diff --git "a/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt" "b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt" @@ -0,0 +1,1786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf,len=1785 +page_content='Spectral analysis and k-spine decomposition of inhomogeneous branching Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Genealogies in fully pushed fronts Emmanuel Schertzer∗ and Julie Tourniaire† January 5, 2023 Abstract We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift −µ and killed upon reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, the particles branch at rate r(x) = (1 + f(x))/2, where f is a compactly supported and non-negative smooth function and the drift µ is chosen in such a way that the system is critical in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi pushed and fully pushed fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Here, we focus on the fully pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We establish a Yaglom law for this branching process and prove that the genealogy of the particles converges to a Brownian Coalescent Point Process using a method of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use spinal decomposition to prove its convergence in the Gromov-weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We also carry the spectral decomposition of a differential operator related to the BBM to determine the invariant measure of the spine as well as its mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Contents 1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1 2 Outline of the proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 30 7 Convergence of metric spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 36 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 The model and assumptions We consider a dyadic branching Brownian motion (Xt)t>0 (BBM) with killing at 0, negative drift −µ and position-dependent branching rate r(x) = 1 2f(x) + 1 2, (1) for some function f : [0, +∞) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We assume that f satisfies the following assumptions: ∗Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria †Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='01697v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='PR] 4 Jan 2023 (A1) the function f is non-negative, continuously differentiable and compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (A2) the support of f is included in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by Nt the set of particles in the system at time t and for all v ∈ Nt, we denote by xv = xv(t) the position of the particle v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Furthermore, we write Zt for the number of particles in the system at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write Px for the law of the process initiated from a point x ≥ 0 and Ex for the corresponding expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We aim at choosing µ in such a way that the number of particles in the system stays roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Fix L > 1 and consider the BBM (XL t )t>0 with branching rate r(x), drift −µ and killed at 0 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Denote by N L t the set of particles in this system at time t and define ZL t = |N L t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By a slight abuse of notations, we will also denote by xv the positions of the particles in the BBM XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (t, x, y) �→ pt(x, y) be the fundamental solution of the linear equation � ∂tu(t, y) = 1 2∂yyu(t, y) + µ∂yu(t, y) + r(y)u(t, y) u(t, 0) = u(t, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (A) We say that pt ≡ pL t is the density of particles in XL in the sense that for any measurable set B ⊂ [0, L], the expected number of particles in B at time t starting from a single particle at x is given by � B pt(x, y)dy [Law18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now define gt(x, y) := eµ(y−x)e µ2−1 2 tpt(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (2) A direct computation shows that gt is the fundamental solution of the self-adjoint PDE � ∂tu(t, y) = 1 2∂yyu(t, y) + 1 2f(y)u(t, y) u(t, 0) = u(t, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (B) Let λ1 = λ1(L) be the maximal eigenvalue [Zet12, Chapter 4] of the Sturm–Liouville problem 1 2v′′(x) + 1 2f(x)v(x) = λ1v(x), (SLP) with boundary conditions v(0) = v(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (BC) It is known that λ1 is an increasing function of L [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1] and that it converges to a finite limit λ∞ 1 ∈ (−∞, +∞) as L → ∞ [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We now choose µ in such a way that the expected number of particles is neither increasing nor decreasing exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to (2), we expect that for large t pt(x, y) ≈ eµ(y−x)e µ2−1 2 teλ∞ 1 t v1(x)v1(y) ||v1||2 , where v1 denotes an eigenfunction associated to λ1 for the Sturm-Liouville problem (SLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 1 (Critical regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is in the critical regime iff µ = � 1 + 2λ∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (3) Pushed and pulled waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The next definitions are motivated by recent numerical simulations and heuristics [BHK18, BHK21] for the noisy F-KPP equation with Allee effect ut = 1 2uxx + u(1 − u)(1 + Bu) + � u N η, where B > 0, N is a large demographic parameter and η is a space-time white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [Tou21] for more details and [EP22] for recent rigorous results on the bistable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2 Definition 2 (Pulled, semi pushed, fully pushed regimes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the BBM (Xt) in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define β := � 2λ∞ 1 , and α := µ + β µ − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 = 0, or equivalently α = 1, the BBM is said to be pulled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 ∈ (0, 1/16) or equivalently α ∈ (1, 2) ⇐⇒ µ > 