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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf,len=1354
+page_content='PLANAR EQUILIBRIUM MEASURE PROBLEM  IN THE QUADRATIC FIELDS WITH A POINT CHARGE  SUNG-SOO BYUN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We consider a two-dimensional equilibrium measure problem under the presence of quadratic po-  tentials with a point charge and derive the explicit shape of the associated droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This particularly shows  that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are dou-  bly connected domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (ii) in the critical case, they contain two merging type singular boundary points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (iii)  in the pre-critical case, they consist of two disconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' From the random matrix theory point  of view, our results provide the limiting spectral distribution of the complex and symplectic elliptic Ginibre  ensembles conditioned to have zero eigenvalues, which can also be interpreted as a non-Hermitian extension of  the Marchenko-Pastur law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Introduction and Main results  In this paper, we study a planar equilibrium problem with logarithmic interaction under the influence of  quadratic potentials with a point charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This problem is purely deterministic, but its motivation comes from  the random world, more precisely, from the random matrix theory or the theory of two-dimensional Coulomb  gases in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' To be more concrete, for given points ζ = (ζj)N  j=1 ∈ CN of configurations, we consider the  Hamiltonians  HC  N(ζ) :=  �  1≤j<k≤N  log  1  |ζj − ζk|2 + N  N  �  j=1  W(ζj),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1)  HH  N(ζ) :=  �  1≤j<k≤N  log  1  |ζj − ζk|2 +  �  1≤j≤k≤N  log  1  |ζj − ¯ζk|2 + 2N  N  �  j=1  W(ζj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2)  Here W : C → R is a suitable function called the external potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' These are building blocks to define joint  probability distributions  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3)  dPC  N,β(ζ) ∝ e− β  2 HC  N(ζ)  N  �  j=1  dA(ζj),  dPH  N,β(ζ) ∝ e− β  2 HH  N(ζ)  N  �  j=1  dA(ζj),  where dA(ζ) = d2ζ/π is the area measure and β > 0 is the inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Both point processes PC  N,β and  PH  N,β represent two-dimensional Coulomb gas ensembles [38, 55, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In particular, if β = 2, they are also called  determinantal and Pfaffian Coulomb gases respectively due to their special integrable structures, see [26, 28] for  recent reviews on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Furthermore, they have an interpretation as eigenvalues of non-Hermitian random  matrices with unitary and symplectic symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For instance, if W(ζ) = |ζ|2, the ensembles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) corresponds  to the eigenvalues of complex and symplectic Ginibre matrices [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  One of the fundamental questions regarding such point processes is their macroscopic/global behaviours as  N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For the case β = 2, this can be regarded as a problem determining the limiting spectral distribution of  given random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The classical results in this direction include the circular law for the Ginibre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Planar equilibrium measure problem, two-dimensional Coulomb gases, elliptic Ginibre ensemble, con-  ditional point process, conformal mapping method, a non-Hermitian extension of the Marchenko-Pastur law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Sung-Soo Byun was partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-51) and by the Na-  tional Research Foundation of Korea (NRF-2019R1A5A1028324) and by a KIAS Individual Grant (SP083201) via the Center for  Mathematical Challenges at Korea Institute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='00324v1  [math-ph]  1 Jan 2023  2  SUNG-SOO BYUN  As expected from the structure of the Hamiltonians (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2), the macroscopic behaviours of the system  can be effectively described using the logarithmic potential theory [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For this purpose, let us briefly recap some basic notions in the logarithmic potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Given a  compactly supported probability measure µ on C, the weighted logarithmic energy IW [µ] associated with the  potential W is given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4)  IW [µ] :=  �  C2 log  1  |z − w| dµ(z) dµ(w) +  �  C  W dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For a general potential W satisfying suitable conditions, there exists a unique probability measure µW which  minimises IW [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Such a minimiser µW is called the equilibrium measure associated with W and its support  SW := supp(µW ) is called the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Furthermore, if W is C2-smooth in a neighbourhood of SW , it follows  from Frostman’s theorem that µW is absolutely continuous with respect to dA and takes the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5)  dµW (z) = ∆W(z) · 1{z∈SW } dA(z),  where ∆ := ∂ ¯∂ is the quarter of the usual Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In relation with the point processes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), it is well known [31, 15, 41] that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6)  µN,W := 1  N  N  �  j=1  δζj → µW  in the weak star sense of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' From the statistical physics viewpoint, this convergence