdirectory) If the core lives in deps/paratoric, add it and link to the same target: add_subdirectory(deps/paratoric) add_executable(myapp main.cpp) target_link_libraries(myapp PRIVATE paratoric::core) 7 SciPost Physics Codebases Submission 3.1.2 Public class ExtendedToricCode The interface class ExtendedToricCode lives in the public header #include . All symbols are in the paratoric namespace. All methods are static, take a single Config object and return a Result object. The required fields in config are documented for each method within the docstrings. Result ExtendedToricCode::get_thermalization(Config config) Run thermalization only. Required fields: lat_spec.{basis,lattice_type,system_size,beta ,boundaries,default_spin}, param_spec.{mu,h,J,lmbda,h_therm,lmbda_therm }, sim_spec.{N_thermalization,N_resamples,custom_therm,observables,seed }, out_spec.{path_out,save_snapshots}. Result ExtendedToricCode::get_sample(Config config) Run a production measurement pass. Returns the observables selected in config. Required fields: lat_spec.{ba sis,lattice_type,system_size,beta,boundaries,default_spin}, param_spec. {mu,h,J,lmbda}, sim_spec.{N_samples,N_thermalization,N_between_samples, N_resamples,observables,seed}, out_spec.{path_out,save_snapshots}. Result ExtendedToricCode::get_hysteresis(Config config) Perform a hysteresis sweep, where the last state of the previous parameter is used as the initial state of the following parameter in h_hys & lmbda_hys. Required fields: lat_spec.{basis,lattice _type,system_size,beta,boundaries,default_spin}, param_spec.{mu,J,h_hys ,lmbda_hys}, sim_spec.{N_samples,N_thermalization,N_between_samples,N_r esamples,observables,seed}, out_spec.{paths_out,save_snapshots}. 3.1.3 Configuration type The struct Config (declared in ) contains multiple nested specifications. Top-level configuration: Config Field Type Purpose sim_spec SimSpec Simulation / MC controls (backend-consumed). param_spec ParamSpec Model couplings / parameters (backend-consumed). lat_spec LatSpec Lattice geometry and basis. out_spec OutSpec Output folders and snapshot toggles. 8 SciPost Physics Codebases Submission Simulation specification (config.sim_spec) Field Type Meaning / Defaults N_samples int Number of recorded snapshots. Default 1000. N_thermalization int Number of warmup steps before sampling. Typically O(Ld), where L is the system size and d is the dimensionality. Default 10000. N_between_samples int Steps between consecutive snapshots. Higher value decreases autocorrelation and improves error bars. Typically O(Ld), where L is the system size and d is the dimensionality. Default 1000. N_resamples int Bootstrap resamples for errors. Default 1000. custom_therm bool Use custom thermalization schedule. Default fa lse. seed int PRNG seed. 0 means \"random seed.\" Default 0. observables vector Names of observables to record each snapshot. For options, see Sec. 2.3. Parameter specification (config.param_spec) Field Type Meaning / Defaults mu double Star term coefficient. Default 1.0. h double Electric field term. Default 0.0. J double Plaquette term. Default 1.0. lmbda double Gauge-field term. Default 0.0. h_therm double Thermalization value for h when using custom schedules. Default NaN (unused). lmbda_therm double Thermalization value for lmbda when using custom schedules. Default NaN (unused). h_hys vector Sweep values of h for hysteresis runs. Default empty. Length must match lmbda_hys. lmbda_hys vector Sweep values of lmbda for hysteresis runs. Default empty. Length must match h_hys. Lattice specification (config.lat_spec) Field Type Meaning / Valid values basis char Spin eigenbasis for the simulation. Must be 'x' or 'z'. lattice_type string The lattice (\"square\", \"triangular\", \"hone ycomb\" or \"cubic\"). system_size int Linear system size (per dimension). beta double Inverse temperature β > 0. boundaries string Boundary condition: \"periodic\" or \"open\". default_spin int Initial link spin, must be +1 or -1. 9 SciPost Physics Codebases Submission Output specification (config.out_spec) Field Type Meaning path_out string Primary output folder name. paths_out vector Hysteresis subfolder names. Length must match h_hys. save_snapshots bool Save snapshots toggle. Default false. 10 SciPost Physics Codebases Submission 3.1.4 Return type Field C++ Type Meaning series vector ,double>>> Time series of all requested observables, thermalization is excluded (except for thermalization simulation). Outer index = observable, inner index = time point. acc_ratio vector Time series of Monte Carlo acceptance ratios. mean vector Bootstrap observable means. mean_std vector Bootstrap standard errors of the mean. binder vector Bootstrap binder ratios. binder_std vector Bootstrap standard errors of the binder ratios. tau_int vector Estimated integrated autocorrelation times. series_hys vector ,double>>>> Hysteresis time series of all requested observables, thermalization is excluded (except for thermalization simulation). Outer vector = hysteresis parameters (order as in h_hys,lm bda_hys), middle vector = observables (order as in observa bles), inner vector = time series. mean_hys vector > Hysteresis bootstrap observable means. Outer vector = hysteresis parameters (order as in h_hys,lm bda_hys), inner vector = observables (order as in observa bles). mean_std_hys vector > Hysteresis bootstrap standard errors of the mean. Indices as above. binder_hys vector > Hysteresis bootstrap binder ratios. Indices as above. binder_std_hys vector > Hysteresis bootstrap standard errors of the binder ratios. Indices as above. tau_int_hys vector > Hysteresis estimated integrated autocorrelation times. Indices as above. 3.1.5 C++ usage examples Listing 1: C++ API - Minimal call 11 SciPost Physics Codebases Submission // C++23 #include #include #include #include #include #include int main() { using namespace paratoric; Config cfg{}; // ---- lattice sub-config (required) ---- cfg.lat_spec.basis = 'z'; // or 'x' cfg.lat_spec.lattice_type = \"square\"; // or \"cubic\", \"honeycomb\", ... cfg.lat_spec.system_size = 16; cfg.lat_spec.beta = 8.0; cfg.lat_spec.boundaries = \"periodic\"; // or \"open\" cfg.lat_spec.default_spin = 1; // ---- Hamiltonian parameters ---- cfg.param_spec.mu = 1.0; // star term cfg.param_spec.J = 1.0; // plaquette term cfg.param_spec.h = 0.20; // electric field term cfg.param_spec.lmbda = 0.00; // gauge-field term // Optional thermalization schedule values (used if custom_therm = true) cfg.param_spec.h_therm = std::numeric_limits ::quiet_NaN(); cfg.param_spec.lmbda_therm = std::numeric_limits ::quiet_NaN(); // (Optional) Hysteresis sweep grids - only read by get_hysteresis(...) cfg.param_spec.h_hys = {}; // e.g. {0.0, 0.1, 0.2, 0.3, 0.2, 0.1, 0.0} cfg.param_spec.lmbda_hys = {}; // e.g. {0.0, 0.1, 0.2, 0.3, 0.2, 0.1, 0.0} // ---- Simulation (MC) controls ---- cfg.sim_spec.N_samples = 0; // 0 => thermalization-only cfg.sim_spec.N_thermalization = 5000; // warmup steps cfg.sim_spec.N_between_samples = 10; // thinning between snapshots cfg.sim_spec.N_resamples = 1000; // bootstrap cfg.sim_spec.custom_therm = false; // set true to use *_therm values cfg.sim_spec.seed = 12345; // 0 => random seed // Observables to record each snapshot (backend-recognized names) cfg.sim_spec.observables = { \"energy\", // total energy \"plaquette_z\", // plaquette energy \"anyon_count\", // number of anyons (x-basis: e-anyons, z-basis: m-anyons ) \"fredenhagen_marcu\" // example: Wilson/'t Hooft loop proxy }; // ---- Output / I/O policy ---- cfg.out_spec.path_out = \"runs/sample\"; // single-run output dir cfg.out_spec.paths_out = {}; // filled only for hysteresis cfg.out_spec.save_snapshots = false; // set true to dump every snapshot 12 SciPost Physics Codebases Submission cfg.out_spec.full_time_series = true; // save full time series (FCS) // 1) Check thermalization Result warmup = ExtendedToricCode::get_thermalization(cfg); std::print(\"Thermalization␣series:␣{}\", warmup.series); // 2) Production sample (set N_samples > 0 and call get_sample) cfg.sim_spec.N_samples = 2000; Result out = ExtendedToricCode::get_sample(cfg); std::print(\"Production␣autocorrelations:␣{}\", out.tau_int); return 0; } 3.2 C++ command-line interface ParaToric ships a C++ command-line interface {CMAKE_INS TALL_BINDIR}/paratoric that orchestrates C++ backends, runs sweeps, and writes HDF5 (observables) and XML (snapshots) outputs. 3.2.1 Build & Installation The command-line interface requires HDF5 ≥1.14.3 (older HDF5 versions may work, but were not tested). The core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions may work, but were not tested). To compile it, run: cmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\ -DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\ -DPARATORIC_BUILD_TESTS=ON -DPARATORIC_BUILD_CLI=ON cmake --build build -jN ctest --test-dir build -jN --output-on-failure cmake --install build Replace N with the number of cores to use, e.g. -j4 for 4 cores. • -DCMAKE_BUILD_TYPE=Release. Only set to Debug if you're a developer. • -DCMAKE_INSTALL_PREFIX. By default, executables install to {CMAKE_INSTALL_BINDIR}/, headers to {CMAKE_SOURCE_DIR}/ {CMAKE_SOURCE_DIR}/bin/; this directory always contains the paratoric executable. To install into a custom directory, pass it via -DCM AKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo ry/. • -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json for tooling. • -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The core itself does not need MPI. • -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and Clang. 13 SciPost Physics Codebases Submission • -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires a CPU which supports AVX2. • -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recommended). • -DPARATORIC_BUILD_CLI=PROJECT_IS_TOP_LEVEL Required for both the C++ and Python command line interface. 14 SciPost Physics Codebases Submission Global options Long flag Short Type Description --simulation -sim string Simulation mode: etc_sample, etc_hysteresis, etc_therm alization. --N_samples -Ns int Number of recorded samples. --N_thermalization -Nth int Thermalization (warmup) steps. --N_between_samples -Nbs int Steps between samples (thinning). --beta -bet double Inverse temperature β = 1/T. --mu_constant -muc double Star-term coupling μ. --J_constant -Jc double Plaquette coupling J. --h_constant -hc double Field h. --lmbda_constant -lmbdac double Field λ. --h_constant_therm -hct double Thermalization value for h (used if custom therm). --lmbda_constant_therm -lmbdact double Thermalization value for λ. --h_hysteresis -hhys list Hysteresis schedule for h (spaceseparated). Length must match l mbdahys. --lmbda_hysteresis -lmbdahys list Hysteresis schedule for λ. Length must match hhys. --N_resamples -Nr int Bootstrap resamples (error bars). --custom_therm -cth bool Use thermalization values (0/1). --observables -obs list Measured observables (spaceseparated). --seed -s int PRNG seed; 0 means random seed. --basis -bas char Spin basis: 'x' or 'z'. --lattice_type -lat string Lattice type (e.g. square, cubi c, ...). --system_size -L int Linear lattice size (per dimension). --boundaries -bound string periodic or open. --default_spin -dsp int Initial link spin (+1 or -1). --output_directory -outdir path Output directory path. --folder_name -fn string Subfolder (of output directory) name for single run. --folder_names -fns list Subfolders (of output directory) for hysteresis steps. Length must match lmbdahys. --snapshots -snap bool Save snapshots into specified subfolders of output directory. --full_time_series -fts bool Save full time series toggle. --process_index -procid int Process identifier (logging/debug). 3.2.2 etc_sample Runs a production measurement pass with the supplied configuration. Listing 2: Example usage 15 SciPost Physics Codebases Submission ./paratoric -sim etc_sample -Ns 2000 -Nth 5000 -Nbs 10 -Nr 1000 -bet 16.0 - muc 1 -Jc 1 -hc 0.2 -lmbdac 0.0 -obs energy plaquette_z anyon_count -bas z -lat square -L 16 -bound periodic -dsp 1 -outdir ./runs/sample -snap =0 -fcs=1 3.2.3 etc_hysteresis Runs a parameter sweep where the last state of step i initializes step i+1. Provide --h_hy steresis and --lmbda_hysteresis as space-separated lists, and --folder_names for per-step outputs. Listing 3: Example usage ./paratoric -sim etc_hysteresis -Ns 1000 -Nth 2000 -Nbs 50 -Nr 500 -bet 12.0 -muc 1 -Jc 1 -lmbdahys 0.2 0.2 0.2 0.2 0.2 0.2 0.2 -hhys 0.0 0.1 0.2 0.3 0.2 0.1 0.0 -obs energy fredenhagen_marcu -bas x -lat square -L 12 -bound periodic -dsp 1 -outdir ./runs/hys -fns step0 step1 step2 step3 step4 step5 step6 3.2.4 etc_thermalization Performs thermalization only (no production sampling). Listing 4: Example usage ./paratoric -sim etc_thermalization -Ns 0 -Nth 5000 -Nbs 10 -Nr 500 -bet 10.0 -muc 1 -Jc 1 -hc 0.3 -lmbdac 0.1 -hct 0.4 -lmbdact 0.2 -cth 1 -obs energy anyon_density -bas z -lat square -L 10 -bound open -dsp 1 -outdir ./runs/therm -snap=1 3.2.5 HDF5 structure The output HDF5 has the structure simulation/results/acc_ratio for an array of the acceptance weights (only for thermalization), simulation/results/observable_name/ series for the time series (if it was enabled), and simulation/results/observable_ name/{mean,mean_error,binder,binder_error,autocorrelation_time}. For the regular sampling, mean, mean_error, binder, binder_error and autocorrelation_ time contain doubles. For the hysteresis, they contain an array of values for the hysteresis parameters (in the order of h_hysteresis and lmbda_hysteresis). 3.3 C interface The C interface enables users to use a ParaToric public header from within another C project or another programming language that supports a C-style interface. The C interface exposes a stable ABI. It mirrors the C++ interface. Include the public header #include . All functions return ptc_status_t. 3.3.1 Build & Installation The code can be compiled in exactly the same fashion as for the C++ interface. 16 SciPost Physics Codebases Submission CMake usage (installed package). cmake_minimum_required(VERSION 3.23) project(my_qmc_c C) find_package(paratoric CONFIG REQUIRED) # provides paratoric::core add_executable(cdemo main.c) target_link_libraries(cdemo PRIVATE paratoric::core) CMake usage (as subdirectory). add_subdirectory(deps/paratoric) add_executable(cdemo main.c) target_link_libraries(cdemo PRIVATE paratoric::core) 3.3.2 Status & error handling Name C type/values Meaning ptc_status_t {PTC_STATUS_OK=0, PTC_STATU S_INVALID_ARGUMENT=1, PTC_S TATUS_RUNTIME_ERROR=2, PTC_ STATUS_NO_MEMORY=3, PTC_STA TUS_INTERNAL_ERROR=4} Return code of every API call. ptc_last_error() const char* Thread-local error string. Valid until next call. 3.3.3 Opaque handle Create and destroy the interface instance. Use ptc_create(ptc_handle_t **out) and p tc_destroy(ptc_handle_t *h). 3.3.4 Configuration type Top-level ptc_config_t aggregates four nested specs. Field names mirror the C++ Config. Top-level configuration: ptc_config_t Field Type Purpose sim ptc_sim_spec_t Monte Carlo parameters. params ptc_param_spec_t Hamiltonian parameters. lat ptc_lat_spec_t Lattice parameters. out ptc_out_spec_t Output paths and snapshot toggle. 17 SciPost Physics Codebases Submission Simulation specification (config.sim) Field Type Meaning N_samples int Number of snapshots. N_thermalization int Thermalization steps. N_between_samples int Thinning between snapshots. N_resamples int Bootstrap resamples. custom_therm bool Custom thermalization schedule. seed int PRNG seed (0 = random). observables const char* const* Array of observable names (nullable). N_observables size_t Length of observables. Parameter specification (config.params) Field Type Meaning mu,h,J,lmbda double Couplings (star, electric, plaquette, gauge). h_therm double Thermalization value for h if custom_therm=true. lmbda_therm double Thermalization value for lmbda if custom_therm=t rue. h_hys const double* Hysteresis schedule for h (nullable). h_hys_len size_t Length of h_hys. Must match lmbda_hys_len. lmbda_hys const double* Hysteresis schedule for lmbda (nullable). lmbda_hys_len size_t Length of lmbda_hys. Must match h_hys_len. Lattice specification (config.lat) Field Type Meaning / Valid values basis char Spin basis: 'x' or 'z'. lattice_type const char* E.g. \"triangular\", \"square\", ... system_size int Linear system size per dimension. beta double Inverse temperature. boundaries const char* \"periodic\" or \"open\". default_spin int Initial link spin: +1 or -1. Output specification (config.out) Field Type Meaning path_out const char* Single output directory (nullable). paths_out const char* const* Output directories for hysteresis steps (nullable). N_paths_out size_t Length of paths_out. Must match h_hys_l en and lmbda_hys_len. save_snapshots bool Toggle snapshot dumping. 3.3.5 Return type All outputs are owned by the caller. Call ptc_result_destroy(&r) to free and zero. 18 SciPost Physics Codebases Submission Field C type Meaning series ptc_series_t Time series (real/complex) of all requested observables, thermalization is excluded (except for thermalization simulation). Outer index = observable, inner index = time point. acc_ratio ptc_dvec_t MC acceptance ratios for each update (thermalization). mean, mean_std ptc_dvec_t Bootstrap mean and standard error (order as in observables). binder, binder_std ptc_dvec_t Binder ratios and standard error (order as in observables). tau_int ptc_dvec_t Integrated autocorrelation time (order as in observables). series_hys ptc_series_block s_t Hysteresis time series. Outer index = hysteresis parameters (order as in h_hys,lmbda_hys), middle index = observables (order as in ob servables), inner vector = time series. mean_hys, mean_std_hys, binder _hys, binder_std_hys, tau_int_ hys ptc_dmat_t Outer index = hysteresis parameters (order as in h_hys,lmbda_h ys), inner index = observables (order as in observables). 