3β, the BBM is said to be semi pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 > 1/16 or equivalently α > 2 ⇐⇒ µ < 3β, (Hfp) the BBM is said to be fully pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that the BBM is pushed if it is either semi or fully pushed, that is when λ∞ 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (Hp) Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ε > 0 and consider the BBM with inhomogeneous branching rate rε(x) = 1 2 + εf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3], for any function f satisfying (A1), there exists 0 < ε1 < ε2 such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is pulled for all ε ∈ (0, ε1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is semi-pushed for all ε ∈ (ε1, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is fully-pushed for all ε > ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It is conjectured that up to rescaling, the size and the genealogy at large time is undistinguishable from the ones of a continuous-state branching process (CSBP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pulled regime, the population size should converge to a Neveu’s continuous-state branching process and the genealogy of the BBM to the Bolthausen–Sznitman coalescent (see [BBS13] in the case f ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the semi pushed regime, the population size should converge to an α-stable CSBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pre- vious example, this has been proved only in the case where f = 1[0,1] [Tou21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Therein, it is also conjectured that the genealogy should converge to a time-changed Beta(α, 2 − α) coalescent [Pit99, Sag99, BBC+05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the fully pushed regime, the rescaled population size should converge to a Feller diffusion and the genealogy should be undistinguishable from the genealogy of a large critical Galton-Watson process with finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This is the content of the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 Comparaison with previous work Branching Brownian motion with inhomogeneous branching rates have received quite a lot of attention in the recent past [HKV20, HHK20, HHKW22, GHK22, FRS22, Tou21, RS21, LS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The general approach always relies on a spinal decomposition of the BBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Roughly speaking, the spine is constructed by conditioning a typical particle to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This conditioning is achieved by a Doob-h transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In our setting, the harmonic function is approximated by h(x) ≈ eµxv1(x) and the resulting h-transform is given by dxt = v′ 1(xt) v1(xt)dt + dBt (5) where Bt is a standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See Section 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A key assumption underlying [Pow19, HKV20, HHK20, HHKW22, GHK22] is that the harmonic function h is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From a technical stand point, we emphasise that this assumption is the one distinguishing our work from the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Indeed, we shall see that v1 decreases exponentially at rate β so that the harmonic function h(x) ∝ e(µ−β)x blows up as x → ∞ since µ = � 1 + β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Due to the explosion of the harmonic function, many of the previously developed technics break down in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3 At first sight, this assumption may only seem technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' However, it is the key assumption which makes possible a transition from the semi to the fully pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us consider the spine dynamics (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pushed regime, the invariant distribution for the spine is given by Π(x) = v2 1(x) ||v1||2 , so that h(x)Π(x) ∝ e(µ−3β)x as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It then becomes clear from Definition 2 that, in the fully pushed regime (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' semi pushed), the harmonic function is integrable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' non-integrable) with respect to the invariant measure of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, relaxing the assumption under which h is bounded is crucial for understanding the transition between these two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This generalisation raises interesting technical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A large fraction of the present work (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1) is dedicated to estimating the speed of convergence to the invariant measure of the spine in the pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, we use a Sturm-Liouville approach in order to understand the spectral decomposition of the differential operator (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The difficulty arises from the fact that the negative part of the spectrum of the Sturm-Liouville problem (SLP) becomes continuous as L → ∞ (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We believe that this contribution is relevant for understanding not only the fully pushed case, but also the semi pushed case which will be the subject of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This continuous spectrum already appeared in the study of homogeneous BBM [BBS13], but the spectral analysis of (SLP) is quite straightforward in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Indeed, when f ≡ 0, the spectral decomposition of (SLP) is given by λi = − iπ2 2L2 and vi(x) = sin � iπx L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' When f is not trivial, the spectrum is not explicit and we use the Prüfer transformation to derive the required estimates on the (vi) and the (λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, one of the main contribution of the present work is the description of the genealogy spanned