is quite natural since  the weighted energy IW in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4) corresponds to the continuum limit of the discrete Hamiltonians (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2)  after proper renormalisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (In the case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2), it is required to further assume that W(ζ) = W(¯ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=')  Contrary to the density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5) of the measure µW , there is no general theory on the determination of its  support SW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (See however [60] for a general theory on the regularity and [49] on the connectivity of the droplet  associated with Hele-Shaw type potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') This leads to the following natural question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For a given potential W, what is the precise shape of the associated droplet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In view of the energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4), this is a typical form of an inverse problem in the potential theory  and is called an equilibrium measure problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Beyond the case when W is radially symmetric (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  [59,  Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6]), this problem is highly non-trivial even for some explicit potentials with a simple form, see  [3, 12, 44, 14, 21, 35, 54, 36] for some recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us also stress that such a problem is important not  only because it provides the intrinsic macroscopic behaviours of the point processes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) but also because it  plays the role of the first step to perform the Riemann-Hilbert analysis which gives rise to a more detailed  statistical information (k-point functions) of the point processes, see [12, 13, 17, 18, 44, 45, 46, 47, 51, 53, 56]  for extensive studies in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In this work, we aim to contribute to the equilibrium problems associated  with the potentials (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14) below, which are of particular interest in the context of non-Hermitian  random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For given parameters τ ∈ [0, 1) and c ≥ 0, we consider the potential  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7)  Q(ζ) :=  1  1 − τ 2  �  |ζ|2 − τ Re ζ2�  − 2c log |ζ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  When β = 2, the ensembles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) associated with Q correspond to the distribution of random eigenvalues of the  elliptic Ginibre matrices of size (c+1)N conditioned to have zero eigenvalues with multiplicity cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We mention  that such a model with c > 0 was also studied in the context of Quantum Chromodynamics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7), the logarithmic term can be interpreted as an insertion of a point charge, see [4, 33, 23, 22, 27]  for recent investigations of the models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Such insertion of a point charge has also been  studied in the theory of planar orthogonal polynomials [12, 13, 17, 51, 52, 53, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' On the other hand, the  parameter τ ∈ [0, 1) captures the non-Hermiticity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' To be more precise, the models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) associated  with Q interpolate the complex/symplectic Ginibre ensembles (τ = 0) with the Gaussian Unitary/Symplectic  ensembles (τ = 1) conditioned to have zero eigenvalues, see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6 for further discussion in relation to our  main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  3  For the case c = 0, the terminology “elliptic” comes from the fact that the limiting spectrum Sτ,0 is given  by the ellipse  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8)  Sτ,0 :=  �  (x, y) ∈ R2 :  �  x  1 + τ  �2  +  �  y  1 − τ  �2  ≤ 1  �  ,  which is known as the elliptic law, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [34, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We refer to [50, 5, 8, 57, 20, 19] and references therein for  more about the recent progress on the complex elliptic Ginibre ensembles and [43, 6, 24, 25] for their symplectic  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For the rotationally invariant case when τ = 0, it is easy to show that the associated droplet S0,c  is given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9)  S0,c :=  �  (x, y) ∈ R2 : c ≤ x2 + y2 ≤ 1 + c  �  ,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [59, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6] and [26, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  The primary goal of this work is to determine the precise shape of the droplet associated with the potential  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7) for general τ ∈ [0, 1) and c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For this, we set some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us write  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10)  τc :=  1  1 + 2c  for the critical non-Hermiticity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For τ ∈ (τc, 1), we define  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11)  f(z) ≡ fτ(z) := (1 + τ)(1 + 2c)  2  (1 − az)(z − aτ)2  z(z − a)  ,  a = −  1  �  τ(1 + 2c)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  We are now ready to present our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let Q be given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then the droplet S ≡ Sτ,c = SQ of the equilibrium measure  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12)  dµQ(z) =  1  1 − τ 2 1S(z) dA(z)  