3.3.6 Procedures (mirror the C++ API) All fill a ptc_result_t *out on success. Return PTC_STATUS_OK on success. ptc_get_thermalization(ptc_handle_t *h, const ptc_config_t *cfg, ptc_r esult_t *out) Run thermalization only. Required fields: cfg->lat.{basis,lattice _type,system_size,beta,boundaries,default_spin}, cfg->params.{mu,h,J,lm bda}, cfg->sim.{N_thermalization,N_resamples,observables,N_observables, seed}, cfg->out.{path_out,save_snapshots}. ptc_get_sample(ptc_handle_t *h, const ptc_config_t *cfg, ptc_result_t *out) Run a production measurement pass. Required fields: cfg->lat.{basis,latti ce_type,system_size,beta,boundaries,default_spin}, cfg->params.{mu,h,J, lmbda,h_therm,lmbda_therm}, cfg->sim.{N_samples,N_thermalization,N_betw een_samples,N_resamples,custom_therm,observables,N_observables,seed}, c fg->out.{path_out,save_snapshots}. ptc_get_hysteresis(ptc_handle_t *h, const ptc_config_t *cfg, ptc_resul t_t *out) Run a hysteresis sweep over h_hys and/or lmbda_hys. The last state of step i initializes step i+1. Required fields: cfg->lat.{basis,lattice_type,system_size ,beta,boundaries,default_spin}, cfg->params.{mu,h_hys,h_hys_len,J,lmbda _hys,lmbda_hys_len}, cfg->sim.{N_samples,N_thermalization,N_between_sam ples,N_resamples,observables,N_observables,seed}, cfg->out.{paths_out,N _paths_out,save_snapshots}. 3.3.7 C usage example 19 SciPost Physics Codebases Submission Listing 5: C API - Minimal call #include #include #include int main(void) { ptc_handle_t* h = NULL; if (ptc_create(&h) != PTC_STATUS_OK) { puts(\"create␣failed\"); return 1; } ptc_lat_spec_t lat = { .basis = 'z', .lattice_type = \"square\", .system_size = 16, .beta = 8.0, .boundaries = \"periodic\", .default_spin = 1 }; ptc_param_spec_t ps = { .mu = 1.0, .h = 0.2, .J = 1.0, .lmbda = 0.0, .h_therm = NAN, .lmbda_therm = NAN, .h_hys = NULL, .h_hys_len = 0, .lmbda_hys = NULL, .lmbda_hys_len = 0 }; const char* obs[] = {\"energy\",\"plaquette_z\",\"anyon_count\"}; ptc_sim_spec_t sim = { .N_samples = 0, /* thermalization-only initially */ .N_thermalization = 5000, .N_between_samples = 10, .N_resamples = 1000, .custom_therm = false, .seed = 12345, .observables = obs, .N_observables = sizeof(obs)/sizeof(obs[0]) }; ptc_out_spec_t outspec = { .path_out = \"runs/sample\", .paths_out = NULL, .N_paths_out = 0, .save_snapshots = false }; ptc_config_t cfg = { .sim = sim, .params = ps, .lat = lat, .out = outspec }; ptc_result_t warm = {0}; ptc_status_t st = ptc_get_thermalization(h, &cfg, &warm); if (st != PTC_STATUS_OK) { puts(ptc_last_error()); ptc_destroy(h); return 2; } ptc_result_destroy(&warm); cfg.sim.N_samples = 2000; ptc_result_t res = {0}; st = ptc_get_sample(h, &cfg, &res); if (st != PTC_STATUS_OK) { puts(ptc_last_error()); ptc_destroy(h); return 3; } 20 SciPost Physics Codebases Submission // use res.mean, res.tau_int, ... ptc_result_destroy(&res); ptc_destroy(h); return 0; } Memory rules. You own all buffers in ptc_result_t. Call ptc_result_destroy once per successful call. 3.4 Python bindings ParaToric exposes a compiled Python extension module _paratoric with a submodule ext ended_toric_code. The bindings convert C++ vectors into NumPy arrays and release the global interpreter lock (GIL) while running the C++ kernels. 3.4.1 Build & Installation The core requires C++23, CMake ≥3.23, and Boost ≥1.87 (older Boost versions may work, but were not tested). The Python bindings require a Python installation with Numpy and PyBind11 (tested with version 3.0.1). Pybind11 is included as a git submodule (you need to pull it!). To compile the Python bindings, run: cmake -S . -B build -DCMAKE_BUILD_TYPE=Release \\ -DPARATORIC_ENABLE_NATIVE_OPT=ON -DPARATORIC_LINK_MPI=OFF \\ -DPARATORIC_BUILD_TESTS=ON -DPARATORIC_BUILD_PYBIND=ON \\ -DPython3_EXECUTABLE=\" {CMAKE_SOURCE_DI R}/ {CMAKE_INSTALL_INCLUDEDIR}/p aratoric, and static libraries to {CMAKE_INSTALL_LIBD IR}/. The Python scripts expect {CMAKE_SOURCE_DI R}/ {CMAKE_INSTALL_INCLUDEDIR}/p aratoric, and static libraries to {CMAKE_INSTALL_LIBD IR}/. The Python scripts expect ${CMAKE_SOURCE_DIR}/bin/; this directory always contains the paratoric executable. To install into a custom directory, pass it via -DCM AKE_INSTALL_PREFIX, e.g. -DCMAKE_INSTALL_PREFIX=/your/custom/directo ry/. • -DPARATORIC_EXPORT_COMPILE_COMMANDS=ON. Export compile_commands.json for tooling. • -DPARATORIC_LINK_MPI=OFF. Link the core to MPI, required on some clusters. The core itself does not need MPI. • -DPARATORIC_ENABLE_NATIVE_OPT=OFF. Turn on -march=native on GCC and Clang. • -DPARATORIC_ENABLE_AVX2=OFF. Enable AVX2 (Haswell New Instructions). Requires a CPU which supports AVX2. • -DPARATORIC_BUILD_TESTS=PROJECT_IS_TOP_LEVEL. Compile the tests (recommended). • -DPARATORIC_BUILD_CLI=PROJECT_IS_TOP_LEVEL Required for both the C++ and Python command line interface. 25 SciPost Physics Codebases Submission General options Long flag Short Description --help -h Show help and exit. --simulation -sim Simulation type selector. --N_thermalization -Nth Thermalization steps (proposed updates). --N_samples -Ns Number of samples/snapshots. --N_between_steps -Nbs Steps between successive samples (thinning). --N_resamples -Nr Bootstrap resamples. --custom_therm -cth Use thermalization values for h,λ (0 or 1). --observables -obs Space-separated list, e.g. fredenhagen_marcu percolation_probability energy. --seed -seed PRNG seed; 0 means random seed. --mu_constant -muc Value of μ. --J_constant -Jc Value of J. --h_constant -hc Value of h. --h_constant_therm -hct Thermalization value of h. --lmbda_constant -lmbdac Value of λ. --lmbda_constant_therm -lmbdact Thermalization value of λ. --output_directory -outdir Output directory. --snapshots -snap Save snapshots toggle (0/1). --full_time_series -fts Save full time series toggle (0/1). --processes -proc Logical CPU count for Python multiprocessing. 0 means all available cores. Negative numbers -x mean use all cores minus x. Default is -4. Lattice-specific options Long flag Short Description --help -h Show help and exit. --basis -bas Spin basis: x or z. --lattice_type -lat square, cubic, triangular, honeycomb, ... --system_size -L Linear size; in 2D, 30 yields a 30×30 lattice (unit cells). --temperature -T Temperature T = 1/β > 0. --boundaries -bound periodic or open. --default_spin -dsp Initial edge spin: 1 or -1. The command line interface offers several sweep modes. All are embarrassingly parallel; set --processes close to the number of steps when possible. 3.5.2 T-sweep Runs T_steps independent Markov chains for evenly spaced temperatures in [T_lower, T_upper] and plots all requested observables. Listing 7: Example usage 26 SciPost Physics Codebases Submission python3 ./python/cli/paratoric.py -sim etc_T_sweep -Ns 1000 -muc 1 -Nth 2000 -Nbs 100 -Tl 0.5 -Tu 5 -Ts 30 -hc 0.1 -Jc 1 -lmbdac 0.1 -Nr 1000 - obs percolation_strength percolation_probability plaquette_percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 - bas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_T_sweep. --T_lower -Tl Lower bound of T. --T_upper -Tu Upper bound of T. --T_steps -Ts Number of temperatures between bounds. 3.5.3 h-sweep Runs h_steps independent chains in parallel for evenly spaced h in [h_lower, h_upper]. Listing 8: Example usage python3 ./python/cli/paratoric.py -sim etc_h_sweep -Ns 1000 -muc 1 -Nth 2000 -Nbs 100 -hl 0.1 -hu 0.5 -hs 8 -T 0.03 -Jc 1 -lmbdac 0.2 -Nr 1000 - obs percolation_strength percolation_probability plaquette_percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 - bas x -lat square -L 6 -bound periodic -dsp 1 -outdir /path/to/out Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_h_sweep. --h_lower -hl Lower bound of h. --h_upper -hu Upper bound of h. --h_steps -hs Number of field steps between bounds. 3.5.4 λ-sweep Runs lmbda_steps independent chains in parallel for evenly spaced λ in [λ_lower, λ_upper]. Listing 9: Example usage python3 ./python/cli/paratoric.py -sim etc_lmbda_sweep -Ns 1000 -muc 1 -Nth 2000 -Nbs 100 -lmbdal 0.01 -lmbdau 1.0 -lmbdas 15 -T 0.1 -hc 0.3 -Jc 1 -Nr 1000 -obs percolation_strength percolation_probability plaquette_percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 - bas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out 27 SciPost Physics Codebases Submission Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_lmbda_sweep. --lmbda_lower -lmbdal Lower bound of λ. --lmbda_upper -lmbdau Upper bound of λ. --lmbda_steps -lmbdas Number of field steps between bounds. 3.5.5 ◦-sweep Runs Theta_steps independent chains in parallel along a circle in (λ,h) centered at (lmbda _constant,h_constant) with radius radius, for angles Θ ∈[Θ_lower, Θ_upper] (angles measured anti-clockwise from the λ-axis). Listing 10: Example usage python3 ./python/cli/paratoric.py -sim etc_circle_sweep -Ns 1000 -muc 1 - Nth 2000 -Nbs 100 -lmbdac 0.4 -rad 0.3 -Thl 0 -Thu 3.141 -Ths 15 -T 0.1 -hc 0.4 -Jc 1 -Nr 1000 -obs percolation_strength percolation_probability plaquette_percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 - bas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_circle_sweep. --lmbda_constant -lmbdac Circle center in λ. --h_constant -hc Circle center in h. --radius -rad Circle radius. --Theta_lower -Thl Lower bound of Θ. --Theta_upper -Thu Upper bound of Θ. --Theta_steps -Ths Number of angles between bounds. 3.5.6 Hysteresis-sweep Use the hysteresis schedule specified in hhys and lmbdahys. This mode will run two Markov chains, one in the original parameter order specified in hhys and lmbdahys, and one with a reversed parameter order, i.e., it calculates both branches of the hysteresis loop. Listing 11: Example usage python3 ./python/cli/paratoric.py -sim etc_hysteresis -Nbs 5000 -Ns 10000 - muc 1 -Nth 20000 -hhys 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -T 0.1 -Jc -1 -lmbdahys 0.5 0.525 0.55 0.575 0.6 0.625 0.65 0.675 0.7 0.725 0.75 -Nr 1000 -obs plaquette_percolation_probability percolation_strength percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 -bas z -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out 28 SciPost Physics Codebases Submission Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_hysteresis. --lmbda_hysteresis -lmbdahys Hysteresis schedule for λ. Length must match hh ys. --h_hysteresis -hhys Hysteresis schedule for h. Length must match lm bdahys. 3.5.7 Thermalization Runs repetitions independent chains in parallel and reports observables and MC acceptance ratios every step, averaged over chains. Listing 12: Example usage python3 ./python/cli/paratoric.py -sim etc_thermalization -muc 1 -Nth 2000 -reps 10 -lmbdac 2 -T 0.1 -hc 0.3 -Jc 1 -Nr 1000 -obs percolation_strength percolation_probability plaquette_percolation_probability largest_cluster string_number energy energy_h energy_mu energy_J energy_lmbda sigma_x sigma_z star_x plaquette_z staggered_imaginary_times delta anyon_count anyon_density fredenhagen_marcu sigma_x_susceptibility sigma_z_susceptibility -s 0 - bas x -lat square -L 4 -bound periodic -dsp 1 -outdir /path/to/out Sweep-specific flags. Long flag Short Description --simulation -sim Use etc_thermalization. --repetitions -reps Number of Markov chains to average. 4 Using ParaToric 4.1 Monte Carlo Updates There is no need for the user to explicitly call specific updates or interact with internal C++ classes when using the documented interfaces. Internally, we use all five updates described in the original algorithm by Wu, Deng, and Prokof'ev [5]. These must furthermore be supplemented with the following two updates: Because for high temperatures and for zero offdiagonal fields the spin at imaginary time 0 = β cannot be flipped, we allow for flipping the spin on the entire imaginary axis on one bond or on a plaquette (star) in the ˆσx-basis (ˆσzbasis). These updates only change the energy terms diagonal in the given basis and is trivial when caching the total integrated diagonal energy (the update locally flips the sign of the total integrated potential energy). Another advantage is that integrated autocorrelation times for observables diagonal in the given basis improve even in regimes that were previously accessible. All seven updates are equally likely to be proposed, and we use a 64-bit Mersenne-Twister for pseudorandom numbers [12] with the ability to externally set the seed. Some updates have early exits for input parameters for which they will always be rejected. 29 SciPost Physics Codebases Submission 4.2 Monte Carlo Diagnostics There are two compilation modes, Release and Debug. In production runs, one should always use the Release mode; however, it still gives the user enough information to diagnose sampling problems without severe performance impacts. 4.2.1 Thermalization mode We provide thermalization routines which should be used before production runs to ensure proper thermalization (also known as burn-in). Thermalization times can vary drastically between different observables and initial conditions. We provide an example of sufficient and insufficient thermalization in Fig. 2. We recommend using the provided Python command-line interface, which will also plot the thermalization of all measured observables for the user. In thermalization runs, we also return the Monte Carlo acceptance ratio of every update. This can also be used to diagnose freezing (in the measurement phase, use the integrated autocorrelation time instead), e.g., when the acceptance ratio is always identical and/or very low. In case one suspects experiencing a serious sampling problem, we recommend recompiling the project in the Debug mode, which provides a wide array of runtime debug information about the proposed steps, acceptance ratios, and intermediate results. However, do not use the Debug mode in production runs, as it negatively impacts performance. 4.2.2 Integrated autocorrelation time When measuring observables, we first thermalize the system with N_thermalization steps, then measure N_samples times with N_between_samples steps between measurements. The normalized autocorrelation function ρO(k) of an observable Ok (observable O measured at time k) applied to a discrete time series of length N is given by: ρO(k) = C(k) C(0), C(k) = 1 N -k N-k-1 X i=0 (Oi - ̄O)(Oi+k - ̄O), ̄O = 1 N N-1 X i=0 Oi. (7) It is a statistical measure of the correlations between measurements of observable O at times i and i + k.2 We define the integrated autocorrelation time τO int = 1 2 + X k≥1 ρO(k). (8) Large τint are generally undesirable since they increase error bars and can lead to bias. In case of perfect sampling, we would have ρO(0) = 1 and ρO(k) = 0 ∀k ≥1, i.e., each measurement is only correlated with itself but not with other measurements and τint = 1/2. In practice, this is usually not feasible, and we have to work with a finite autocorrelation time τint > 1/2. When using ParaToric, we strongly recommend monitoring τint for all simulations and all observables. It is automatically calculated for every observable based on the full time series. As a rule of thumb, the autocorrelation is fine as long as τint ≪Nsamples, otherwise it leads to bias and seriously underestimated error bars. In the vicinity of phase transitions, τint dramatically increases (\"critical slowing down\") [13]. Importantly, τint can differ vastly between different observables! If the autocorrelation is too high, increase the number of steps between samples. In more complicated cases, one may need to adapt the update proposal distributions and/or the updates themselves as a last resort. 2In ParaToric, the autocorrelation function is calculated efficiently using fast Fourier transforms. 30 SciPost Physics Codebases Submission It is also important to mention that ParaToric only computes a statistical estimate of τint. Many factors determine how accurate this estimate is, and crucially, the system needs to be properly thermalized. In principle, one can use τint in the way it is computed above directly for calculating error bars of correlated time series; however, ParaToric uses a more robust bootstrapping approach. 4.2.3 Error bars ParaToric applies the stationary bootstrap [14-16] for all error bars, thus capturing autocorrelation effects. Large τint will lead to worse error bars. The only parameter that the user can change is the number of bootstrap resamples N_between_steps. The default is 1000, which is enough in most cases. Note that a too low value of N_between_steps increases the relative computational cost of performing the measurements, which may negatively affect the code efficiency at no statistical gain. If the error bars are too large, either the number of samples is too low (in which case one should increase N_samples) or the autocorrelation is too large (in which case one could additionally increase N_between_samples). 4.3 Tips & tricks 4.3.1 Probing ground state physics The algorithm implemented by ParaToric fundamentally requires a finite temperature T > 0. However, in QMC simulations, there is always a finite-size energy gap (the difference between the energy of the ground state and the first excited state). Additionally, some phases like the topological ground state of the toric code have a physical bulk gap (even at L →∞). As long as the temperature is well below the total gap, we are exponentially close to the ground state. Usually, a temperature T ∼1/L suffices for the toric code although other situations may arise. 