by the population at a large time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Beyond our Kolmogorov estimate and the Yaglom law reminiscent of [Pow19, HKV20, HHK20, HHKW22, GHK22], we use k-spine decompositions [HR17] and moment methods developed in [FRS22] to prove convergence of the genealogy in the Gromov weak topology to a continuum random metric space known as the Brownian Coalescent Point Process [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This approach will be further explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider f = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It was calculated in [Tou21] that the negative part of the spectrum of (SLP) with boundary conditions (BC) consists in the solutions of tan( √ 9 − 2λ) √ 9 − 2λ = −tan( √ −2λ(L − 1)) √ −2λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (6) The solutions of this equation are plotted on Figure 1 4 Figure 1: Negative spectrum of the Sturm–Liouville problem (SLP) with boundary conditions (BC) for f defined as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 and different values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The blue line corresponds to RHS of (6) and the red line to the LHS of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 Main results Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let v1 be the eigenfunction associated to the eigenvalue λ1 for the Sturm-Liouville problem (SLP) with boundary conditions (BC), normalised in such a way that v1(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Under (Hp), v1 converges to a positive limiting function v∞ 1 as L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Further, if in addition (Hfp) holds, then � R+ eµx(v∞ 1 )3(x)dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To see why the latter proposition may hold true, recall that f ≡ 0 on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Hence, on this interval, the problem reduces to 1 2v′′ 1 (x) = λv1(x), x ∈ [1, L], v1(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we impose the condition v1(1) = 1, a direct computation shows that v1(x) = sinh(√2λ1(L−x)) sinh(√2λ1(L−1)) on [1, L] so that, for all x ∈ [1, ∞), v1(x) → v∞ 1 (x) = e−β(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The integrability condition then holds under the extra assumption (Hfp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, we define h∞(x) := ceµxv∞ 1 (x), and ˜h∞(x) := ˜ce−µxv∞ 1 (x), (7) where ˜c := �� ∞ 0 e−µxv∞ 1 (x)dx �−1 and c := (˜c∥v∞ 1 ∥2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The constants c and ˜c are thought as Perron-Frobenius renormalisation constants (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' [AN72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='185]), in the sense that h∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ˜h∞) is a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) eigenfunction associated to the maximal eigenvalue of the differential operator Lu = 1 2∂xxu + µ∂xu + r(x)u, (8) 5 L=10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 =50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 L=100normalised in such a way that � ∞ 0 ˜h∞(x)dx = 1 and � ∞ 0 h∞(x)˜h∞(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From this perspective, the function ˜h∞ should correspond to the stable configuration of the system and the function h∞ to the reproductive values of the individuals as a function of their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 1 (Kolmogorov estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As N → ∞, for all x, t > 0, NPx (ZtN > 0) → 2 Σ2th∞(x), where Σ2 2 := � R+ r(z)(h∞(z))2˜h∞(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This theorem is the continuous analogous of Kolmogorov estimate for multi-type Galton-Watson process [AN72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='187]: in our case, the variance in the offspring distribution is given by Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We now turn to the description of the genealogy and the Yaglom law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Intuitively, the next result states that the genealogy is asymptotically identical to the one of a critical Galton Watson [Lam18, HJR20, Joh19], whereas the marks are assigned independently according to ˜h∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now give a more precise description of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From now on, we condition on the event {ZtN > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' , vk) be k individuals chosen uniformly at random from NtN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Denote by dT (vi, vj) the time to the most recent common ancestor of vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let xvi be the position of the ith individual at time tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let U be a uniform r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' on [0, t] and θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define U θ such that ∀s ≤ t, P(U θ ≤ s) := (1 + θ)P(U ≤ s) 1 + θP(U ≤ s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (9) Let (U θ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [k]) be k i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' copies of U θ and set ∀1 ≤ i < j ≤ k, U θ i,j = U θ j,i := max{U θ k : k ∈ {i, · · · , j − 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, define the random distance matrix (Hi,j) := (Hi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k]) such that for every bounded and continuous function ϕ : Rk2 → R, E � ϕ � (Hi,j) �� = k � ∞ 0 1 (1 + θ)2 � θ 1 + θ �k−1 E � ϕ � (U θ i,j) �� dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (10) Finally, (Wi) := (Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [k]) will denote a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' copies of a random variable W with law ˜h∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 2 (Yaglom law and limiting genealogies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Start with a single particle at x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Conditional on ZtN > 0, as N → ∞, (i) we have ZtN N → Σ2t 2 E, in distribution, where E is a standard exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) �� dT (vi,vj) N � , � xvi �� converges to the distribution of � (Hσi,σj), (Wσi) � where σ is a random uniform permutation {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' , k} and σ, (Hi,j) and (Wi) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The random distance matrix (Hi,j) is the one obtained from a critical Galton Watson with finite second moment conditioned on surviving up to a large time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [Lam18, HJR20, Joh19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 Notation Given two sequences of positive real numbers (aN) and (bN), we write aN ≪ bn if aN/bN → 0 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write aN ≲ bN if aN/bN is bounded in absolute value by a positive constant and aN ≍ bN if aN ≲ bN and bN ≲ aN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write O(·) to refer to a quantity bounded in absolute value by a constant times what the quantity inside the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Unless otherwise specified, these constants only depend on λ∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 6 2 Outline of the proof Our approach relies on a method of moments devised in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To illustrate the approach, let us first think about the Yaglom law of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To prove this result, one needs to show that the moments of ZtN/N converge to the moments of an exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It turns out that this approach can be extended to genealogies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1, following the approach in [DGP11], we encode the genealogy at time tN as a random marked measured metric space (mmm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In turn, the moments of a random mmm are obtained by biasing the population by its kth moment and then picking k individuals uniformly at random (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2, we introduce a limiting random mmm called the marked Coalescent Point Process (CPP) which corresponds to the limiting genealogy of a critical Galton-Watson process [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The remainder of the section is dedicated to the sketch of the proof for the convergence of the moments of our BBM to the moments of the marked CPP using the spinal decomposition introduced in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 Marked Metric Spaces Let (E, dE) be a fixed complete separable metric space, referred to as the mark space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In our application, E = (0, ∞) is endowed with the usual distance on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A marked metric measure space (mmm- space for short) is a triple [X, d, µ], where (X, d) is a complete separable metric space, and µ is a finite measure on X × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To define a topology on the set of mmm-spaces, for each k ≥ 1, we consider the map Rk : � (X × E)k → Rk2 + × Ek � (xi, ui);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ≤ k � �→ � d(xi, xj), ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ≤ k � that maps k points in X×E to the matrix of pairwise distances and marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by νk,X = µ⊗k◦R−1 k , the marked distance matrix distribution of [X, d, µ], which is the pushforward of µ⊗k by the map Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Note that µ is not necessarily a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let k ≥ 1 and consider a measurable bounded test function ϕ: Rk2 + × Ek → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' One can define a functional Φ � X, d, µ � = ⟨νk,X, ϕ⟩ = � X×E ϕ � d(vi, vj), xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k] � k � i=1 µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (11) Functionals of the previous form are called polynomials, and the set of all polynomials, obtained by varying k and ϕ, is denoted by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the moment of [X, d, µ] associated to Φ is defined as E(Φ � X, d, µ � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ϕ be of the form ϕ � di,j, xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k] � = � i,j ψi,j(di,j) � i ϕi(xi) where ψi,j, ϕi are bounded measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that Φ(X, d, µ) is a product polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by ˜Π the set of product polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The moments of a random mmm can be rewritten as E � Φ � X, d, µ �� = E(|X|k) × 1 E(|X|k)E � |X|kϕ(d(vi, vj), Xvi, i ̸= j ∈ [k]) � , where (vi, Xvi) are k points sampled uniformly at random with their marks and |X| = µ(X ×E) is thought as the total size of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, the moments of a random mmm are obtained by biasing the population size by its kth moment and then picking k individuals uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The marked Gromov-weak (MGW) topology is the topology on mmm-spaces induced by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A random mmm-space is a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' with values in M – the set of (equivalence classes of) mmm-spaces – endowed with the Gromov-weak topology and the associated Borel σ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the marked Gromov-weak (MGW) topology is identical to the topology induced by the product polynomials ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 7 Many properties of the marked Gromov-weak topology are derived in [DGP11] under the further assumption that µ is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In particular, the following result shows that Π forms a convergence determining class only when the limit satisfies a moment condition, which is a well-known criterion for a real variable to be identified by its moments, see for instance [Dur19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This result was already stated for metric measure spaces without marks in [DG19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='7] and