is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (Post-critical case) If τ ∈ (0, τc], we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13)  Sτ,c =  �  (x, y) ∈ R2 :  �  x  (1 + τ)√1 + c  �2  +  �  y  (1 − τ)√1 + c  �2  ≤ 1 ,  x2 + y2  (1 − τ 2)c ≥ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (Pre-critical case) If τ ∈ [τc, 1), the droplet Sτ,c is the closure of the interior of the real analytic  Jordan curves given by the image of the unit circle with respect to the map z �→ ±  �  f(z), where f is  given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (a) τ = 1/6 < τc  (b) τ = 1/3 = τc  (c) τ = 1/2 > τc  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The droplet Sτ,c, where c = 1 and a Fekete point configuration with N = 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  2  1  0  1  2  2  1  1  22  1  1  2  2  1  1  22  1  0  1  2  2  1  1  24  SUNG-SOO BYUN  Note that if c = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=', τ = 0), the droplet (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13) corresponds to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We mention that the  post-critical case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 is indeed shown in a more general setup, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2 (Phase transition of the droplet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1, we observe that if c > 0, the topology of  the droplet reveals a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Namely, for the post-critical case when τ < τc, the droplet is a doubly  connected domain, whereas for the pre-critical case τ > τc, it consists of two disconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' At  criticality when τ = τc, the droplet contains two symmetric double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We refer to [12, 14, 3, 36] for further  models whose droplets reveal various phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us also mention that recently, there have been several  works on the models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) with multi-component droplets, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [13, 30, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In this pre-critical regime,  some theta-function oscillations are expected to appear for various kinds of statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [32, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The precise  asymptotic behaviours of the partition function would also be interesting in connection with the conjecture that  these depend on the Euler index of the droplets, see [42, 29] and [26, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3] for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3 (Fekete points and numerics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' A configuration {ζj}N  j=1 which makes the Hamiltonians (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) or  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2) minimal is known as a Fekete configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This can be interpreted as the ensembles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) with low  temperature limit β = ∞, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [61, 58, 7, 11] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Since the droplet is independent of  the inverse temperature β > 0 (excluding the high-temperature regime [2] when β = O(1/N)), the Fekete points  are useful to numerically observe the shape of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In Figures 1 and 2, Fekete configurations associated  with the Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) are also presented, which show good fits with Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Notice that the potential (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7) and the droplet Sτ,c are invariant under the map ζ �→ −ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We now discuss an  equivalent formulation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 under the removal of such symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (See [36, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3] for a similar  discussion in a vector equilibrium problem on a sphere with point charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') The motivation for this formulation  will be clear in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For this purpose, we denote  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14)  �Q(ζ) :=  2  1 − τ 2  �  |ζ| − τ Re ζ  �  − 2c log |ζ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By definition, the potentials Q in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7) and �Q in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14) are related as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='15)  Q(ζ) = 1  2  �Q(ζ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Denoting by �S the droplet associated with �Q, it follows from [14, Lemma 1] that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16)  S = {ζ ∈ C : ζ2 ∈ �S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Due to the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16) and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='17)  ∆ �Q(ζ) =  1  2(1 − τ 2)  1  |ζ|,  we have the following equivalent formulation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let �Q be given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then the droplet �S ≡ �Sτ,c = S �  Q of the equilibrium measure  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18)  dµ �  Q(ζ) =  1  2(1 − τ 2)  1  |ζ|1�S(ζ) dA(ζ)  is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (i) (Post-critical case) If τ ∈ (0, τc], we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='19)  �Sτ,c =  �  (x, y) ∈ R2 :  � x − 2τ(1 + c)  (1 + τ 2)(1 + c)  �2  +  �  y  (1 − τ 2)(1 + c)  �2  ≤ 1 ,  x2 + y2  (1 − τ 2)2c2 ≥ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (ii) (Pre-critical case) If τ ∈ [τc, 1), the droplet �Sτ,c is the closure of the interior of the real analytic  Jordan curve