4.3.2 Probing first-order transitions ParaToric provides functionalities to probe weak and strong first-order phase transitions. The hysteresis mode can be used to probe hysteresis loops in the vicinity of strong first-order phase transitions, by repeating the simulation two times and mirroring the order of the parameters in h_hysteresis and lmbda_hysteresis. Weak first-order transitions can be detected by plotting a time series histogram of an observable (it exhibits a double-peak structure). Both approaches have been used in the context of the toric code [11]. 4.3.3 Choosing the basis Sometimes, one can work in both the ˆσx and the ˆσz-basis. The performance can vary drastically! Generally, the ˆσx-basis is more efficient for h/J > λ/μ and vice versa. 4.3.4 Choosing N_thermalization Based on our experience, we can say that Nthermalization = 500Ld/T is a sensible choice for small fields, where d is the dimensionality of the system. Nevertheless, one should make use of the provided tools to benchmark thermalization, see Sec. 4.2, and rather err on the side of safety. 4.3.5 Choosing N_samples Neglecting autocorrelation effects, the error of an observable ∆O scales as ∆O ∼1/ Æ Nsamples. More samples are, in principle, always better and lead to lower error 31 SciPost Physics Codebases Submission bars. Smoothness of a curve of statistical results also requires that error bars be small in relation to the parameter grid size. If one increases the parameter resolution (e.g., in the field h), then one typically also increases N_samples. 4.3.6 Choosing N_between_samples The optimal choice for N_samples is the integrated autocorrelation time. A good guess of the autocorrelation time based on previous simulations for smaller system sizes or nearby parameter points can result in substantial computational saving costs in production runs for large system sizes. Near continuous phase transitions, the integrated autocorrelation time has an additional dependence τint ∼Lz, where z is the dynamical exponent of the universality class of the transition. For a 2D system in the vicinity of a continuous phase transition, a sensible scaling for N_between_samples could be O(L2 × β × Lz) (O(L2) links, each has off-diagonal spin flips O(β)). 4.3.7 Choosing N_resamples As with N_samples, more is better (but also more costly). Usually N_resamples ≈1000 is a sensible choice. 4.3.8 Extracting snapshots When the option save_snapshots is enabled, ParaToric will write the snapshots into the directory specified in path_out (or in the paths paths_out for hysteresis sweeps). The snapshots are saved in the GraphML format (XML-based), which is supported by many major graph libraries. One snapshot will be saved for every measurement of observables, i.e., N_s amples snapshots in total. All snapshots are written into a single file to save disk space and simultaneously offer a structured, self-documenting format. Every edge stores a list of spins. The first spin belongs to the first snapshot, the second one to the second snapshot, and so on. There are no special requirements for disks or memory bandwidth; the snapshots are kept in RAM and are only written to disk after the simulation has finished. 4.3.9 Adding new observables/lattices/updates After adding features to the code, always benchmark them using analytical results, other numerical methods (exact diagonalization, tensor networks, ...), and unit tests. We advise using a fixed seed during development, e.g., when checking whether two methods produce the exact same result. The code has some built-in features to check self-consistency, e.g., at the end of each simulation, the code checks whether the cached total energy is numerically close to the total energy calculated from scratch. Do not turn off these features, as they will point you toward bugs! 4.4 Benchmarks 4.4.1 Thermalization In Fig. 2 (which we already discussed before and repeat here for completeness) we plot the gauge field energy ∝λ for two systems: one is sufficiently thermalized, the other one is not. The plots are a direct output of the Python command-line interface. Always make sure that the system is well thermalized. 32 SciPost Physics Codebases Submission (b) (a) Figure 2: Good and bad thermalization plots produced by the Python commandline interface. We show the gauge field energy ∝λ. (a) The system is well thermalized; after its initial drop, the energy fluctuates around the expectation value. (b) The system is not yet thermalized; the floating average is still decreasing. 4.4.2 Integrated autocorrelation time Here, we demonstrate how the integrated autocorrelation time τint grows with decreasing N_ between_samples. We use the following setup: Listing 13: N_between_samples benchmarking setup >>> import numpy as np >>> from paratoric import extended_toric_code as etc >>> series, mean, std, binder, binder_std, tau_int = extended_toric_code. get_sample(N_samples=100000, N_thermalization=10000, N_between_samples =1, N_resamples=1000, custom_therm=False, observables=[\"energy\"], seed =0, mu=1.0, h=0.0, h_therm=0.0, J=1.0, lmbda=0, lmbda_therm=0.0, basis=' x', lattice_type=\"square\", system_size=4, beta=10, boundaries=\"periodic\" , default_spin=1, save_snapshots=False) We only run the simulation once per N_between_samples. The results are: N_between_samples 1 10 100 500 1000 τint (energy) 1895 141.4 19.7 3.24 1.64 For very small N_between_samples, τint is very high: in cases where the update is rejected, the configuration is identical to the one measured before! A choice of N_between_s amples between 500 and 1000 would be a good tradeoff between τint and runtime for this example. Increasing N_between_samples to well over 1000 would be a waste of CPU time. 4.4.3 Run-time We benchmark the run-time for two realistic parameter sets on the square lattice and varying system size. The first setup simulates the toric code without fields:3 Listing 14: L benchmarking setup 1 >>> import numpy as np >>> from paratoric import extended_toric_code as etc 3All tests were run on a laptop; some conditions, like the CPU temperature, were not identical for all simulations. The benchmarks are therefore only an approximation. 33 SciPost Physics Codebases Submission >>> L=20 >>> series, mean, std, binder, binder_std, tau_int = extended_toric_code. get_sample(N_samples=10000, N_thermalization=500*L*L*L, N_between_samples=8*L*L*L, N_resamples=1000, custom_therm=False, observables=[\"energy\", \"sigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.0, h_therm=0.0, J=1.0, lmbda=0, lmbda_therm=0.0, basis='x', lattice_type=\" square\", system_size=L, beta=L, boundaries=\"periodic\", default_spin=1, save_snapshots=False) We only run one test per system size. The results are: L 4 8 12 16 20 Runtime (s) 3.1 21.3 75 197 379 From our experience, for large L the update complexity is approximately O(L3 logβ), owing to the chosen cubic dependency of N_thermalization and N_between_samples and a O(logβ) dependence of operations on the imaginary time axis, see β benchmark below. The system size itself does not impact the performance, as the interactions are local. On computing clusters, we have realized system sizes of up to L = 80 for the square lattice; this number will only increase in the future as CPUs get faster. The second setup simulates the toric code with fields in both ˆσx and ˆσz-direction: Listing 15: L benchmarking setup 2 >>> import numpy as np >>> from paratoric import extended_toric_code as etc >>> L=20 >>> series, mean, std, binder, binder_std, tau_int = extended_toric_code. get_sample(N_samples=10000, N_thermalization=500*L*L*L, N_between_samples=8*L*L*L, N_resamples=1000, custom_therm=False, observables=[\"energy\", \"sigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.2, J =1.0, lmbda=0.2, basis='x', lattice_type=\"square\", system_size=L, beta=L , boundaries=\"periodic\", default_spin=1, save_snapshots=False) We only run one test per system size. The results are: L 4 8 12 16 20 Runtime (s) 3.9 34.1 133 323 689 We also test the run-time dependence of the inverse temperature β, with the following setup: Listing 16: β benchmarking setup >>> import numpy as np >>> from paratoric import extended_toric_code as etc >>> series, mean, std, binder, binder_std, tau_int = extended_toric_code. get_sample(N_samples=10000, N_thermalization=20000, N_between_samples =2000, N_resamples=1000, custom_therm=False, observables=[\"energy\", \" sigma_x\", \"sigma_z\"], seed=0, mu=1.0, h=0.2, J=1.0, lmbda=0.2, basis='x' , lattice_type=\"square\", system_size=10, beta=20, boundaries=\"periodic\", default_spin=1, save_snapshots=False) We only run one test per β. The results are: β 4 8 12 16 20 Runtime (s) 14.9 17.2 19.1 20.0 22.1 34 SciPost Physics Codebases Submission 0.20 0.25 0.30 0.35 0.40 h 0 1 Ox FM square, ⁄ = 0.2 L = 10 L = 20 L = 30 L = 40 0.20 0.25 0.30 0.35 0.40 h 0.0 0.5 Ox SI square, ⁄ = 0.2 L = 10 L = 20 L = 30 L = 40 0.31 0.32 0.33 0.34 h 100 101 102 UΠx square, ⁄ = 0.2 L = 10 L = 20 L = 30 L = 40 0.20 0.25 0.30 0.35 0.40 h 0 1 Πx square, ⁄ = 0.2 L = 10 L = 20 L = 30 L = 40 (b) (a) (c) (d) Figure 3: Topological phase transition of the extended toric code (1) on the square lattice. The critical field is known and located at hc(λ = 0.2) ≈0.33 [5,9]. Our results agree with the value published in the literature, within error bars. (a) The percolation probability Binder ratio UΠx [9] features a crossing point around h = 0.33. (b) The percolation probability Πx is non-zero in the topological phase and zero in the trivial phase. The transition gets sharper with increasing system size. (c) The staggered imaginary time order parameter Ox SI is zero in the topological phase and non-zero in the trivial phase. (d) The Fredenhagen-Marcu order parameter Ox FM is zero in the topological phase and non-zero in the trivial phase. The loop length grows with O(L). Compared to the other order parameters, it is very noisy because it is a multi-body correlator and on top of that a division of two exponentially small numbers. This benchmark illustrates an appealing feature of our implementation: there is almost no slowing down when increasing β implying that very low temperatures are within reach with ParaToric. This paradoxical result (because the number n of off-diagonal star/plaquette and magnetic field operators must physically scale linearly in β) is explained by the fact that most searches within the imaginary time axis scale as O(log n) by making use of binary searches. 4.4.4 Topological phase transition We probe the well-known topological phase transition in the ground state of the extended toric code (1) on the square lattice, where we have a gapped Z2 quantum spin liquid for small fields h,λ and a topologically trivial phase for high fields. We set J = μ = 1, λ = 0.2 and sweep h over the known critical value hc(λ = 0.2) ≈0.33 [5,9] for L ∈{10,20,30,40} in the ˆσx-basis. The temperature is set to T = 1/L to capture ground state physics. We take 30000 snapshots, with 8L3 steps in between snapshots and 500L3 thermalization steps.4. We confirm that the system is well thermalized and that all integrated autocorrelation times are below 10, i.e., the produced snapshots can safely be considered identically and independently distributed. We show the percolation probability, the Fredenhagen-Marcu string order parameter, and the staggered imaginary time order parameter in Fig. 3. All of them reproduce the known phase boundary. 4If we were interested in quantities like critical exponents and we need to go very close to the critical field, we should take into account the dynamical exponent z (τint ∼Lz) in the number of steps between snapshots to account for critical slowing down. Now the error bars are just larger near the critical field. 35 SciPost Physics Codebases Submission 5 Conclusion & Outlook We have presented ParaToric, a continuous-time quantum Monte Carlo solver for the toric code in a parallel field. ParaToric builds on the existing work by Wu, Deng, and Prokof'ev [5] and is also applicable to high temperature and low off-diagonal couplings. ParaToric can store snapshots, which makes it ideally suited to generate training/benchmarking data for applications in other fields, such as lattice gauge theories, cold atom or other quantum simulators, quantum spin liquids, artificial intelligence, and quantum error correction. We believe it also serves a pedagogical purpose. Another strength of ParaToric is its interoperability with other programming languages. The C interface is compatible with virtually all programming languages, thus ParaToric can be seamlessly integrated into other projects. ParaToric comes with an MIT license. For future release of ParaToric we plan extensions along the following lines: • Additional lattices such as the kagome and the ruby lattice. Given the underlying graph structure used in ParaToric, such extensions are straightforward. • Additional observables: we think here of, for instance, the finite temperature extension of the fidelity susceptibility to diagnose the phase transitions in the absence of a local order parameter. It would be worthwhile to have additional off-diagonal observables such as the off-diagonal Fredenhagen-Marcu string operators, or correlation functions between off-diagonal operators in space and or time. Measurements of the Renyi entropy are also high on the to-do list. The latter two classes require however major changes to the code, and testing. • Additional interaction types. There are many classes of models in which topological order may be emergent instead of explicit as in the toric code. Such models typically have additional interactions than the ones covered in ParaToric, such as longer-range Ising interactions, and miss some others (typically the plaquette type interactions, and sometimes even the star terms). It is in general an open problem how to efficiently simulate such models at the lowest temperatures (even for sign-free models). Extending ParaToric to dealing with other types of interactions can thus serve as an additional tool for benchmarking purposes and algorithmic exploration. Acknowledgements The authors acknowledge fruitful discussions with A. Bohrdt, G. De Paciani, G. Dünnweber, F. Grusdt, L. Homeier, and N. V. Prokof'ev. Author contributions SML did the main coding and planning work with input from LP. All authors contributed to the writing of the manuscript. Funding information This research was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program - ERC Starting Grant SimUcQuam (Grant Agreement No. 948141), and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2111 - project number 390814868. 36 SciPost Physics Codebases Submission References [1] A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303(1), 2 (2003), [2] E. Dennis, A. Kitaev, A. Landahl and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43(9), 4452 (2002), https://pubs.aip. org/aip/jmp/article-pdf/43/9/4452/19183135/4452_1_online.pdf. [3] A. G. Fowler, M. Mariantoni, J. M. Martinis and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012), [4] J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979), [5] F. Wu, Y. Deng and N. Prokof'ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B 85, 195104 (2012), [6] K. Fredenhagen and M. Marcu, Charged states in z2 gauge theories, Communications in Mathematical Physics 92(1), 81 (1983), [7] K. Fredenhagen and M. Marcu, Confinement criterion for qcd with dynamical quarks, Phys. Rev. Lett. 56, 223 (1986), [8] K. Fredenhagen and M. Marcu, Dual interpretation of order parameters for lattice gauge theories with matter fields, Nuclear Physics B - Proceedings Supplements 4, 352 (1988), [9] S. M. Linsel, A. Bohrdt, L. Homeier, L. Pollet and F. Grusdt, Percolation as a confinement order parameter in z2 lattice gauge theories, Phys. Rev. B 110, L241101 (2024), [10] G. Dünnweber, S. M. Linsel, A. Bohrdt and F. Grusdt, Percolation renormalization group analysis of confinement in z2 lattice gauge theories, Phys. Rev. B 111, 024314 (2025), [11] S. M. Linsel, L. Pollet and F. Grusdt, Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices, (2025). [12] M. Matsumoto and T. Nishimura, Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul. 8, 3 (1998). [13] U. Wolff, Critical slowing down, Nuclear Physics B - Proceedings Supplements 17, 93 (1990), [14] D. N. Politis and J. P. Romano, The stationary bootstrap, Journal of the American Statistical Association 89(428), 1303 (1994), [15] D. N. Politis and H. White, Automatic block-length selection for the dependent bootstrap, Econometric Reviews 23(1), 53 (2004), [16] A. Patton, D. N. Politis and H. White, Correction to \"automatic block-length selection for the dependent bootstrap\" by d. politis and h. white, Econometric Reviews 28(4), 372 (2009), 37"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":40074,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NzI4OTg4NSwic3ViIjoiL2RhdGFzZXRzL2RhbmtlZy9BcnhpdkJ1bGtEYXRhc2V0IiwiZXhwIjoxNzc3MjkzNDg1LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.7v8cbyvrjgDl2GqbSFKtiWCYUdV6iXmyGRS9ItwBOgo-YCwUqa6ZIc10UP7ZCyiO9blob-7DeDqqiqJz8RQLBg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":40074,"numBytesParquet":2657400671}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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2510.14883 | arXiv:2510.14883v1 [hep-th] 16 Oct 2025
Low energy dynamics of vibrating Kinks.