was proved in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Suppose that [X, d, µ] is a random mmm-space verifying lim sup p→∞ E[µ(X × E)p]1/p p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (12) Then, for a sequence [Xn, dn, µn] of random mmm-spaces to converge in distribution for the marked Gromov-weak topology to [X, d, µ] it is sufficient that lim n→∞ E � Φ � Xn, dn, µn �� = E � Φ � X, d, µ �� for all Φ ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 Marked Brownian Coalescent Point Process (CPP) Let T > 0 and m be a measure on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Assume that |m| := m(R+) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider P a PPP � dx x2 ⊗ dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define YT = inf {y : (y, t) ∈ P, t ≥ T} , and dT (x, y) = sup{t : (z, t) ∈ P and x ≤ z ≤ y}, 0 < x < y < YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The marked Brownian Coalescent Point Process (CPP) is defined as MT := � [0, YT ], dT , dv ⊗ m(dx) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This object is a natural extension of the standard Brownian CPP [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A direct computation shows that YT m(R+) (which can be thought as the population size at time T) is distributed as an exponential random variable with mean T|m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we think as the CPP as the genealogy of critical branching processes, this consistent with Yaglom’s law for critical branching processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Figure 2: Simulation of the unmarked Brownian CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' On the RHS, a vertical line of height x at location s represents an atom (x, s) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' On the LHS, the tree corresponding to the right CPP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' the distance dT is the tree distance of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 8 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider an arbitrary product polynomial of the form ∀M = [X, d, µ], Ψ(M) := � � i,j ψi,j(dT (vi, vj)) � i ϕi(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then E [Ψ(MT )] = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='T KE �� i,j ψi,j(Uσi,σj) � �� i � m(dx)ϕi(x) �K , where (Ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K − 1]) is a vector of uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables on [0, T] and for i < j Ui,j = Uj,i = max{Uk : k = i, · · · , j − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof is identical to Proposition 4 in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 (Sampling from the CPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let k ∈ N and sample k points (v1, · · · , vk) uniformly at random from the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let xv1 · · · , xvk be the corresponding types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then ((d(vi, vj))i,j , (xvi)i) is identical in law to � (Hσi,σj), (Wσi) � where (Hi,j) is defined as in Theorem 2 (ii) and (Wi) are independent random variables with law m |m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof is identical to the one in the case of the unmarked CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [BFRS22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 Convergence of mmm Fix t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that Nt refers the set of particles alive at time t in the BBM X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Set µt := � v∈Nt δv,xv, ∀v, v′ ∈ Nt, d(v, v′) = � t − |v ∧ v′| � , where v∧v′ denotes the MRCA of v and v′, and |v| denotes the generation of vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let Mt = [Nt, d, µt] be the resulting random mmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, set ¯µt := 1 N � v∈NtN δv,xv, ∀v, v′ ∈ NtN, ¯d(v, v′) = � t − 1 N |v ∧ v′| � , and define the rescaled metric space ¯ Mt := [NtN, ¯d, ¯µt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The main idea underlying Theorem 2 is to prove the convergence of ¯ Mt to a limiting CPP whose size and sampling structure coincides with (i) and (ii) in Theorem 2 — See Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Conditional on the event {ZtN > 0}, ( ¯ Mt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' N ∈ N) converges in distribution for the Gromov weak topology to a marked Brownian CPP with parameters (t, Σ2 2 ˜h∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof of the theorem relies on a cut-off procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Namely, let L := 1 µ − β log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (13) Recall that XL refers to the BMM killed at 0 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let µL t (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯µL t ) be the empirical measure obtained by replacing N by N L in the definition of µt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯µt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let M L t be the mmm obtained from XL, that is M L t = [N L t , d, µL t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t is defined analogously to ¯ Mt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' accelerating time by N and rescaling the empirical measure by 1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For our choice of L, we will show that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t converges to the limit described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t and ¯ Mt converge to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The choice for L will be motivated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We start by motivating the fact that ¯ M L t converges to the desired limit using a spinal decomposition introduced in [FRS22] in a discrete time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 The K-spine Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine is the stochastic process on [0, L] with generator 1 2∂xxu + v′ 1(x) v1(x)∂xu, u(0) = u(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, qt(x, y) will denote the probability kernel of the 1-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We