given by the image of the unit circle with respect to the rational map z �→ f(z), where f  is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  5  (a) τ = 1/6 < τc  (b) τ = 1/3 = τc  (c) τ = 1/2 > τc  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The droplet �Sτ,c, where c = 1 and a Fekete point configuration with N = 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5 (Joukowsky transform in the critical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' If τ = τc with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10), we have a = 1/a = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Thus  in this critical case, the rational function fτ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11) is simplified as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='20)  fτc(z) = (1 + c)(z + τ)2  z  = (1 + c)  �  z + 2τ + τ 2  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that compared to the general case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11), there is one less zero and one less pole in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Indeed, in the  critical case, the rational map fτc is a (shifted) Joukowsky transform  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='21)  fτc : Dc →  �  (x, y) ∈ R2 :  � x − 2τ(1 + c)  (1 + τ 2)(1 + c)  �2  +  �  y  (1 − τ 2)(1 + c)  �2  ≥ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In [3], a similar type of Joukowsky transform was used to solve an equilibrium measure problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For the models  under consideration in the present work, due to a more complicated form of the rational function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11), the  required analysis for the associated equilibrium problem turns out to be more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6 (A non-Hermitian extension of the Marchenko-Pastur distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In the Hermitian limit τ ↑ 1,  by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14), we have  lim  τ↑1 Q(x + iy) = V (x) :=  �  �  �  x2  2 − 2c log |x|,  if y = 0,  +∞  otherwise,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='22)  lim  τ↑1  �Q(x + iy) = �V (x) :=  �  x − 2c log |x|,  if y = 0, x > 0,  +∞  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='23)  Then the associated equilibrium measures are given by the well-known Marchenko-Pastur law (with squared  variables) [38, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  dµV (x) =  1  2π|x|  �  (λ2  + − x2)(x2 − λ2  −) · 1[−λ+,−λ−]∪[λ−,λ+] dx,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24)  dµ�V (x) =  1  2πx  �  (λ2  + − x)(x − λ2  −) · 1[λ2  −,λ2  +] dx,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='25)  where λ± := √2c + 1 ± 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Therefore one can interpret Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=', Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4) as  a non-Hermitian generalisation of the Marchenko-Pastur distribution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='25)), see [8, Section 2]  for more about the geometric meaning with the notion of the statistical cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We also refer to [3] for  another non-Hermitian extension of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='25) in the context of the chiral Ginibre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  3  2  1  0  1  2  3  2  1  0  1  2  m  43  2  1  0  1  2  E-  2  1  0  1  2  m  43  2  1  0  1  2  E-  2  I-  0  1  2  m  46  SUNG-SOO BYUN  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7 (Inclusion relations of the droplets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us write  S1 =  �  (x, y) ∈ R2 :  �  x  1 + τ  �2  +  �  y  1 − τ  �2  ≤ 1 + c  �  ,  S2 :=  �  (x, y) ∈ R2 : x2 + y2 ≤ (1 − τ 2)c  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='26)  and denote by �Sj (j = 1, 2) the image of Sj under the map z �→ z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then it follows from the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10)  that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27)  τ ∈ (0, τc)  if and only if  Sc  1 ∩ S2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4, for general τ ∈ [0, 1) and c ≥ 0, one can observe that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='28)  Sτ,c ⊆ S1 ∩ (Int S2)c,  �Sτ,c ⊆ �S1 ∩ (Int �S2)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Here, equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='28) holds if and only if in the post-critical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (This property holds in a more general  setup, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') On the other hand, in the pre-critical case one can interpret that the particles in  Sc  1 ∩ S2 are smeared out to S1 ∩ Sc  2 which makes the inclusion relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='28) strictly hold, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (a) Sτ,c  (b) �Sτ,c  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The droplets Sτ,c and �Sτ,c in the pre-critical case, where c = 2 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7 > τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Here, the dashed lines display the boundaries of Sj and �Sj (j = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Recall that µW is a unique minimiser of the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' It is well known that  the equilibrium measure µW is characterised by the variational conditions (see [59, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27])  �  log  1  |ζ − z|2 dµW (z) + W(ζ) = C,  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  if ζ ∈ SW ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29)  �  log  1  |ζ − z|2 dµW (z) + W(ζ) ≥ C,  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  if ζ /∈ SW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30)  Here, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' stands for quasi-everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (Nevertheless, this notion is