J. Mateos Guilarte(a)
(a) Instituto Universitario de F´ısica Fundamental y Matem´aticas
Universidad de Salamanca, SPAIN
ABSTRACT
Low energy dynamics of Kinks and Kink-AntiKink configurations in the Jackiw-Rebbi model is fully
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2510.14877 | Prepared for submission to JCAP
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Rua Dr.
Xavier Sigaud 150,
22290-180 Rio de Janeiro, RJ, Brazil
Rodrigo da Mata, Juan C. Rodr´ıguez-Ram´ırez, and Cl´ecio R. Bom
Centro Brasileiro de Pesquisas F´ısicas,
... | Multi-wavelength analysis of the progenitor of GRB 230307A via Bayesian model comparison Viviane Alfradique∗ Centro Brasileiro de Pesquisas F ́ısicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil Rodrigo da Mata, Juan C. Rodr ́ıguez-Ram ́ırez, and Cl ́ecio R. Bom Centro Brasileiro de Pesquisas F ́ısi... |
2510.14882 | Preprint
SCALEWEAVER:
WEAVING EFFICIENT CONTROL-
LABLE T2I GENERATION WITH MULTI-SCALE REFER-
ENCE ATTENTION
Keli Liu*, Zhendong Wang*, Wengang Zhou†, Shaodong Xu, Ruixiao Dong, Houqiang Li
University of Science and Technology of China
{sa23006063, zhendongwang, xiodon, dongruixiaoyx}@mail.ustc.edu.cn
{zhwg,lihq}@ustc.... | Preprint SCALEWEAVER: WEAVING EFFICIENT CONTROLLABLE T2I GENERATION WITH MULTI-SCALE REFERENCE ATTENTION Keli Liu*, Zhendong Wang*, Wengang Zhou†, Shaodong Xu, Ruixiao Dong, Houqiang Li {sa23006063, zhendongwang, xiodon, { canny depth colorization palette sketch deblur Figure 1: Images generated by ScaleWeaver. Our Sca... |
2510.14878 | PREDICTING KERNEL REGRESSION LEARNING CURVES
FROM ONLY RAW DATA STATISTICS
Dhruva Karkada∗,1, Joseph Turnbull∗,1, Yuxi Liu1, James B. Simon1,
{dkarkada,joeyturnbull,yuxi liu,jsi}@berkeley.edu
ABSTRACT
We study kernel regression with common rotation-invariant kernels on real datasets
including CIFAR-5m, SVHN, and ImageN... | PREDICTING KERNEL REGRESSION LEARNING CURVES FROM ONLY RAW DATA STATISTICS Dhruva Karkada∗,1, Joseph Turnbull∗,1, Yuxi Liu1, James B. Simon1, {dkarkada,joeyturnbull,yuxi ABSTRACT We study kernel regression with common rotation-invariant kernels on real datasets including CIFAR-5m, SVHN, and ImageNet. We give a theoreti... |
2510.14872 | Strategic Behavior in Crowdfunding: Insights
from a Large-Scale Online Experiment.
Din Amir1, Bar Hoter1, and Moran Koren1[0000−0003−0012−0208]
Ben-Gurion University of the Negev, Department of Industrial Engineering and
Management
Abstract. This study examines strategic behavior in crowdfunding us-
ing a large-scale o... | Strategic Behavior in Crowdfunding: Insights from a Large-Scale Online Experiment. Din Amir1, Bar Hoter1, and Moran Koren1[0000-0003-0012-0208] Ben-Gurion University of the Negev, . This study examines strategic behavior in crowdfunding using a large-scale online experiment. Building on the model of [4], we test predic... |
2510.14874 | Under review as a conference paper at ICLR 2026
TOUCH: TEXT-GUIDED CONTROLLABLE GENERA-
TION OF FREE-FORM HAND-OBJECT INTERACTIONS
Guangyi Han1, Wei Zhai1,†, Yuhang Yang1, Yang Cao1, Zheng-Jun Zha1
1 University of Science and Technology of China
{hanguangyi@mail., wzhai056@, yyuhang@mail., forrest@, zhazj@}ustc.edu.cn
... | Under review as a conference paper at ICLR 2026 TOUCH: TEXT-GUIDED CONTROLLABLE GENERATION OF FREE-FORM HAND-OBJECT INTERACTIONS Guangyi Han1, Wei Zhai1,†, Yuhang Yang1, Yang Cao1, Zheng-Jun Zha1 1 {hanguangyi@mail., wzhai056@, yyuhang@mail., forrest@, ABSTRACT Hand-object interaction (HOI) is fundamental for humans to... |
2510.14873 | Sampling Density Compensation using Fast
Fourier Deconvolution
Rui Luo, Peng Hu, Haikun Qi ∗
Abstract
Density Compensation Function (DCF) is widely used in non-Cartesian
MRI reconstruction, either for direct Non-Uniform Fast Fourier Trans-
form (NUFFT) reconstruction or for iterative undersampled reconstruc-
tion.
Curr... | Sampling Density Compensation using Fast Fourier Deconvolution Rui Luo, Peng Hu, Haikun Qi ∗ Abstract Density Compensation Function (DCF) is widely used in non-Cartesian MRI reconstruction, either for direct Non-Uniform Fast Fourier Transform (NUFFT) reconstruction or for iterative undersampled reconstruction. Current ... |
2510.14876 | BADAS: Context Aware Collision Prediction Using Real-World
Dashcam Data
Roni Goldshmidt
roni.goldshmidt@getnexar.com
Hamish Scott
hamish.scott@getnexar.com
Lorenzo Niccolini
lorenzo.niccolini@getnexar.com
Shizhan Zhu
shizhan.zhu@getnexar.com
Daniel Moura
daniel.moura@getnexar.com
Orly Zvitia
orly.zvitia@getnexar.com
Oc... | BADAS: Context Aware Collision Prediction Using Real-World Dashcam Data Roni Goldshmidt Hamish Scott Lorenzo Niccolini Shizhan Zhu Daniel Moura Orly Zvitia October 17, 2025 Abstract Existing collision prediction methods often fail to distinguish between ego-vehicle threats and random accidents non-involving ego-vehicle... |
2510.14869 | Tight bounds towards Zarankiewicz problem in
hypergraph
Guorong Gaoa,b*, Jianfeng Houb†, Shuping Huangc‡, Hezhi Wanga§
aCenter for Discrete Mathematics and Theoretical Computer Science,
Fuzhou University, Fuzhou, Fujian, China
bSchool of Mathematics and Statistics, Fuzhou University, Fuzhou, Fujian, China
cSchool of Ma... | Tight bounds towards Zarankiewicz problem in hypergraph Guorong Gaoa,b*, Jianfeng Houb†, Shuping Huangc‡, Hezhi Wanga§ aCenter for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou, Fujian, China b . Let Ks1,...,sr be the complete r-partite r-graph such that the i-th part has si vertices.... |
2510.14871 | From Loop Nests to Silicon: Mapping AI Workloads onto
AMD NPUs with MLIR-AIR
ERWEI WANG, SAMUEL BAYLISS, ANDRA BISCA, ZACHARY BLAIR, SANGEETA CHOWD-
HARY, KRISTOF DENOLF, JEFF FIFIELD, BRANDON FREIBERGER, ERIKA HUNHOFF,
PHIL JAMES-ROXBY, JACK LO, JOSEPH MELBER, STEPHEN NEUENDORFFER, ED-
DIE RICHTER, ANDRÉ RÖSTI, JAVIER... | From Loop Nests to Silicon: Mapping AI Workloads onto AMD NPUs with MLIR-AIR ERWEI WANG, SAMUEL BAYLISS, ANDRA BISCA, ZACHARY BLAIR, SANGEETA CHOWDHARY, KRISTOF DENOLF, JEFF FIFIELD, BRANDON FREIBERGER, ERIKA HUNHOFF, PHIL JAMES-ROXBY, JACK LO, JOSEPH MELBER, STEPHEN NEUENDORFFER, EDDIE RICHTER, ANDRÉ RÖSTI, JAVIER SET... |
2510.14868 | Electron transport in junctions between altermagnets
Shubham Ghadigaonkar,1, ∗Sachchidanand Das,1, ∗and Abhiram Soori1, †
1School of Physics, University of Hyderabad, Prof. C. R. Rao Road, Gachibowli, Hyderabad-500046, India
We theoretically investigate electron transport in junctions between the two AMs in strong and
... | Electron transport in junctions between altermagnets Shubham Ghadigaonkar,1, ∗Sachchidanand Das,1, ∗and Abhiram Soori1, † 1 . C. R. Rao Road, Gachibowli, Hyderabad-500046, India We theoretically investigate electron transport in junctions between the two AMs in strong and weak altermagnetic phases. The charge and spin ... |
2510.14867 | Disorder-assisted Spin-Filtering at Metal/Ferromagnet Interfaces: An Alternative
Route to Anisotropic Magnetoresistance
Ivan Iorsh1 and Mikhail Titov2
1Department of Physics, Engineering Physics & Astronomy, Queen’s Universiy, Kingston, Canada
2Institute for Molecules and Materials, Radboud University, Nijmegen, The Ne... | Disorder-assisted Spin-Filtering at Metal/Ferromagnet Interfaces: An Alternative Route to Anisotropic Magnetoresistance Ivan Iorsh1 and Mikhail Titov2 1 's Universiy, Kingston, Canada 2Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands (Dated: October 17, 2025) We introduce a minimal i... |
2510.14870 | Computation of attractor dimension and maximal sums of
Lyapunov exponents using polynomial optimization
Jeremy P. Parker1, David Goluskin2
1Division of Mathematics, University of Dundee, Dundee, DD1 4HN, United
Kingdom
2Department of Mathematics and Statistics, University of Victoria, Victoria, BC,
V8P 5C2, Canada
Abst... | Computation of attractor dimension and maximal sums of Lyapunov exponents using polynomial optimization Jeremy P. Parker1, David Goluskin2 1Division of Mathematics, 1 4HN, United Kingdom 2 8P 5C2, Canada Abstract Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on Lyapun... |
2510.14866 | BENCHMARKING MULTIMODAL LARGE LANGUAGE MODELS
FOR FACE RECOGNITION
Hatef Otroshi Shahreza and S´ebastien Marcel
Idiap Research Institute, Switzerland
{hatef.otroshi, sebastien.marcel}@idiap.ch
ABSTRACT
Multimodal large language models (MLLMs) have achieved
remarkable performance across diverse vision-and-language
tasks... | BENCHMARKING MULTIMODAL LARGE LANGUAGE MODELS FOR FACE RECOGNITION Hatef Otroshi Shahreza and S ́ebastien Marcel Idiap Research Institute, Switzerland {hatef.otroshi, ABSTRACT Multimodal large language models (MLLMs) have achieved remarkable performance across diverse vision-and-language tasks. However, their potential... |
2510.14875 | Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX631
AREPO-RSG: Aspherical Circumstellar Material and Winds from Pulsating Dusty Red Supergiants
in Global 3D Radiation Hydrodynamic Simulations
Jing-Ze Ma (马竟泽)
,1 Stephen Justham
,1 R¨udiger Pakmor
,1 Andrea Chiavassa
,2, 1 Taeho Ryu
,3, 4, 1
... | Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX631 AREPO-RSG: Aspherical Circumstellar Material and Winds from Pulsating Dusty Red Supergiants in Global 3D Radiation Hydrodynamic Simulations Jing-Ze Ma (马竟泽) ,1 Stephen Justham ,1 R ̈udiger Pakmor ,1 Andrea Chiavassa ,2, 1 Taeho Ryu ,3, 4, 1... |
2510.14864 | The Whole Is Less than the Sum of Parts: Subsystem
Inconsistency in Partial Information Decomposition
Aobo Lyu⋆, Andrew Clark⋆, and Netanel Raviv†
⋆Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, USA
†Department of Computer Science and Engineering, Washington Univers... | The Whole Is Less than the Sum of Parts: Subsystem Inconsistency in Partial Information Decomposition Aobo Lyu⋆, Andrew Clark⋆, and Netanel Raviv† ⋆ . Louis, St. Louis, MO, USA † . Louis, St. Louis, MO, USA , , Abstract-Partial Information Decomposition (PID) was proposed by Williams and Beer in 2010 as a tool for anal... |
2510.14861 |
LabOS: The AI-XR Co-Scientist That Sees and Works With Humans
Le Cong¹,²,*,**, Zaixi Zhang³,*, Xiaotong Wang¹,²,*, Yin Di¹,²,*, Ruofan Jin³, Michal Gerasimiuk¹,², Yinkai
Wang¹,², Ravi K. Dinesh¹,², David Smerkous4, Alex Smerkous5, Xuekun Wu2,6, Shilong Liu³, Peishan Li¹,²,
Yi Zhu¹,², Simran Serrao¹,², Ning Z... | LabOS: The AI-XR Co-Scientist That Sees and Works With Humans Le Cong1,2,*,**, Zaixi Zhang3,*, Xiaotong Wang1,2,*, Yin Di1,2,*, Ruofan Jin3, Michal Gerasimiuk1,2, Yinkai Wang1,2, Ravi K. Dinesh1,2, David Smerkous4, Alex Smerkous5, Xuekun Wu2,6, Shilong Liu3, Peishan Li1,2, Yi Zhu1,2, Simran Serrao1,2, Ning Zhao1,2, Imr... |
2510.14859 | Draft version October 17, 2025
Typeset using LATEX preprint2 style in AASTeX7.0.1
Deuterated water ice on the satellites of Saturn
Michael E. Brown,1 Samantha K. Trumbo,2 M. Ryleigh Davis,1 and Swaroop Chandra1
1Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
2... | Draft version October 17, 2025 Typeset using LATEX preprint2 style in AASTeX7.0.1 Deuterated water ice on the satellites of Saturn Michael E. Brown,1 Samantha K. Trumbo,2 M. Ryleigh Davis,1 and Swaroop Chandra1 1Division of Geological and Planetary Sciences, California 91125, USA 2 92093, USA ABSTRACT The deuterium to ... |
2510.14865 | Preprint
MIDTRAINING BRIDGES PRETRAINING AND
POSTTRAINING DISTRIBUTIONS
Emmy Liu, Graham Neubig, Chenyan Xiong
Language Technologies Institute
Carnegie Mellon University
Pittsburgh, PA 15213, USA
{mengyan3, gneubig, cx}@cs.cmu.edu
ABSTRACT
Recently, many language models have been pretrained with a “midtraining” phase,
... | Preprint MIDTRAINING BRIDGES PRETRAINING AND POSTTRAINING DISTRIBUTIONS Emmy Liu, Graham Neubig, Chenyan Xiong Language Technologies Institute Carnegie Mellon University Pittsburgh, PA 15213, USA {mengyan3, gneubig, ABSTRACT Recently, many language models have been pretrained with a "midtraining" phase, in which higher... |
2510.14863 | SINGULARITIES OF CURVE SHORTENING FLOW WITH
CONVEX PROJECTIONS
QI SUN
Abstract. We show that any closed immersed curve in Rn with a one-to-
one convex projection onto some 2-plane develops a Type I singularity and
becomes asymptotically circular under Curve Shortening flow in Rn.