can also directly determine the invariant distribution of the spine Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine has a unique invariant measure given by Π(dx) = � v1(x) ||v1|| �2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', UK−1) be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables uniformly distributed on [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define ∀ 1 ≤ i < j ≤ K − 1, d(i, j) = d(j, i) = max{Ui, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', Uj−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (14) Let T be the unique planar ultrametric tree of depth t with K leaves labeled by {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', K} such that the tree distance between the leaves is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We then assign marks on this tree such that, on each branch of the tree, marks evolve according to the 1-spine (on [0, L]) and branch into independent particles at the branching points of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [FRS22] for a more formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' t 0 U1 U2 V1 V2 V3 t 0 x x0 ζv1 qt(x0, ·) qt−U1(ζv1, ·) Figure 3: K-spine with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Left panel: planar tree T generated from 2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' uniform random variables (U1, U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Right panel: branching 1-spines running along the branches of the tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The resulting planar marked ultrametric tree will be referred to as the K-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by QK,t x the distribution of the K-spine rooted at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The t superscript refers to the depth of the underlying genealogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Note that QK,t x has an implicit dependence on N by our choice of L – see (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To ease the notation, this dependence will be dropped in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, B will denote the set consisting of the K − 1 branching points of the K-spine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' L will denote the set consisting of the K leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by ζv the mark (or the position) of the spine at a node v ∈ B ∪ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For v ∈ B, |v| will denote the time component of the branching point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, (Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) is the enumeration of the leaves from left to right in the K-spine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', Vi is the leaf with label i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will also need the accelerated version of the K-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 5 (Accelerated K-spine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the 1-spine accelerated by N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' the transition kernel of the 1-spine is now given by qtN(x, y) ≡ qL tN(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote this kernel by ¯qt(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the same planar structure as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', the depth is t and the distance between points at time t is given by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by ¯QK,t x the distribution of the K-spine obtained by running accelerated spines along the branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For any vertex v in the K-spine, ¯ζv will denote the mark of the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 10 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='7 (Rescaled many-to-few).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions and define ∀M = [X, d, µ], Ψ(M) = � � i,j 1(vi ̸= vj)ψi,j(d(vi, vj)) � i ϕ(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define h(t, x) := ce(λ∞ 1 −λ1)teµxv1(x) where c is the renormalisation constant in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then Ex � Ψ( ¯ M L t ) � = 1 N K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' h(0, x) tK−1 ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ϕi(¯ζVσi ) � , where ¯∆ := � v∈B(U) r(¯ζv)h(|v|N, ¯ζv) � v∈L(U) 1 h(tN, ¯ζv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The second crucial result is the following convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ( ˜ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Assume further that the ˜ϕi’s are compactly supported in (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As N → ∞, ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi )h∞(¯ζVσi ) � → �Σ2 2 �K−1 E �� i,j ψi,j(Uσi,σj) � � i � R+ ˜ϕi(x)Π∞(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now give a brief heuristics underlying the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By definition ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi )h∞(¯ζVσi ) � = ¯QK,t x �� v∈B r(¯ζv)h(|v|N, ¯ζv) � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi ) h∞(¯ζVσi ) h(tN, ¯ζVσi ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The branching structure for the K-spine is binary a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' whereas the spines running along the branches are accelerated by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that the invariant measure for the 1-spine is Π ≈ Π∞ as L → ∞ (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Moreover, we will show later on that under (Hfp), h(tN, x) ≈ h∞(x) for N large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It is now reasonable to believe that, provided enough mixing, the RHS can be approximated by �� L 0 h∞(x)r(x)Π∞(dx) �K−1 E �� i,j ψi,j(Uσi,σj) � � i � ˜ϕi(x)Π∞(dx), assuming the values of the spine at the branching points and the leaves converge to a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables with law Π∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This yields the content of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The previous argument relies on a k-mixing property of the 