not important in the sequel as we will show  that for the models we consider the conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) indeed hold everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=')  Due to the uniqueness of the equilibrium measure, all we need to show is that if W = Q, then µQ in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12) satisfies the variational principles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Equivalently, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16), it also suffices to show the  variational principles for the equilibrium measure µ �  Q in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  However, it is far from being obvious to obtain the “correct candidate” of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Perhaps one may  think that at least for the post-critical case, the shape of the droplet (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13) is quite natural given the well-known  cases (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9) as well as the fact that the area of Sτ,c should be (1 − τ 2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' On the other hand, for the  pre-critical case, one can easily notice that there is some secret behind deriving the explicit formula of the  rational function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' To derive the correct candidate, we use the conformal mapping method with the help  of the Schwarz function, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8 (Removal of symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We emphasise that the conformal mapping method does not work for  the multi-component droplet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' the pre-critical case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This is essentially due to the lack of  the Riemann mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Nevertheless, one can observe that once we remove the symmetry ζ �→ −ζ, the  droplet in the pre-critical case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This explains the reason why we need the  idea of removing symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' 1  1  1  1  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  7  The rest of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In Section 2, we prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1, we show the post-critical case of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 in a more general setup, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' On the other hand, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2, we  show the pre-critical case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then by the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16), these complete the proof of our  main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  This article contains two appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In Appendix A, we explain the conformal mapping method  to derive the “correct candidate” of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In Appendix B, we present a way to solve a one-  dimensional equilibrium problem in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3, which shares a common feature with the conformal  mapping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' These appendices are made only for instructive purposes and the readers who only  want the proof of the main theorems may stop at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Proof of main theorem  In this section, we show Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Post critical cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Extending (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7), we consider the potential  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1)  Qp(ζ) :=  1  1 − τ 2  �  |ζ|2 − τ Re ζ2�  − 2c log |ζ − p|,  p ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For the case τ = 0, the shape of the droplet associated with the potential (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) was fully characterised in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (In this case, it suffices to consider the case p ≥ 0 due to the rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') In particular, it was shown  that if  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2)  c < (1 − p2)2  4p2  ,  τ = 0,  p ≥ 0,  the droplet is given by S = D(0, √1 + c) \\ D(p, √c), where D(p, R) is the disc with centre p and radius R, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5 for the other case c > (1 − p2)2/(4p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  To describe the droplets associated with Qp, we denote  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3)  S1 :=  �  (x, y) ∈ R2 :  �  x  1 + τ  �2  +  �  y  1 − τ  �2  ≤ 1 + c  �  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4)  S2 :=  �  (x, y) ∈ R2 : (x − Re p)2 + (y − Im p)2 ≤ (1 − τ 2)c  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Suppose that the parameters τ, c ∈ R and p ∈ C are given to satisfy  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5)  S2 ⊂ S1,  where S1 and S2 are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then the droplet S ≡ SQp associated with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6)  S = S1 ∩ (Int S2)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  See Figure 4 for the shape of the droplets and numerical simulations of Fekete point configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We  remark that with slight modifications, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 can be further extended to the case with multiple point  charges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' the potential of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7)  1  1 − τ 2  �  |ζ|2 − τ Re ζ2�  − 2  �  cj log |ζ − pj|,  pj ∈ C,  cj ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (See Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 for a related discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') Let us also mention that a similar statement