As an application, we prove an analog o... | SINGULARITIES OF CURVE SHORTENING FLOW WITH CONVEX PROJECTIONS QI SUN Abstract. We show that any closed immersed curve in Rn with a one-toone convex projection onto some 2-plane develops a Type I singularity and becomes asymptotically circular under Curve Shortening flow in Rn. As an application, we prove an analog of ... |
2510.14860 | Differential equations for intertwining operators
among untwisted and twisted modules
Daniel Tan
Abstract
Given any vertex operator algebra V with an automorphism g, we derive a Jacobi identity
for an intertwining operator Y of type
W3
W1 W2
when W1 is an untwisted V -module, and
W2 and W3 are g-twisted V -modules.... | Differential equations for intertwining operators among untwisted and twisted modules Daniel Tan Abstract Given any vertex operator algebra V with an automorphism g, we derive a Jacobi identity for an intertwining operator Y of type W3 W1 W2 when W1 is an untwisted V -module, and W2 and W3 are g-twisted V -modules. We ... |
2510.14862 | U.P.B. Sci. Bull., Series A, Vol. 88, Iss. 3, 2025
ISSN 1223-7027
MULTI-MODAL VIDEO DATA-PIPELINES FOR MACHINE
LEARNING WITH MINIMAL HUMAN SUPERVISION
Pˆırvu Mihai-Cristian1 and Marius Leordeanu2
The real-world is inherently multi-modal at its core. Our tools ob-
serve and take snapshots of it, in digital form, such as... | U.P.B. Sci. Bull., Series A, Vol. 88, Iss. 3, 2025 ISSN 1223-7027 MULTI-MODAL VIDEO DATA-PIPELINES FOR MACHINE LEARNING WITH MINIMAL HUMAN SUPERVISION Pˆırvu Mihai-Cristian1 and Marius Leordeanu2 The real-world is inherently multi-modal at its core. Our tools observe and take snapshots of it, in digital form, such as v... |
2510.14858 | Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from
the IEEE by sending a request to pubs-permissions@ieee.org.
Abstract— Non-diffracting (ND) beams are often cited as a
promising solution to mitigate blockage i... | IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to . Abstract- Non-diffracting (ND) beams are often cited as a promising solution to mitigate blockage in millimeter-wave (mmWave) systems. However, a qua... |
2510.14857 | A SIMULATION FRAMEWORK FOR STUDYING SYSTEMIC
EFFECTS OF FEEDBACK LOOPS IN RECOMMENDER SYSTEMS
Gabriele Barlacchi
Scuola Normale Superiore
Pisa, Italy
gabriele.barlacchi@sns.it
Margherita Lalli
Scuola Normale Superiore
Pisa, Italy
margherita.lalli@sns.it
Emanuele Ferragina
Sciences Po
Paris
emanuele.ferragina@sciencespo... | A SIMULATION FRAMEWORK FOR STUDYING SYSTEMIC EFFECTS OF FEEDBACK LOOPS IN RECOMMENDER SYSTEMS Gabriele Barlacchi Scuola Normale Superiore Pisa, Italy Margherita Lalli Scuola Normale Superiore Pisa, Italy Emanuele Ferragina Sciences Po Paris Fosca Giannotti Scuola Normale Superiore Pisa, Italy Luca Pappalardo CNR and Sc... |
2510.14855 | A Multi-Task Deep Learning Framework for Skin Lesion
Classification, ABCDE Feature Quantification, and Evolution
Simulation
Harsha Kotlaa, Arun Kumar Rajasekarana, Hannah Ranab
aDepartment of Chemistry, University of Cambridge, Cambridge, United Kingdom
bHarvard Ophthalmology AI Lab, Harvard Medical School, Boston, MA,... | A Multi-Task Deep Learning Framework for Skin Lesion Classification, ABCDE Feature Quantification, and Evolution Simulation Harsha Kotlaa, Arun Kumar Rajasekarana, Hannah Ranab a . ABCDE, which stands for Asymmetry, Border irregularity, Color variation, Diameter, and Evolving, is a well-known classification method for ... |
2510.14856 | Rate-Adaptive Protograph-Based
MacKay-Neal Codes
Ayman Zahr, Graduate Student Member, IEEE, Emna Ben Yacoub, Graduate Student Member, IEEE, Bal´azs
Matuz, Senior Member, IEEE, Gianluigi Liva Senior Member, IEEE
Abstract—Rate-adaptive MacKay-Neal (MN) codes based on
protographs are analyzed. The code construction employ... | Rate-Adaptive Protograph-Based MacKay-Neal Codes Ayman Zahr, Graduate Student Member, IEEE, Emna Ben Yacoub, Graduate Student Member, IEEE, Bal ́azs Matuz, Senior Member, IEEE, Gianluigi Liva Senior Member, IEEE Abstract-Rate-adaptive MacKay-Neal (MN) codes based on protographs are analyzed. The code construction emplo... |
2510.14849 | Multi Agent Switching Mode Controller for
Sound Source Localization
Marcello Sorge ∗Nicola Cigarini ∗Riccardo Lorigiola ∗∗
Giulia Michieletto ∗∗∗,∗∗Andrea Masiero ∗
Angelo Cenedese ∗∗,∗∗∗∗Alberto Guarnieri ∗
∗Department of Land, Environment, Agriculture and Forestry,
University of Padova, Legnaro (PD), Italy.
∗∗Departm... | Multi Agent Switching Mode Controller for Sound Source Localization Marcello Sorge ∗Nicola Cigarini ∗Riccardo Lorigiola ∗∗ Giulia Michieletto ∗∗∗,∗∗Andrea Masiero ∗ Angelo Cenedese ∗∗,∗∗∗∗Alberto Guarnieri ∗ ∗ (PD), Italy. ∗∗ . ∗∗∗ . ∗∗∗∗ . Abstract: Source seeking is an important topic in robotic research, especially ... |
2510.14853 | REWIRING EXPERTS ON THE FLY:
CONTINUOUS
REROUTING
FOR
BETTER
ONLINE
ADAPTATION IN MIXTURE-OF-EXPERT MODELS
Guinan Su1*
Yanwu Yang4*
Li Shen5
Lu Yin6
Shiwei Liu1,2,3
Jonas Geiping1,2,3
1Max Planck Institute for Intelligent Systems
2ELLIS Institute T¨ubingen
3T¨ubingen AI Center
4University of T¨ubingen
5Sun Yat-sen Univ... | REWIRING EXPERTS ON THE FLY: CONTINUOUS REROUTING FOR BETTER ONLINE ADAPTATION IN MIXTURE-OF-EXPERT MODELS Guinan Su1* Yanwu Yang4* Li Shen5 Lu Yin6 Shiwei Liu1,2,3 Jonas Geiping1,2,3 1Max Planck Institute for Intelligent Systems 2ELLIS Institute T ̈ubingen 3T ̈ubingen AI Center 4 ̈ubingen 5Sun Yat-sen University 6 -of... |
2510.14854 | ACCEPTED BY IEEE COMMUNICATIONS SURVEYS & TUTORIALS FOR PUBLICATION, DOI: 10.1109/COMST.2025.3623258
1
Through-the-Earth Magnetic Induction
Communication and Networking: A Comprehensive
Survey
Honglei Ma, Erwu Liu, Senior Member, IEEE, Wei Ni, Fellow, IEEE, Zhijun Fang, Rui Wang, Yongbin Gao,
Dusit Niyato, Fellow, IEEE... | ACCEPTED BY IEEE COMMUNICATIONS SURVEYS & TUTORIALS FOR PUBLICATION, 1 Through-the-Earth Magnetic Induction Communication and Networking: A Comprehensive Survey Honglei Ma, Erwu Liu, Senior Member, IEEE, Wei Ni, Fellow, IEEE, Zhijun Fang, Rui Wang, Yongbin Gao, Dusit Niyato, Fellow, IEEE, and Ekram Hossain, Fellow, IEE... |
2510.14850 | General-relativistic radiation magnetohydrodynamics simulations of binary neutron star
mergers: The influence of spin on the multi-messenger picture
Anna Neuweiler1 , Henrique Gieg1 , Henrik Rose1 , Hauke Koehn1 , Ivan Markin1 , Federico Schianchi2,1 ,
Liam Brodie3 , Alexander Haber3,4 , Vsevolod Nedora1,5 , Mattia Bul... | General-relativistic radiation magnetohydrodynamics simulations of binary neutron star mergers: The influence of spin on the multi-messenger picture Anna Neuweiler1 , Henrique Gieg1 , Henrik Rose1 , Hauke Koehn1 , Ivan Markin1 , Federico Schianchi2,1 , Liam Brodie3 , Alexander Haber3,4 , Vsevolod Nedora1,5 , Mattia Bul... |
2510.14848 | A Geometric Approach to Optimal Experimental Design
Gavin Kerrigan⋆
Christian A. Naesseth
Tom Rainforth
University of Oxford
University of Amsterdam
University of Oxford
Abstract
We introduce a novel geometric framework
for optimal experimental design (OED).
Traditional OED approaches, such as those
based on mutual inf... | A Geometric Approach to Optimal Experimental Design Gavin Kerrigan⋆ Christian A. Naesseth Tom Rainforth (OED). Traditional OED approaches, such as those based on mutual information, rely explicitly on probability densities, leading to restrictive invariance properties. To address these limitations, we propose the mutua... |
2510.14852 | Unidirectional Zero Reflection and Perfect Absorption
via Exceptional Points in Active Piezoelectric Willis
Metamaterials
Hrishikesh Danawe1 and Serife Tol1,*
1Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI USA 48109
*Corresponding author(s): Email(s) stol@umich.edu
ABSTRACT
Electro-momentu... | Unidirectional Zero Reflection and Perfect Absorption via Exceptional Points in Active Piezoelectric Willis Metamaterials Hrishikesh Danawe1 and Serife Tol1,* 1 48109 *Corresponding author(s): Email(s) ABSTRACT Electro-momentum coupling in piezoelectric metamaterials with broken inversion symmetry enables asymmetric el... |
2510.14846 | Where to Search:
Measure the Prior-Structured Search Space of LLM Agents
Zhuo-Yang Song1
1School of Physics, Peking University, Beijing 100871, China
Abstract
The generate-filter-refine (iterative paradigm) based on large language models (LLMs) has
achieved progress in reasoning, programming, and program discovery in A... | Where to Search: Measure the Prior-Structured Search Space of LLM Agents Zhuo-Yang Song1 1 100871, China Abstract The generate-filter-refine (iterative paradigm) based on large language models (LLMs) has achieved progress in reasoning, programming, and program discovery in AI+Science. However, the effectiveness of sear... |
2510.14845 | Preprint
BACKDOOR UNLEARNING BY LINEAR TASK
DECOMPOSITION
Amel Abdelraheem∗† 1
Alessandro Favero∗1
G´erˆome Bovet2
Pascal Frossard1
1EPFL, Lausanne, Switzerland
2Cyber-Defence Campus, armasuisse, Switzerland
ABSTRACT
Foundation models have revolutionized computer vision by enabling broad general-
ization across diverse... | Preprint BACKDOOR UNLEARNING BY LINEAR TASK DECOMPOSITION Amel Abdelraheem∗† 1 Alessandro Favero∗1 G ́erˆome Bovet2 Pascal Frossard1 1EPFL, Lausanne, Switzerland 2Cyber-Defence Campus, armasuisse, Switzerland ABSTRACT Foundation models have revolutionized computer vision by enabling broad generalization across diverse ... |
2510.14842 | Boosting Instruction Following at Scale
Ben Elder, Evelyn Duesterwald, Vinod Muthusamy
IBM T.J. Watson Research
Yorktown Heights, NY
Abstract
A typical approach developers follow to influence an
LLM’s behavior in an application is through careful ma-
nipulation of the prompt, such as by adding or modify-
ing instructio... | Boosting Instruction Following at Scale Ben Elder, Evelyn Duesterwald, Vinod Muthusamy IBM T.J. Watson Research Yorktown Heights, NY Abstract A typical approach developers follow to influence an LLM's behavior in an application is through careful manipulation of the prompt, such as by adding or modifying instructions. ... |
2510.14851 | SADCHER: Scheduling using Attention-based Dynamic Coalitions of
Heterogeneous Robots in Real-Time
Jakob Bichler1, Andreu Matoses Gimenez1, Javier Alonso-Mora1
Abstract— We present Sadcher, a real-time task assign-
ment framework for heterogeneous multi-robot teams that
incorporates dynamic coalition formation and task ... | SADCHER: Scheduling using Attention-based Dynamic Coalitions of Heterogeneous Robots in Real-Time Jakob Bichler1, Andreu Matoses Gimenez1, Javier Alonso-Mora1 Abstract- We present Sadcher, a real-time task assignment framework for heterogeneous multi-robot teams that incorporates dynamic coalition formation and task pr... |
2510.14844 | Provable Unlearning with Gradient Ascent on
Two-Layer ReLU Neural Networks
Odelia Melamed *
Gilad Yehudai†
Gal Vardi ‡
Abstract
Machine Unlearning aims to remove specific data from trained models, addressing growing privacy and ethical
concerns. We provide a theoretical analysis of a simple and widely used method—gradi... | Provable Unlearning with Gradient Ascent on Two-Layer ReLU Neural Networks Odelia Melamed * Gilad Yehudai† Gal Vardi ‡ Abstract Machine Unlearning aims to remove specific data from trained models, addressing growing privacy and ethical concerns. We provide a theoretical analysis of a simple and widely used method-gradi... |
2510.14843 | Rate-Adaptive Spatially Coupled MacKay-Neal
Codes with Thresholds Close to Capacity
Ayman Zahr, Student Member, IEEE, and Gianluigi Liva, Senior Member, IEEE
Abstract—We analyze by density evolution the asymptotic per-
formance of rate-adaptive MacKay-Neal (MN) code ensembles,
where the inner code is a protograph spati... | Rate-Adaptive Spatially Coupled MacKay-Neal Codes with Thresholds Close to Capacity Ayman Zahr, Student Member, IEEE, and Gianluigi Liva, Senior Member, IEEE Abstract-We analyze by density evolution the asymptotic performance of rate-adaptive MacKay-Neal (MN) code ensembles, where the inner code is a protograph spatial... |
2510.14841 | On the order of lazy cellular automata
Edgar Alcal´a-Arroyo∗1 and Alonso Castillo-Ramirez†1
1Centro Universitario de Ciencias Exactas e Ingenier´ıas, Universidad de Guadalajara,
M´exico.