1-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This analysis will be carried out in Section 4 using Sturm–Liouville theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 Limiting moments Let us now demonstrate the importance of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ∀M = [X, d, µ], ˜Ψ(M) := � X �� i,j ψi,j(d(vi, vj)) � i ˜ϕi(xi)h∞(xi) µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 11 for any bounded measurable functions ˜ϕi, ψi,j such that the ˜ϕi are compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From the many-to-few formulae, our result entails Ex � ˜Ψ( ¯ M L t ) � = Ex � � � i,j ψi,j( ¯d(vi, vj)) � i ˜ϕi(xi)h∞(xi) ¯µL t (dvi ⊗ dxi) � ≈ Ex � � � i,j 1(vi ̸= vj)ψi,j( ¯d(vi, vj)) � i ˜ϕi(xi)h∞(xi) ¯µL t (dvi ⊗ dxi) � ≈ 2h∞(x) NΣ2t × K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' � tΣ2 2 �K E �� i,j ψi,j(Uσi,σj) � � i � R+ ˜ϕi(x)Π∞(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us formally take ψi,j ≡ 1 and ˜ϕi ≡ 1/h∞ in the previous expression (note that this is problematic since ˜ϕi is neither bounded nor compactly supported, see Challenge 4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then for large N, Ex �� 1 N ZL tN �K� ≈ 2h∞(x) NΣ2t × K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' � tΣ2 2 �K � �� � exponential moments × � ∞ 0 ˜h∞(x)dx � �� � =1 , where we used the fact that ˜h∞ = Π∞/h∞ under our Perron–Frobenius renormalisation (see (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Now, the RHS coincides with the moments of a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' with law � 1 − 2h∞(x) NΣ2t � δ0(dx) + 2h∞(x) NΣ2t exp � − 2x Σ2t � 2dx Σ2t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we identify the Dirac measure at 0 with the extinction probability, this suggests the Kolmogorov estimate and the Yaglom law exposed in Theorem 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Further, if we replace ˜ϕi = ϕi/h∞ in Theorem 4 (again a problematic step), the previous estimates entail Ex � Ψ( ¯ M L t ) ��ZL tN > 0 � ≈ K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='E �� i,j ψi,j(Uσi,σj) � � i � R+ ϕi(x) � tΣ2 2 � ˜h∞(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' where Ψ(M) is now an arbitrary polynomial of the form Ψ(M) := � � i,j ψi,j(d(vi, vj)) � i ϕi(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4, this coincides with the moments of the Brownian CPP described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The previous computation only suggests that the probability for the population to be o( 1 N ) is given by the Kolmogorov estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Intuitively, the Dirac mass above corresponds to a population whose size becomes invisible at the limit after rescaling the population by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It thus remains to show that if the population is small compared to N then it must be extinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This will be carried out in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Going from Theorem 4 to the convergence of the M L tN requires to use test functions ex- ploding at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To overcome this technical difficultly, we will impose an extra thinning of the population by killing all the particles close to the boundaries at time tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This final technical step will be carried in Section 7 using some general property of the Gromov-weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6 Choosing the cutoff L We now motivate our choice for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to the previous arguments, we want to choose L large enough such that (i) The particles do not reach L with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This will imply that ¯ M L t and ¯ Mt coincide with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) The 1-spine reaches equilibrium in a time o(N) regardless of its initial position on [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This is needed to justify the calculations of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 12 (i) Hitting the right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let E be a compact set in the vicinity of the boundary L (say [L − 2, L − 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall from the discussion after Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 that for x ≥ 1, h∞(x) = c eµxv∞ 1 (x) ≈ c eβe(µ−β)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A direct application of the many-to-few lemma with K = 1 (many-to-one) and ϕ1(x) = 1x∈E implies that Ex � � � v∈N L tN 1xv∈E � � = � E h(0, x) h(tN, y)qtN(x, y)dy ≈ � E h∞(x) h∞(y) Π∞(dy) ≈ h∞(x)O( =N−α � �� � e− µ+β µ−β log(N)), where the last approximation holds under the assumption that E is a compact set close to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Integrating on [0, tN], this yields that the occupation time of the set E on the time interval [0, tN] is O(N 1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that the probability of survival is of order 1/N so that the occupation time of the conditioned process is O(N 2−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recalling that α > 2 under (Hfp), this yields the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) Mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall from the discussion after Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 that v1(x) ≈ e−β(x−1) for x ≥ 1 and L large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, the 1-spine (see Definition 4) is well approximated by the diffusion dzt = −βdt + dwt for zt ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A good proxy for the mixing time is the first returning time at 1 which is of the order log(N) = o(N) for every x ∈ [1, L], as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A more refined analysis will be carried in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3 The many-to-few theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 The general case In this section, we consider a general BBM killed at the boundary of a regular open domain Ω ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Unless otherwise specified, we used the same notation