for an equilibrium  problem on the sphere was shown in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For a treatment of a more general case, we refer the reader to  [35, 54, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' If p = 0, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5) corresponds to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8)  τ <  1  1 + 2c = τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 for the special value p = 0 gives Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  As a consequence, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16),  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 (i) also follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We also mention that if τ = 0 and p > 0, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5) coincides with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  8  SUNG-SOO BYUN  (a) p =  2  21  √  14 i  (b) p = 2  7  √  14  (c) p = 3  5 + 1  5i  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The droplet S in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1, where τ = 1/3 and c = 1/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Here, a Fekete point  configuration with N = 2048 is also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3 (Equilibrium measure in the Hermitian limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Before moving on to the planar equilibrium problem  for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1), we first discuss the one-dimensional problem arising in the Hermitian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For p ∈ R, the Hermitian  limit τ ↑ 1 of the potential Qp is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9)  lim  τ↑1 Qp(x + iy) = Vp(x) :=  �  �  �  x2  2 − 2c log |x − p|,  if y = 0,  +∞  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then one can show that the associated equilibrium measure µV ≡ µVp is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10)  dµV (x)  dx  =  �  − �4  j=1(x − λj)  2π|x − p|  1[λ1,λ2]∪[λ3,λ4](x),  where  λ1 = p − 2 −  �  (p + 2)2 + 8c  2  ,  λ2 = p + 2 −  �  (p − 2)2 + 8c  2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11)  λ3 = p − 2 +  �  (p + 2)2 + 8c  2  ,  λ4 = p + 2 +  �  (p − 2)2 + 8c  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12)  We remark that when p = 0, it recovers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' See Figure 5 for the graphs of the equilibrium measure µVp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The  equilibrium measure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10) follows from the standard method using the Stieltjes transform and the Sokhotski-  Plemelj inversion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For reader’s convenience, we provide a proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (a) p = 0  (b) p = 1  (c) p = 4  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Graphs of the equilibrium measure µVp, where c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  15  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content="5  0'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content="5  10  1'5 15  10  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content="5  0'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  10  150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content='05  2  0  2  415  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content="5  0'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5  10  15EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  9  In the rest of this subsection, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' First, let us show the following elementary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For a, b > 0, let  K :=  �  (x, y) ∈ R2 :  �x  a  �2  +  �y  b  �2  ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13)  �  K  1  ζ − z dA(z) =  �  �  �  �  �  ¯ζ − a − b  a + b ζ  if ζ ∈ K,  2ab  a2 − b2  �  ζ −  �  ζ2 − a2 + b2  �  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In particular, for ζ ∈ K, there exists a constant c0 ∈ R such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14)  �  K  log |ζ − z|2 dA(z) = |ζ|2 − a − b  a + b Re ζ2 + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The Cauchy transform in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13) is useful to explicitly compute the moments of the equilibrium  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Namely, by definition, we have  �  K  1  ζ − z dA(z) = 1  ζ  ∞  �  k=0  1  ζk  �  K  zk dA(z),  ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand, we have  ζ −  �  ζ2 − a2 + b2 = a2 − b2  ζ  ∞  �  k=0  � 1/2  k + 1  �(b2 − a2)k  ζ2k  ,  ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining the above equations with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13), we obtain that for any non-negative integer k,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='15)  1  ab  �  K  z2k dA(z) = 2  � 1/2  k + 1  �  (b2 − a2)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Recall that D is the unit disc with centre the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then the Joukowsky transform  f : ¯Dc → Kc is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16)  f(z) = a + b  2  z + a − b  2  1  z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By applying Green’s formula, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='17)  �  K  1  ζ − z dA(z) =  1  2πi  �  ∂K  ¯z  ζ − z dz + ¯ζ · 1{ζ∈K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Furthermore, by the change of variable z = f(w), it follows that  �  ∂K  ¯z  ζ − z dz =  �  ∂D  f(1/ ¯w)  ζ − f(w)f ′(w) dw =  �  ∂D  gζ(w) dw,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18)  where gζ is the rational function given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='19)  gζ(w) :=  1  ζ − f(w)  �a + b  2  1  w + a − b  2  w  ��a + b  2  − a − b  2  1  w2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Observe that  ζ = f(w)  if and only if  w = w±  ζ := ζ ±  �  ζ2 − a2 + b2  a + b  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' the points w±  ζ are solutions