October 17, 2025
Abstract
We study the most elementary family of cellular automata defined over an arbitrary
group universe G and an ... | On the order of lazy cellular automata Edgar Alcal ́a-Arroyo∗1 and Alonso Castillo-Ramirez†1 1Centro Universitario de Ciencias Exactas e Ingenier ́ıas, Universidad de Guadalajara, M ́exico. October 17, 2025 Abstract We study the most elementary family of cellular automata defined over an arbitrary group universe G and ... |
2510.14840 | NORMAL AND PRIMITIVE NORMAL ELEMENTS WITH PRESCRIBED
TRACES IN INTERMEDIATE EXTENSIONS OF FINITE FIELDS
ARPAN CHANDRA MAZUMDER, GIORGOS KAPETANAKIS, AND DHIREN KUMAR BASNET
Abstract. In this article, we study the existence and distribution of elements in finite
field extensions with prescribed traces in several interme... | NORMAL AND PRIMITIVE NORMAL ELEMENTS WITH PRESCRIBED TRACES IN INTERMEDIATE EXTENSIONS OF FINITE FIELDS ARPAN CHANDRA MAZUMDER, GIORGOS KAPETANAKIS, AND DHIREN KUMAR BASNET Abstract. In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several interme... |
2510.14838 | 1
Dynamic-Key–Aware Co-Simulation Framework for
Next Generation of SCADA Systems Encrypted by
Quantum-Key-Distribution Techniques
Ziqing Zhu, Member, IEEE
Abstract—To address growing cybersecurity challenges in
modern power dispatch systems, this paper proposes a multi-
layer modeling and optimization framework for SCA... | 1 Dynamic-Key-Aware Co-Simulation Framework for Next Generation of SCADA Systems Encrypted by Quantum-Key-Distribution Techniques Ziqing Zhu, Member, IEEE Abstract-To address growing cybersecurity challenges in modern power dispatch systems, this paper proposes a multilayer modeling and optimization framework for SCADA... |
2510.14837 | Reinforcement Learning with Stochastic Reward
Machines⋆
Jan Corazza1,2
Ivan Gavran2
Daniel Neider2
corazzajan@gmail.com
gavran@mpi-sws.org
neider@mpi-sws.org
1 University of Zagreb
2 Max Planck Institute for Software Systems
Abstract. Reward machines are an established tool for dealing with re-
inforcement learning pro... | Reinforcement Learning with Stochastic Reward Machines⋆ Jan Corazza1,2 Ivan Gavran2 Daniel Neider2 1 2 Max Planck Institute for Software Systems Abstract. Reward machines are an established tool for dealing with reinforcement learning problems in which rewards are sparse and depend on complex sequences of actions. Howe... |
2510.14835 | OBVIOUS AND NON-OBVIOUS ASPECTS OF DIGITAL
SELF-EXCITED-LOOPS FOR SRF CAVITY CONTROL
Larry Doolittle∗, Shreeharshini Murthy, LBNL;
Matei Guran, Lennon Reyes, Shrividhyaa Sankar Raman, Philip Varghese, FNAL
Abstract
In 1978, Delayen showed how Self-Excited Loops (SEL)
can be used to great advantage for controlling narro... | OBVIOUS AND NON-OBVIOUS ASPECTS OF DIGITAL SELF-EXCITED-LOOPS FOR SRF CAVITY CONTROL Larry Doolittle∗, Shreeharshini Murthy, LBNL; Matei Guran, Lennon Reyes, Shrividhyaa Sankar Raman, Philip Varghese, FNAL Abstract In 1978, Delayen showed how Self-Excited Loops (SEL) can be used to great advantage for controlling narro... |
2510.14847 | Preprint
ImagerySearch:
Adaptive Test-Time Search for
Video Generation Beyond Semantic Dependency Con-
straints
Meiqi Wu1,3∗
Jiashu Zhu2
Xiaokun Feng1,3
Chubin Chen4
Chen Zhu5
Bingze Song2
Fangyuan Mao2
Jiahong Wu2†
Xiangxiang Chu2
Kaiqi Huang1,3‡
1UCAS 2AMAP, Alibaba Group 3CRISE 4THU 5SEU
GitHub: https://github.com/A... | Preprint ImagerySearch: Adaptive Test-Time Search for Video Generation Beyond Semantic Dependency Constraints Meiqi Wu1,3∗ Jiashu Zhu2 Xiaokun Feng1,3 Chubin Chen4 Chen Zhu5 Bingze Song2 Fangyuan Mao2 Jiahong Wu2† Xiangxiang Chu2 Kaiqi Huang1,3‡ 1UCAS 2AMAP, Alibaba Group 3CRISE 4THU 5SEU GitHub: https://github.com/AMA... |
2510.14832 | Intelligent Dynamic Handover via AI-assisted
Signal Quality Prediction in 6G Multi-RAT
Networks
Maria Lamprini A. Bartsioka
R&D Department
Four Dot Infinity
Athens, Greece
mbarts@fourdotinfinity.com
Anastasios Giannopoulos
R&D Department
Four Dot Infinity
Athens, Greece
angianno@fourdotinfinity.com
Sotirios Spantideas
... | Intelligent Dynamic Handover via AI-assisted Signal Quality Prediction in 6G Multi-RAT Networks Maria Lamprini A. Bartsioka R&D Department Four Dot Infinity Athens, Greece Anastasios Giannopoulos R&D Department Four Dot Infinity Athens, Greece Sotirios Spantideas R&D Department Four Dot Infinity Athens, Greece Abstract... |
2510.14836 | QDepth-VLA: Quantized Depth Prediction as Auxiliary
Supervision for Vision–Language–Action Models
Yixuan Li
School of Artificial Intelligence, University of the Chinese
Academy of Sciences
Beijing, China
liyixuan223@mails.ucas.ac.cn
Yuhui Chen
Institute of Automation, Chinese Academy of Science
Beijing, China
chenyuhui... | QDepth-VLA: Quantized Depth Prediction as Auxiliary Supervision for Vision-Language-Action Models Yixuan Li * -LanguageAction (VLA) models to accomplish fine-grained manipulation tasks. However, existing approaches often lack the ability to understand and reason over the essential 3D structures necessary for precise co... |
2510.14833 | Non-exotic traversable wormholes with strong deflection angle in King and
Dekel-Zhao dark matter halos under f(R, Lm) gravity
Susmita Sarkar
,1, ∗Nayan Sarkar
,2, † Abdelmalek Bouzenada
,3, ‡ and Farook Rahaman
4, §
1Department of Applied Science and Humanities,
Haldia Institute of Technology, Haldia-721606, West Bengal... | Non-exotic traversable wormholes with strong deflection angle in King and Dekel-Zhao dark matter halos under f(R, Lm) gravity Susmita Sarkar ,1, ∗Nayan Sarkar ,2, † Abdelmalek Bouzenada ,3, ‡ and Farook Rahaman 4, § 1 -721606, West Bengal, India 2 741152, West Bengal, India 3Laboratory of Theoretical and Applied Physic... |
2510.14839 | Draft version October 17, 2025
Typeset using LATEX default style in AASTeX7.0.1
Antarctic Infrared Binocular Telescope. I.
System Overview, Laboratory Testing, and On-Sky Performance Evaluation
Zhongnan Dong
,1 Bin Ma
,1, 2 Haoran Zhang,1 Jinji Li,1 Xu Yang,3 Yi Hu,3 Zhaohui Shang3
And Michael C. B. Ashley4
The Terra M... | Draft version October 17, 2025 Typeset using LATEX default style in AASTeX7.0.1 Antarctic Infrared Binocular Telescope. I. System Overview, Laboratory Testing, and On-Sky Performance Evaluation Zhongnan Dong ,1 Bin Ma ,1, 2 Haoran Zhang,1 Jinji Li,1 Xu Yang,3 Yi Hu,3 Zhaohui Shang3 And Michael C. B. Ashley4 The Terra M... |
2510.14834 | Improved Voltage Regulation with Optimal Design
of Decentralized Volt-VAr Control
Daniel Russell, Dakota Hamilton, Mads R. Almassalkhi, and Hamid R. Ossareh1
Abstract— Integration of distributed energy resources has
created a need for autonomous, dynamic voltage regulation.
Decentralized Volt-VAr Control (VVC) of grid-... | Improved Voltage Regulation with Optimal Design of Decentralized Volt-VAr Control Daniel Russell, Dakota Hamilton, Mads R. Almassalkhi, and Hamid R. Ossareh1 Abstract- Integration of distributed energy resources has created a need for autonomous, dynamic voltage regulation. Decentralized Volt-VAr Control (VVC) of grid-... |
2510.14828 | RoboGPT-R1: Enhancing Robot Planning with Reinforcement
Learning
Jinrui Liu∗
Institute of Automation, CASIA
School of Artificial Intelligence, UCAS
China
liujinrui2024@ia.ac.cn
Bingyan Nie∗
Institute of Automation, CASIA
School of Artificial Intelligence, UCAS
China
niebingyan2025@ia.ac.cn
Boyu Li
Institute of Automati... | RoboGPT-R1: Enhancing Robot Planning with Reinforcement Learning Jinrui Liu∗ ∗ ., Ltd China Shunsen He Huawei Cloud Technology Co., Ltd China Haoran Li† -view manipulation tasks successfully. Despite the success of large language models and vision language models based on Supervised FineTuning (SFT) in planning tasks, ... |
2510.14827 | NEURAL
IMPLICIT
FLOW
FIELDS
FOR
SPATIO-
TEMPORAL MOTION MAPPING
Yufei Zhu∗
Shih-Min Yang∗
Andrey Rudenko†
Tomasz P. Kucner‡
Achim J. Lilienthal∗†
Martin Magnusson∗
ABSTRACT
Safe and efficient robot operation in complex human environments can benefit
from good models of site-specific motion patterns. Maps of Dynamics (M... | NEURAL IMPLICIT FLOW FIELDS FOR SPATIOTEMPORAL MOTION MAPPING Yufei Zhu∗ Shih-Min Yang∗ Andrey Rudenko† Tomasz P. Kucner‡ Achim J. Lilienthal∗† Martin Magnusson∗ ABSTRACT Safe and efficient robot operation in complex human environments can benefit from good models of site-specific motion patterns. Maps of Dynamics (MoD... |
2510.14829 | MNRAS 000, 1–18 (2025)
Preprint 17 October 2025
Compiled using MNRAS LATEX style file v3.3
The Launching of Galactic Winds from a Multiphase ISM
Fernando Hidalgo-Pineda,1★Max Gronke2,1 and Philipp Grete3
1Max Planck Institute for Astrophysics, Garching D-85748, Germany
2Astronomisches Rechen-Institut, Zentrum für Astro... | MNRAS 000, 1-18 (2025) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.3 The Launching of Galactic Winds from a Multiphase ISM Fernando Hidalgo-Pineda,1★Max Gronke2,1 and Philipp Grete3 1Max Planck Institute for Astrophysics, Garching D-85748, Germany 2Astronomisches Rechen-Institut, Zentrum für Astro... |
2510.14831 | Scaling Tumor Segmentation:
Best Lessons from Real and Synthetic Data
Qi Chen1,2
Xinze Zhou1
Chen Liu1,3
Hao Chen4
Wenxuan Li1
Zekun Jiang5
Ziyan Huang6,7
Yuxuan Zhao8
Dexin Yu8
Junjun He7
Yefeng Zheng9
Ling Shao2
Alan Yuille1
Zongwei Zhou1,*
1Johns Hopkins University
2UCAS-Terminus AI Lab, University of Chinese Academ... | Scaling Tumor Segmentation: Best Lessons from Real and Synthetic Data Qi Chen1,2 Xinze Zhou1 Chen Liu1,3 Hao Chen4 Wenxuan Li1 Zekun Jiang5 Ziyan Huang6,7 Yuxuan Zhao8 Dexin Yu8 Junjun He7 Yefeng Zheng9 Ling Shao2 Alan Yuille1 Zongwei Zhou1,* 1Johns Hopkins University 2UCAS-Terminus AI Lab, 3Hong Kong Polytechnic Unive... |
2510.14824 | Supervised Fine-Tuning or Contrastive Learning?