as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will assume that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The generator of a single particle is given by a differential operator Lf(x) = 1 2 � i,j aij(x)∂xixjf(x) + � i bi(x)∂xif(x), x ∈ Ω, (15) f(x) = 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We assume that (aij) is uniformly elliptic, which means that there exists a constant θ > 0 such that for all ξ ∈ Rd and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' x ∈ Ω, �d i,j=1 aij(x)ξi, ξj ≥ θ∥ξ∥2 (see [Eva10, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In addition, we assume that ai,j ∈ C1(Ω) and supx∈Ω |bi(x)| < ∞ for all 1 ≤ i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A particle at location x branches into two particles at rate r(x) (we only consider binary branching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by Nt the set of particles alive at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For any pair of particles v, w ∈ Nt, d(u, w) will denote the time to their most recent common ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, we define the random mmm space Mt = [Nt, d, µt], where µt = � v∈Nt δv,xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that a function h(x) is harmonic if and only h satisfies the Dirichlet problem Bh(x) := Lh(x) + r(x)h(x) = 0, for every x ∈ Ω, h(x) = 0, for x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine whose generator is given by the Doob h-transform of the differential operator B B(hf) h (x) = 1 2 � i,j aij(x) � ∂xixjf(x) + ∂xih h ∂xjf(x) + ∂xjh h ∂xif(x) � + � i bi(x)∂xif(x), x ∈ Ω, f(x) = 0, x ∈ ∂Ω, where the first equality is a direct consequence of the fact that h is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by qt(x, y) the transition probability at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The K-spine distribution QK,t x is defined analogously to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 13 Our many-to-few formulae rely on a uniform planarisation of the BBM that we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' At every time t > 0, every particle is now endowed with two marks (xv, pv) where pv ∈ ∪n∈N{0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As before, xv denotes the position of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The planarisation marks pv are assigned recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We mark the root with ∅ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' At every branching point v, we distribute the marks (pv, 0) and (pv, 1) uniformly among the two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (pv, 0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (pv, 1)) is said to the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) child of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The mark pv does not vary between two branching points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', pv1 = pv2 if the trajectory connecting v1 and v2 does not encounter any branching points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let N pl t be the the set particles at time t in the planar BBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For every K-uplet v1, v2 · · · , vK in N pl t , let U(⃗v) = [X(⃗v), d(⃗v), µ(⃗v)] be the planar (ultra-)mmm space induced by this set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The space consists of the set of vertices ancestral to some vertex in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In particular, U(⃗v) is binary and made of K leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the measure is given by the counting measure on the leaves µ(⃗v) = � ⃗v δv,(xv,pv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now introduce some definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider a planar ultra-mmm U made of finitely many leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We define σ(U) to be the diameter U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that U has no simultaneous branching iff for every distinct pair of leaves (v1, v2), (v3, v4), we have d(v1, v2) ̸= d(v3, v4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For such a tree, there exists a unique element v ∈ U such that v = Argmax d(w, ¯w) where the maximum is taken over the pair of leaves in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The individual v is said to be the MRCA of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Moreover, we define T0(U) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' T1(U)) to be the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) subtree attached to the MRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, B(U) will refer to the set of branching points and L(U) will refer to the set of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, xMRCA(U) will be the spatial position of the MRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 (Many-to-one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For every bounded continuous function f Ex � � v∈Nt f(xv) � = � Ω f(y)qt(x, y)h(x) h(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' One can readily check that pt(x, y) := qt(x, y) h(x) h(y) are the fundamental solutions of the same PDE � ∂tu(t, y) = L∗u(t, y), y ∈ Ω, u(t, y) = 0, y ∈ ∂Ω, where L∗ is the adjoint of the differential operator (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let K ∈ N and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define the measure RK,t x on the set of planar mmm so that for every bounded measurable function F, RK,t x (F) := Ex � � � � v1̸=···̸=vK, vi∈N pl t F(U(⃗v)) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then RK,t x (F) = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' h(x)tK−1 QK,t x (∆F) with ∆ = � v∈B(U) r(ζv)h(ζv) � v∈L(U) 1 h(ζv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will show the result by an induction of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The case K = 1 is the many-to-one lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now consider F of the following product form F(U) = f(σ(U))ψ0(T0(U))ψ1(T1(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 14 Then RK,t x (F) = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='Ex � � � v1<···