to the quadratic equation  (a + b)w2 − 2ζ w + (a − b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Here, the branch of the square root is chosen such that  w−  ζ → 0  ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By above observation, the function gζ has poles only at  0,  w+  ζ ,  w−  ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  10  SUNG-SOO BYUN  Moreover note that  ζ ∈ K  if and only if  w±  ζ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Notice that if ζ ∈ Kc, then w−  ζ ∈ D and w+  ζ ∈ Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Using the residue calculus, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='20)  Res  w=0  �  gζ(w)  �  = a + b  a − b ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='21)  Res  w=w±  ζ  �  gζ(w)  �  = −f(1/ ¯w±  ζ ) = −a + b  2  1  w±  ζ  − a − b  2  w±  ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In particular,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='22)  Res  w=w+  ζ  �  gζ(w)  �  + Res  w=w−  ζ  �  gζ(w)  �  = −2a2 + b2  a2 − b2 ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining all of the above, we obtain the desired identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The second assertion immediately follows  from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13) and the real-valuedness of ζ �→  �  K log |ζ − z|2 dA(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For R > 0 and p ∈ C we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='23)  �  D(p,R)  log |ζ − z| dA(z) =  �  �  �  R2 log |ζ − p|  if ζ /∈ D(p, R),  R2 log R − R2  2 + |ζ − p|2  2  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' First, recall the well-known Jensen’s formula: for r > 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24)  1  2π  � 2π  0  log |ζ − reiθ| dθ =  �  log r  if r > |ζ|,  log |ζ|  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By the change of variables, we have  �  D(p,R)  log |ζ − z| dA(z) =  �  D(0,R)  log |ζ − p − z| dA(z) = 1  π  � R  0  r  � 2π  0  log |ζ − p − reiθ| dθ dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Suppose that ζ /∈ D(p, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then by applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24), we have  1  π  � R  0  r  � 2π  0  log |ζ − p − reiθ| dθ dr = 2  � R  0  r log |ζ − p| dr = R2 log |ζ − p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand if ζ ∈ D(p, R), we have  1  π  � R  0  r  � 2π  0  log |ζ − p − reiθ| dθ dr = 2  � |ζ−p|  0  r log |ζ − p| dr + 2  � R  |ζ−p|  r log r dr  = R2 log R − R2  2 + |ζ − p|2  2  ,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  We are now ready to complete the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Note that by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5), the equilibrium measure µ associated with Qp is of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='25)  dµ(z) := ∆Qp(z) · 1S(z) dA(z) =  1  1 − τ 2 · 1S(z) dA(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Due to the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5), we have  �  log  1  |ζ − z|2 dµ(z) =  1  1 − τ 2  � �  S1  log  1  |ζ − z|2 dA(z) −  �  S2  log  1  |ζ − z|2 dA(z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  11  Note that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4, there exists a constant c0 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='26)  �  S1  log  1  |ζ − z|2 dA(z) = −|ζ|2 + τ Re ζ2 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27)  �  S2  log  1  |ζ − z|2 dA(z) = −2(1 − τ 2)c log |ζ − p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='26), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1), we obtain  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='28)  �  log  1  |ζ − z|2 dµ(z) = −Qp(ζ) −  c0  1 − τ 2 ,  which leads to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Next, we show the variational inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Note that if ζ ∈ S2, it immediately follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Thus it is enough to verify (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) for the case ζ ∈ Sc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29)  Hp(ζ) :=  �  log  1  |ζ − z|2 dµ(z) + Qp(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Suppose that the variational inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then since Hp(ζ) → ∞ as ζ → ∞, there exists  ζ∗ ∈ Sc  1 such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30)  ∂ζHp(ζ)|ζ=ζ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand, by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6, if ζ ∈ Sc  1, the Cauchy transform of the measure µ is computed as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='31)  � dµ(z)  ζ − z = 1  2τ  �  ζ −  �  ζ2 − 4τ(1 + c)  �  −  c  ζ − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1), this gives rise to  ∂ζQp(ζ) −  � dµ(z)  ζ − z =  1  1 − τ 2  �  ¯ζ − τζ  �  − 1  2τ  �  ζ −  �  ζ2 − 4τ(1 + c)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='32)  Then it follows that the condition ∂ζHp(ζ) = 0 is equivalent to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='33)  (1 + τ 2)|ζ|2 − τ(ζ2 + ¯ζ2) = (1 − τ 2)2(1 + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), one can notice that ∂ζHp(ζ) = 0 if and only if ζ ∈ ∂S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This yields a contradiction with  the assumption ζ∗ ∈ Sc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Therefore we conclude that the variational inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) holds for ζ ∈ Sc, which  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us denote by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='34)  mk :=  �  zk dµ(z)  the k-th moment of the equilibrium measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Notice that the Cauchy transform of µ satisfies the asymptotic  expansion  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='35)  � dµ(z)  ζ − z = 1  ζ  ∞  �  k=0  mk  ζk ,  ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Using this property and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='31), after straightforward computations, one