Towards Better Multimodal LLM Reranking
Ziqi Dai1∗, Xin Zhang1,2∗, Mingxin Li, Yanzhao Zhang, Dingkun Long
Pengjun Xie, Meishan Zhang1, Wenjie Li2, Min Zhang1
1Harbin Institute of Technology, Shenzhen
2The Hong Kong Polytechnic University
{ziqi.dai, zhangxin2023}@stu.hit.... | Supervised Fine-Tuning or Contrastive Learning? Towards Better Multimodal LLM Reranking Ziqi Dai1∗, Xin Zhang1,2∗, Mingxin Li, Yanzhao Zhang, Dingkun Long Pengjun Xie, Meishan Zhang1, Wenjie Li2, Min Zhang1 1Harbin 2The Hong Kong Polytechnic University {ziqi.dai, Release at https://github.com/vec-ai/lychee-rerank-mm AB... |
2510.14826 | To Infinity and Beyond: Tool-Use Unlocks
Length Generalization in State Space Models
Eran Malach, Omid Saremi, Sinead Williamson, Arwen Bradley, Aryo Lotfi, Emmanuel Abbe, Josh Susskind, Etai
Littwin
Apple
State Space Models (SSMs) have become the leading alternative to Transformers for sequence modeling. Their primary... | To Infinity and Beyond: Tool-Use Unlocks Length Generalization in State Space Models Eran Malach, Omid Saremi, Sinead Williamson, Arwen Bradley, Aryo Lotfi, Emmanuel Abbe, Josh Susskind, Etai Littwin Apple State Space Models (SSMs) have become the leading alternative to Transformers for sequence modeling. Their primary... |
2510.14822 | Regression Model Selection Under General
Conditions
Amaze Lusompa *
Federal Reserve Bank of Kansas City
October 17, 2025
Abstract
Model selection criteria are one of the most important tools in statistics. Proofs
showing a model selection criterion is asymptotically optimal are tailored to the type
of model (linear reg... | Regression Model Selection Under General Conditions Amaze Lusompa * Federal Reserve Bank of Kansas City October 17, 2025 Abstract Model selection criteria are one of the most important tools in statistics. Proofs showing a model selection criterion is asymptotically optimal are tailored to the type of model (linear reg... |
2510.14825 | Preprint
PROGRAMMATIC REPRESENTATION LEARNING WITH
LANGUAGE MODELS
Gabriel Poesia∗
Kempner Institute at Harvard University
gabriel poesia@fas.harvard.edu
Georgia Gabriela Sampaio∗
Stanford University
gsamp@stanford.edu
ABSTRACT
Classical models for supervised machine learning, such as decision trees, are ef-
ficient an... | Preprint PROGRAMMATIC REPRESENTATION LEARNING WITH LANGUAGE MODELS Gabriel Poesia∗ Kempner Institute at Harvard University gabriel Georgia Gabriela Sampaio∗ Stanford University ABSTRACT Classical models for supervised machine learning, such as decision trees, are efficient and interpretable predictors, but their qualit... |
2510.14821 | THE ORDERING PRINCIPLE AND HIGHER DEPENDENT
CHOICE
PETER HOLY AND JONATHAN SCHILHAN
Abstract. We provide, for any regular uncountable cardinal κ, a new argu-
ment for Pincus’ result on the consistency of ZF with the higher dependent
choice principle DC<κ and the ordering principle in the presence of a fail-
ure of the ... | THE ORDERING PRINCIPLE AND HIGHER DEPENDENT CHOICE PETER HOLY AND JONATHAN SCHILHAN Abstract. We provide, for any regular uncountable cardinal κ, a new argument for Pincus' result on the consistency of ZF with the higher dependent choice principle DC κ+ will produce new models for the above described consistency result... |
2510.14820 | Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX7.0.1
Exploring a cosmic ray inverse-Compton origin to the SZ-to-X-ray pressure deficit in the cool core
cluster ZwCl 3146
Emily M. Silich,1 Jack Sayers,1 Philip F. Hopkins,1 Charles Romero,2 Brian Mason,3 John Orlowski-Scherer,2
and Craig L. S... | Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX7.0.1 Exploring a cosmic ray inverse-Compton origin to the SZ-to-X-ray pressure deficit in the cool core cluster ZwCl 3146 Emily M. Silich,1 Jack Sayers,1 Philip F. Hopkins,1 Charles Romero,2 Brian Mason,3 John Orlowski-Scherer,2 and Craig L. S... |
2510.14823 | FraQAT: Quantization Aware Training with
Fractional bits
Luca Morreale
Alberto Gil C. P. Ramos
Malcolm Chadwick
Mehid Noroozi
Ruchika Chavhan
Abhinav Mehrotra
Samsung AI Center
Cambridge, UK
Sourav Bhattacharya
Sana 600M
FraQAT
SD3.5-M
FraQAT
Flux-schnell
FraQAT
Figure 1: FraQAT is a Quantization aware Training (QAT) t... | FraQAT: Quantization Aware Training with Fractional bits Luca Morreale Alberto Gil C. P. Ramos Malcolm Chadwick Mehid Noroozi Ruchika Chavhan Abhinav Mehrotra Samsung AI Center Cambridge, UK Sourav Bhattacharya Sana 600M FraQAT SD3.5-M FraQAT Flux-schnell FraQAT Figure 1: FraQAT is a Quantization aware Training (QAT) t... |
2510.14817 | Signatures of Topological Symmetries on a Noisy Quantum Simulator
Christopher Lamb,1, ∗Robert M. Konik,2 Hubert Saleur,3, 4 and Ananda Roy1, †
1Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA
2Division of Condensed Matter Physics and Material Science,
Brookhaven National Laborator... | Signatures of Topological Symmetries on a Noisy Quantum Simulator Christopher Lamb,1, ∗Robert M. Konik,2 Hubert Saleur,3, 4 and Ananda Roy1, † 1 08854-8019 USA 2Division of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA 3Institut de physique th ́eorique, CEA, CN... |
2510.14815 | STABLE TYPE I BLOW-UP FOR THE ONE-DIMENSIONAL WAVE EQUATION
WITH TIME-DERIVATIVE NONLINEARITY
OLIVER GOUGH
Abstract. We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic
time-derivative nonlinearity,
utt −uxx = (ut)2,
(x, t) ∈R × [0, T).
Building on the work of Ghoul, Liu, and M... | STABLE TYPE I BLOW-UP FOR THE ONE-DIMENSIONAL WAVE EQUATION WITH TIME-DERIVATIVE NONLINEARITY OLIVER GOUGH Abstract. We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, utt -uxx = (ut)2, (x, t) ∈R × [0, T). Building on the work of Ghoul, Liu, and M... |
2510.14830 | RL-100: Performant Robotic Manipulation with Real-World
Reinforcement Learning
Kun Lei1,2,*, Huanyu Li1,2,*, Dongjie Yu1,3,*, Zhenyu Wei5,*, Lingxiao Guo6, Zhennan Jiang7,
Ziyu Wang4, Shiyu Liang2 and Huazhe Xu1,4
Abstract
Real-world robotic manipulation in homes and factories demands reliability, efficiency, and robus... | RL-100: Performant Robotic Manipulation with Real-World Reinforcement Learning Kun Lei1,2,*, Huanyu Li1,2,*, Dongjie Yu1,3,*, Zhenyu Wei5,*, Lingxiao Guo6, Zhennan Jiang7, Ziyu Wang4, Shiyu Liang2 and Huazhe Xu1,4 Abstract Real-world robotic manipulation in homes and factories demands reliability, efficiency, and robus... |
2510.14814 | Pre-print
TACKLING TIME-SERIES FORECASTING GENERALIZA-
TION VIA MITIGATING CONCEPT DRIFT
Zhiyuan Zhao, Haoxin Liu, B. Aditya Prakash
School of Computational Science & Engineering
Georgia Institute of Technology
Atlanta, GA 30332, USA
{leozhao1997@, hliu763@, badityap@cc.}gatech.edu
ABSTRACT
Time-series forecasting find... | Pre-print TACKLING TIME-SERIES FORECASTING GENERALIZATION VIA MITIGATING CONCEPT DRIFT Zhiyuan Zhao, Haoxin Liu, B. Aditya Prakash 30332, USA {leozhao1997@, hliu763@, ABSTRACT Time-series forecasting finds broad applications in real-world scenarios. Due to the dynamic nature of time series data, it is important for tim... |
2510.14816 | POLYNOMIAL PRECONDITIONING FOR INDEFINITE MATRICES
HAYDEN HENSON† AND RONALD B. MORGAN‡
Abstract. Polynomial preconditioning is an important tool in solving large linear systems and
eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and
restarted Arnoldi. Here we give methods for i... | POLYNOMIAL PRECONDITIONING FOR INDEFINITE MATRICES HAYDEN HENSON† AND RONALD B. MORGAN‡ Abstract. Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for i... |
2510.14818 | Trends of Pink Slime Journalism Advertisement
Expenditure and Spread on Facebook from
2019-2024
Christine Sowa Lepird1, Lynnette Hui Xian Ng1, and Kathleen M. Carley1
Carnegie Mellon University, USA
csowa@andrew.cmu.edu
Abstract. Pink slime journalism is a practice where news outlets pub-
lish low-quality or inflammato... | Trends of Pink Slime Journalism Advertisement Expenditure and Spread on Facebook from 2019-2024 Christine Sowa Lepird1, Lynnette Hui Xian Ng1, and Kathleen M. Carley1 Carnegie Mellon University, USA Abstract. Pink slime journalism is a practice where news outlets publish low-quality or inflammatory partisan articles, c... |
2510.14812 | Published as a conference paper at ICLR 2026
EFFICIENT DYNAMIC STRUCTURED SPARSE TRAINING
WITH LEARNED SHUFFLES
Abhishek Tyagi1 Arjun Iyer2 Liam Young2 William H. Renninger2 Christopher Kanan1 Yuhao Zhu1
1Department of Computer Science
2The Institute of Optics
University of Rochester, NY, USA
{atyagi2,aiyer2,lyoung12,w... | Published as a conference paper at ICLR 2026 EFFICIENT DYNAMIC STRUCTURED SPARSE TRAINING WITH LEARNED SHUFFLES Abhishek Tyagi1 Arjun Iyer2 Liam Young2 William H. Renninger2 Christopher Kanan1 Yuhao Zhu1 1 2The { , ABSTRACT Structured sparsity accelerates training and inference on modern GPUs, yet it still trails unstr... |
2510.14811 | Efficient adaptive control strategy for multi-parameter quantum metrology in
two-dimensional systems
Qifei Wei1 and Shengshi Pang1,2∗
1School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China and
2Hefei National Laboratory, Hefei 230088, China
Quantum metrology leverages quantum resources such as e... | Efficient adaptive control strategy for multi-parameter quantum metrology in two-dimensional systems Qifei Wei1 and Shengshi Pang1,2∗ 1 -sen University, Guangzhou, Guangdong 510275, China and 2Hefei National Laboratory, Hefei 230088, China Quantum metrology leverages quantum resources such as entanglement and squeezing... |
2510.14813 | Quantum Fisher Information as a Thermal and Dynamical Probe in Frustrated Magnets:
Insights from Quantum Spin Ice
Chengkang Zhou,1, ∗Zhengbang Zhou,2, ∗F´elix Desrochers,2, 3 Yong Baek Kim,2 and Zi Yang Meng1
1Department of Physics and HK Institute of Quantum Science & Technology,
The University of Hong Kong, Pokfulam ... | Quantum Fisher Information as a Thermal and Dynamical Probe in Frustrated Magnets: Insights from Quantum Spin Ice Chengkang Zhou,1, ∗Zhengbang Zhou,2, ∗F ́elix Desrochers,2, 3 Yong Baek Kim,2 and Zi Yang Meng1 1 2 5S 1A7, Canada 3 02138, USA (Dated: October 17, 2025) Quantum Fisher information (QFI) is a novel measure ... |
2510.14808 | Agentic NL2SQL to Reduce Computational Costs
Dominik Jehle
jehled@cs.uni-freiburg.de
University of Freiburg
Freiburg im Breisgau, Germany
Lennart Purucker
purucker@cs.uni-freiburg.de
University of Freiburg
Freiburg im Breisgau, Germany
Frank Hutter
Prior Labs
ELLIS Institute Tübingen
University of Freiburg
Abstract
Tra... | Agentic NL2SQL to Reduce Computational Costs Dominik Jehle übingen (NL2SQL or Text-to-SQL) has recently been empowered by large language models (LLMs). Using LLMs to perform NL2SQL methods on a large collection of SQL databases necessitates processing large quantities of meta-information about the databases, which in t... |
2510.14819 | Unifying Environment Perception and Route Choice Modeling for
Trajectory Representation Learning
Ji Cao1,3 ∗, Yu Wang1 ∗, Tongya Zheng2 †, Zujie Ren3,
Canghong Jin2, Gang Chen1, Mingli Song1
1Zhejiang University
2Hangzhou City University
3Zhejiang Lab
{caoj25, yu.wang, renzju, cg, brooksong}@zju.edu.cn
doujiang_zheng@1... | Unifying Environment Perception and Route Choice Modeling for Trajectory Representation Learning Ji Cao1,3 ∗, Yu Wang1 ∗, Tongya Zheng2 †, Zujie Ren3, Canghong Jin2, Gang Chen1, Mingli Song1 1Zhejiang University 2Hangzhou City University 3Zhejiang Lab {caoj25, yu.wang, renzju, cg, , Abstract Trajectory Representation L... |
2510.14807 | Technical Report
SIMKO: SIMPLE PASS@K POLICY OPTIMIZATION
Ruotian Peng1,*
Yi Ren2,*
Zhouliang Yu3
Weiyang Liu3
Yandong Wen1,†
1Westlake University
2University of British Columbia
3CUHK
ABSTRACT
Reinforcement learning with verifiable rewards (RLVR) has advanced the reason-
ing capabilities of large language models (LLMs... | Technical Report SIMKO: SIMPLE PASS@K POLICY OPTIMIZATION Ruotian Peng1,* Yi Ren2,* Zhouliang Yu3 Weiyang Liu3 Yandong Wen1,† 1Westlake University 2 3CUHK ABSTRACT Reinforcement learning with verifiable rewards (RLVR) has advanced the reasoning capabilities of large language models (LLMs). However, prevailing RLVR meth... |
2510.14809 | Astronomy & Astrophysics manuscript no. articlev3
©ESO 2025
October 17, 2025
The Nearby Evolved Stars Survey III: First data release of JCMT
CO-line observations
S. H. J. Wallström1, 2, P. Scicluna3, 4, 5, 2, S. Srinivasan6, 2⋆, J. G. A. Wouterloot7, I. McDonald8, 9, L. Decock1, M.
Wijshoff1, R. Chen10, 2, D. Torres11,... | Astronomy & Astrophysics manuscript no. articlev3 October 17, 2025 The Nearby Evolved Stars Survey III: First data release of JCMT CO-line observations S. H. J. Wallström1, 2, P. Scicluna3, 4, 5, 2, S. Srinivasan6, 2⋆, J. G. A. Wouterloot7, I. McDonald8, 9, L. Decock1, M. Wijshoff1, R. Chen10, 2, D. Torres11, L. Umans1... |
2510.14804 | On cliques in hypergraphs
Jun Gao∗
Mathematics Institute and DIMAP, University of Warwick, Coventry, UK
October 17, 2025
Abstract
We prove that for any k ≥3, every k-uniform hypergraph on n vertices contains at
most n −o(n) different sizes of cliques (maximal complete subgraphs). In particular,
the 3-uniform case answe... | On cliques in hypergraphs Jun Gao∗ Mathematics Institute and DIMAP, 17, 2025 Abstract We prove that for any k ≥3, every k-uniform hypergraph on n vertices contains at most n -o(n) different sizes of cliques (maximal complete subgraphs). In particular, the 3-uniform case answers a question of Erdős. 1 Introduction Given... |
2510.14805 | Augmented Lagrangian Method based adjoint space
framework for sparse reconstruction of acoustic source with
boundary measurements
Nirui Tan∗
Hongpeng Sun†‡
October 17, 2025
Abstract
We propose a semismooth Newton-based augmented Lagrangian method for re-
constructing sparse sources in inverse acoustic scattering proble... | Augmented Lagrangian Method based adjoint space framework for sparse reconstruction of acoustic source with boundary measurements Nirui Tan∗ Hongpeng Sun†‡ October 17, 2025 Abstract We propose a semismooth Newton-based augmented Lagrangian method for reconstructing sparse sources in inverse acoustic scattering problems... |
2510.14802 | A Scalable MVDR Beamforming Algorithm That is
Linear in the Number of Antennas
Sanjaya Herath∗, Armin Gerami∗, Kevin Wagner†, Ramani Duraiswami ∗, Christopher A. Metzler∗
Email: sanjayah@umd.edu, agerami@umd.edu, kevin.wagner@nrl.navy.mil, ramanid@umd.edu, metzler@umd.edu
∗University of Maryland, College Park, MD
† Nav... | A Scalable MVDR Beamforming Algorithm That is Linear in the Number of Antennas Sanjaya Herath∗, Armin Gerami∗, Kevin Wagner†, Ramani Duraiswami ∗, Christopher A. Metzler∗ Email: , , , , ∗ † Naval Research Laboratory, Washington, DC Abstract-The Minimum Variance Distortionless Response (MVDR) beamforming technique is wi... |
2510.14810 | Rethinking Hebbian Principle: Low-Dimensional
Structural Projection for Unsupervised Learning
Shikuang Deng1∗, Jiayuan Zhang2∗, Yuhang Wu1, Ting Chen1, Shi Gu3,4B
1School of Computer Science and Engineering, UESTC
2Glasgow College, UESTC
3 School of Computer Science and Technology, Zhejiang University
4 State Key Lab o... | Rethinking Hebbian Principle: Low-Dimensional Structural Projection for Unsupervised Learning Shikuang Deng1∗, Jiayuan Zhang2∗, Yuhang Wu1, Ting Chen1, Shi Gu3,4B 1 2Glasgow College, UESTC 3 4 State Key Lab of Brain-Machine Intelligence, Zhejiang University, China , Abstract Hebbian learning is a biological principle t... |
2510.14803 | Scaling Artificial Intelligence for Multi-Tumor
Early Detection with More Reports, Fewer Masks
Pedro R. A. S. Bassi1,2,3, Xinze Zhou1†, Wenxuan Li1†,
Szymon Płotka4†, Jieneng Chen1, Qi Chen1, Zheren Zhu5,14,
Jakub Prządo6, Ibrahim E. Hamaci7,8, Sezgin Er9,
Yuhan Wang10, Ashwin Kumar11, Bjoern Menze9,
Jarosław B. Ćwikła... | Scaling Artificial Intelligence for Multi-Tumor Early Detection with More Reports, Fewer Masks Pedro R. A. S. Bassi1,2,3, Xinze Zhou1†, Wenxuan Li1†, Szymon Płotka4†, Jieneng Chen1, Qi Chen1, Zheren Zhu5,14, Jakub Prządo6, Ibrahim E. Hamaci7,8, Sezgin Er9, Yuhan Wang10, Ashwin Kumar11, Bjoern Menze9, Jarosław B. Ćwikła... |
2510.14799 | Error analysis of Abate–Whitt methods for Inverse
Laplace Transforms and a new algorithm for
queuing theory applications
Nikita Deniskin1* and Federico Poloni2*
1Classe di Scienze, Scuola Normale Superiore, Pisa, Italy.
2Dipartimento di Informatica, Universit`a di Pisa, Pisa, Italy.
*Corresponding author(s). E-mail(s):... | Error analysis of Abate-Whitt methods for Inverse Laplace Transforms and a new algorithm for queuing theory applications Nikita Deniskin1* and Federico Poloni2* 1Classe di Scienze, Scuola Normale Superiore, Pisa, Italy. 2Dipartimento di Informatica, Universit`a di Pisa, Pisa, Italy. *Corresponding author(s). E-mail(s):... |
2510.14800 | Morphology-Aware Prognostic model for Five-Year Survival Prediction in
Colorectal Cancer from H&E Whole Slide Images
Usama Sajjad1*, Abdul Rehman Akbar1, Ziyu Su1, Deborah Knight1, Wendy L. Frankel1, Metin N. Gurcan2, Wei
Chen1, Muhammad Khalid Khan Niazi1
1Department of Pathology, The Ohio State University Wexner ... | Morphology-Aware Prognostic model for Five-Year Survival Prediction in Colorectal Cancer from H&E Whole Slide Images Usama Sajjad1*, Abdul Rehman Akbar1, Ziyu Su1, Deborah Knight1, Wendy L. Frankel1, Metin N. Gurcan2, Wei Chen1, Muhammad Khalid Khan Niazi1 1 . 2Center for Artificial Intelligence Research, Wake Forest U... |
2510.14806 | JOINT CHANNEL AND CFO ESTIMATION FROM BEAM-SWEPT SYNCHRONIZATION
SIGNAL UNDER STRONG INTER-CELL INTERFERENCE
Bowen Li1,2,3, Junting Chen1,2 and Nikolaos Pappas3
1School of Science and Engineering and 2Shenzhen Future Network of Intelligence Institute,
The Chinese University of Hong Kong, Shenzhen, China
3Department of ... | JOINT CHANNEL AND CFO ESTIMATION FROM BEAM-SWEPT SYNCHRONIZATION SIGNAL UNDER STRONG INTER-CELL INTERFERENCE Bowen Li1,2,3, Junting Chen1,2 and Nikolaos Pappas3 1 2Shenzhen Future Network of Intelligence Institute, The Chinese 3 ̈oping University, Link ̈oping, Sweden ABSTRACT Complete awareness of the wireless environm... |
2510.14801 | MNRAS 000, 1–7 (2025)
Preprint 17 October 2025
Compiled using MNRAS LATEX style file v3.3
A new photometric ephemeris for the 2M1510 AB double brown dwarf
eclipsing binary system
Seb T. Millward1★, Vedad Kunovac1,2
1Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
2Centre for Exoplanets and Habitabil... | MNRAS 000, 1-7 (2025) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.3 A new photometric ephemeris for the 2M1510 AB double brown dwarf eclipsing binary system Seb T. Millward1★, Vedad Kunovac1,2 1 4 7AL, UK 2Centre for Exoplanets and Habitability, 4 7AL, UK Accepted XXX. Received YYY; in original fo... |
2510.14795 | ON THE FINITENESS OF LOG SURFACES
DANIIL SEREBRENNIKOV
Abstract. We prove that a log surface has only finitely many weakly log canonical projective
models with klt singularities up to log isomorphism, by reducing the problem to the boundedness
of their polarization.
Contents
1.
Introduction
1
2.
Preliminary
2
3.
Isotri... | ON THE FINITENESS OF LOG SURFACES DANIIL SEREBRENNIKOV Abstract. We prove that a log surface has only finitely many weakly log canonical projective models with klt singularities up to log isomorphism, by reducing the problem to the boundedness of their polarization. Contents 1. Introduction 1 2. Preliminary 2 3. Isotri... |
2510.14796 | NOVIKOV COHOMOLOGY, FINITE DOMINATION, AND
COHOMOLOGICAL DIMENSION
SAM P. FISHER
Abstract. We introduce the Σ∗-invariant of a group of finite type, which
is defined to be the subset of non-zero characters χ ∈H1(G; R) with vanish-
ing associated top-dimensional Novikov cohomology. We prove an analogue of
Sikorav’s Theor... | NOVIKOV COHOMOLOGY, FINITE DOMINATION, AND COHOMOLOGICAL DIMENSION SAM P. FISHER Abstract. We introduce the Σ∗-invariant of a group of finite type, which is defined to be the subset of non-zero characters χ ∈H1(G; R) with vanishing associated top-dimensional Novikov cohomology. We prove an analogue of Sikorav's Theorem... |
2510.14798 | Balls and Bins and the Infinite Process with Random Deletions
Petra Berenbrink∗
Tom Friedetzky†
Peter Kling‡
Lars Nagel§
Abstract
We consider an infinite balls-into-bins process with deletions where in each discrete step t
a coin is tossed as to whether, with probability β(t) ∈(0, 1), a new ball is allocated using the
... | Balls and Bins and the Infinite Process with Random Deletions Petra Berenbrink∗ Tom Friedetzky† Peter Kling‡ Lars Nagel§ Abstract We consider an infinite balls-into-bins process with deletions where in each discrete step t a coin is tossed as to whether, with probability β(t) ∈(0, 1), a new ball is allocated using the ... |
2510.14793 | Reflections of quantum educators on strategies to diversify the second quantum
revolution
Apekshya Ghimire* and Chandralekha Singh
Department of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA, 15260
*apg61@pitt.edu
Introduction
The second quantum revolution focuses on quantum ... | Reflections of quantum educators on strategies to diversify the second quantum revolution Apekshya Ghimire* and Chandralekha Singh 3941 O'Hara St, Pittsburgh, PA, 15260 * Introduction The second quantum revolution focuses on quantum information science and technology (QIST). It promises to bring about unprecedented imp... |
2510.14797 | Grain volume distribution alters the critical phenomena in complex granular systems
Teng Man,1, ∗Yimin Lu,2 Zhongrong Wang,3 Herbert Huppert,4 Alessio Zaccone,5 and Honglei Sun1, †
1College of Civil Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China
2Department of Civil, Environmental, and... | Grain volume distribution alters the critical phenomena in complex granular systems Teng Man,1, ∗Yimin Lu,2 Zhongrong Wang,3 Herbert Huppert,4 Alessio Zaccone,5 and Honglei Sun1, † 1 310023, China 2 79409, United States 3 3001, Australia 4 's College, 's Parade, Cambridge CB2 1ST, United Kingdom 5 "A. Pontremoli," 16, ... |
2510.14794 | 1
Bridging Theory and Practice in
Reconfigurable Fluid Antenna Systems
Halvin Yang, Member, IEEE, Yizhe Zhao, Member, IEEE, Kai-Kit Wong, Fellow, IEEE,
Hsiao-Hwa Chen, Life Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE
Abstract—Fluid antennas, including those based on liquid, me-
chanical, and pixel-based technologi... | 1 Bridging Theory and Practice in Reconfigurable Fluid Antenna Systems Halvin Yang, Member, IEEE, Yizhe Zhao, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Hsiao-Hwa Chen, Life Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE Abstract-Fluid antennas, including those based on liquid, mechanical, and pixel-based technologies... |
2510.14790 | ACTIVE JAMMER LOCALIZATION VIA ACQUISITION-AWARE PATH PLANNING
Luis Gonz´alez-Gudi˜no1, Mariona Jaramillo-Civill2, Pau Closas2, Tales Imbiriba1
1Dept. of Computer Science, University of Massachusetts Boston, Boston, MA, USA
2Dept. of Electrical & Computer Engineering, Northeastern University, Boston, MA, USA
ABSTRACT
W... | ACTIVE JAMMER LOCALIZATION VIA ACQUISITION-AWARE PATH PLANNING Luis Gonz ́alez-Gudi ̃no1, Mariona Jaramillo-Civill2, Pau Closas2, Tales Imbiriba1 1Dept. of Computer Science, 2Dept. of Electrical & Computer Engineering, Northeastern University, Boston, MA, USA ABSTRACT We propose an active jammer localization framework ... |
2510.14791 | MNRAS 000, 1–15 (2025)
Preprint 17 October 2025
Compiled using MNRAS LATEX style file v3.3
Stellar population astrophysics (SPA) with the TNG. The Phosphorus
abundance on the young side of Milky Way★
Mingjie Jian (简明杰) ,1† Xiaoting Fu (符晓婷) ,2,3 Valentina D’Orazi,4,5 Angela Bragaglia,3
S. Bijavara Seshashayana,6,7 He Z... | MNRAS 000, 1-15 (2025) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.3 Stellar population astrophysics (SPA) with the TNG. The Phosphorus abundance on the young side of Milky Way★ Mingjie Jian (简明杰) ,1† Xiaoting Fu (符晓婷) ,2,3 Valentina D'Orazi,4,5 Angela Bragaglia,3 S. Bijavara Seshashayana,6,7 He Z... |
2510.14787 | A Human-Vector Susceptible–Infected–Susceptible Model for Analyzing
and Controlling the Spread of Vector-Borne Diseases
Lorenzo Zino, Alessandro Casu, and Alessandro Rizzo
Abstract— We propose an epidemic model for the spread
of vector-borne diseases. The model, which is built extending
the classical susceptible–infect... | A Human-Vector Susceptible-Infected-Susceptible Model for Analyzing and Controlling the Spread of Vector-Borne Diseases Lorenzo Zino, Alessandro Casu, and Alessandro Rizzo Abstract- We propose an epidemic model for the spread of vector-borne diseases. The model, which is built extending the classical susceptible-infect... |
2510.14789 | arXiv:2510.14789v1 [gr-qc] 16 Oct 2025
Quantum confinement of scalar bosons in the Bonnor-Melvin spacetime: uniform magnetic
field and rainbow gravity effects
Omar Mustafa∗
Department of Physics, Eastern Mediterranean University,
99628, G. Magusa, north Cyprus, Mersin 10 - T¨urkiye
Abdullah Guvendi†
Department of Basic ... | 16 Oct 2025 Quantum confinement of scalar bosons in the Bonnor-Melvin spacetime: uniform magnetic field and rainbow gravity effects Omar Mustafa∗ 99628, G. Magusa, north Cyprus, Mersin 10 - T ̈urkiye Abdullah Guvendi† 25050, Erzurum, T ̈urkiye (Dated: October 17, 2025) We present an exact analytical study of Klein-Gord... |
2510.14784 | Topological bands in metals
Yu.
B. Kudasov1, ∗
1Sarov Physics and Technology Institute NRNU ”MEPhI”, 6, str. Dukhov, Sarov, 607186, Russia
In crystalline systems with a superstructure, the electron dispersion can form a nontrivial covering
of the Brillouin zone. It is proved that the number of sheets in this covering a... | Topological bands in metals Yu. B. Kudasov1, ∗ 1Sarov Physics and Technology Institute NRNU "MEPhI", 6, str. Dukhov, Sarov, 607186, Russia In crystalline systems with a superstructure, the electron dispersion can form a nontrivial covering of the Brillouin zone. It is proved that the number of sheets in this covering a... |
2510.14786 | SCALING LIMITS FOR THE CRITICAL LEVEL-SET PERCOLATION
OF THE GAUSSIAN FREE FIELD ON REGULAR TREES
JIˇR´I ˇCERN´Y, RAMON LOCHER
Abstract. We continue the study of the level-set percolation of the discrete Gaussian
free field (GFF) on regular trees in the critical regime, initiated in [ˇCL25]. First, we
derive a sharp as... | SCALING LIMITS FOR THE CRITICAL LEVEL-SET PERCOLATION OF THE GAUSSIAN FREE FIELD ON REGULAR TREES JIˇR ́I ˇCERN ́Y, RAMON LOCHER Abstract. We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in [ˇCL25]. First, we derive a sharp ... |
2510.14785 | Generalized Reduced Jacobian Method
M. El Maghri∗1 and Y. Elboulqe†2
1Department of Mathematics and Computer, Faculty of Sciences A¨ın Chock, Hassan II
University, Casablanca, BP. 5366, Morocco
2Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech,
BP. 2390, Morocco
October 17, 2025... | Generalized Reduced Jacobian Method M. El Maghri∗1 and Y. Elboulqe†2 1 ̈ın Chock, Hassan II University, Casablanca, BP. 5366, Morocco 2 . 2390, Morocco October 17, 2025 Abstract In a recent work, we presented the reduced Jacobian method (RJM) as an extension of Wolfe's reduced gradient method to multicriteria (multiobj... |
2510.14780 | Causal Discovery for Linear DAGs with Dependent
Latent Variables via Higher-order Cumulants
Ming Caia, Penggang Gaoa, Hisayuki Haraa,b
aGraduate School of Informatics, Kyoto University, Yoshida
Konoe-cho, Kyoto, 606-8501, Japan
bInstitute for Liberal Arts and Sciences, Kyoto University, Yoshida
Nihonmatsu-cho, Kyoto, 6... | Causal Discovery for Linear DAGs with Dependent Latent Variables via Higher-order Cumulants Ming Caia, Penggang Gaoa, Hisayuki Haraa,b aGraduate -cho, Kyoto, 606-8501, Japan bInstitute for Liberal Arts and Sciences, Kyoto University, Yoshida Nihonmatsu-cho, Kyoto, 606-8501, Japan Abstract This paper addresses the probl... |
2510.14781 | SciPost Physics Codebases
Submission
ParaToric 1.0-beta: Continuous-time quantum Monte Carlo
for the toric code in a parallel field
Simon M. Linsel⋆and Lode Pollet†
Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),
Ludwig-Maximilians-Universität München, Theresienstr. 37, München D-80333... | SciPost Physics Codebases Submission ParaToric 1.0-beta: Continuous-time quantum Monte Carlo for the toric code in a parallel field Simon M. Linsel⋆and Lode Pollet† (ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37, München D-80333, Germany Munich Center for Quantum Science and Technology (MCQST), Schelli... |
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