can verify that the equilibrium measure  µ in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 has the moments  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='36)  m2k = 2  (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='τ k(1 + c)k+1 − c p2k,  m2k+1 = −c p2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Notice in particular that if p = 0, all odd moments vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  12  SUNG-SOO BYUN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Pre-critical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In this subsection, we show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16), Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Recall that �Q is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14) and that all we need to show is the variational  principles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) for W = �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For this, similar to above, let  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='37)  H(ζ) :=  �  log  1  |ζ − z|2 d�µ(z) + �Q(ζ),  where �µ is the equilibrium measure associated with �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='38)  ∂ζH(ζ) = ∂ζ �Q(ζ) − C(ζ) =  1  1 − τ 2  ��  ¯ζ  ζ − τ  �  − c  ζ − C(ζ),  where C(ζ) is the Cauchy transform of �µ given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='39)  C(ζ) =  1  2(1 − τ)2  �  �S  1  ζ − z  1  |z| dA(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Here, we have used (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Applying Green’s formula, we have  (1 − τ 2)C(ζ) =  1  2πi  �  ∂ �S  1  ζ − z  �  ¯z  z dz +  �  ¯ζ  ζ · 1{ζ∈Int(�S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='40)  Recall that f is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let  g(w) :=  �  f(1/ ¯w)  f(w) f ′(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='41)  Since f ′(aτ) = 0, the function g(w) has poles only at 0, 1/a, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We also write  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='42)  hζ(w) :=  g(w)  ζ − f(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Using the change of variable z = f(w),  1  2πi  �  ∂ �S  1  ζ − z  �  ¯z  z dz =  1  2πi  �  ∂D  1  ζ − f(w)  �  f(1/ ¯w)  f(w) f ′(w) dw =  1  2πi  �  ∂D  hζ(w) dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='43)  By the residue calculus, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='44)  Res  w=0  �  hζ(w)  �  = 1  τ ,  Res  w=a  �  hζ(w)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that ζ = f(w) is equivalent to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='45)  d(1 − aw)(w − aτ)2 = w(w − a)ζ,  d = (1 + τ)(1 + 2c)  2  ,  which can be rewritten as a cubic equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='46)  adw3 − (d + 2a2dτ − ζ)w2 + a(2dτ + a2τ 2d − ζ)w − a2τ 2d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  For given ζ ∈ C, there exist w(j)  ζ  (j = 1, 2, 3) such that f(w(j)  ζ ) = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Note that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='46), we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='47)  w(1)  ζ w(2)  ζ w(3)  ζ  = aτ 2 ∈ (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Furthermore, since f is a conformal map from Dc onto �Sc, we have the following:  (1) If ζ ∈ Int(�S), then all w(j)  ζ ’s are in D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2) If ζ ∈ �Sc, then two of w(j)  ζ ’s are in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  13  By the residue calculus using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='41), for each j,  Res  w=w(j)  ζ  �  hζ(w)  �  = −  g(w(j)  ζ )  f ′(w(j)  ζ )  = −  (w(j)  ζ  − a)(1 − aτw(j)  ζ )  (w(j)  ζ  − aτ)(1 − aw(j)  ζ )  = d  ζ  (aτw(j)  ζ  − 1)(w(j)  ζ  − aτ)  w(j)  ζ  = d  ζ  �  aτw(j)  ζ  − (a2τ 2 + 1) + aτ  w(j)  ζ  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='48)  where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' On the other hand, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='46) that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='49)  3  �  j=1  w(j)  ζ  = d + 2a2dτ − ζ  ad  ,  3  �  j=1  1  w(j)  ζ  = −ζ + 2dτ + a2τ 2d  aτ 2d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  These relations give rise to  d  3  �  j=1  �  aτw(j)  ζ  − (a2τ 2 + 1) + aτ  w(j)  ζ  �  = dτ + 2a2dτ 2 − τζ − ζ  τ + 2d + a2τd − 3d(a2τ 2 + 1)  = −  �  τ + 1  τ  �  ζ − c(1 − τ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='50)  Combining all of the above, we have shown that if ζ ∈ Int(�S),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='51)  3  �  j=1  Res  w=w(j)  ζ  �  hζ(w)  �  = −  �  τ + 1  τ  �  − c(1 − τ 2)  ζ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore if ζ ∈ Int(�S), we obtain  (1 − τ 2)C(ζ) =  �  ¯ζ  ζ − 1  τ − c(1 − τ 2) = (1 − τ 2)∂ζ �Q(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='52)  Then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='40), the variational equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Now it remains to show the variational inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Note that by definition, H(ζ) → ∞ as ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Suppose that the variational inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Then there exists ζ∗ ∈ �Sc such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='53)  ∂ζH(ζ)|ζ=ζ∗ = ∂ �Q(ζ∗) − C(ζ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Recall that if ζ ∈ �Sc, then only one of w(j)  ζ ’s, say wζ, is in Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' By combining the above computations, we have  that for ζ ∈ �Sc,  (1 − τ 2)  �  ∂ζ �Q(ζ) − C(ζ)  �  =  �  ¯ζ  ζ − Res  w=wζ  �  hζ(w)  �  =  �  ¯ζ  ζ − d  ζ  �  aτwζ − (a2τ 2 + 1) + aτ  wζ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='54)  Therefore the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='53) holds if and only if  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='55)  |ζ∗| = d  �  aτwζ∗ − (a2τ 2 + 1) + aτ  wζ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11),  −  d  f(x)  �  aτx − (a2τ 2 + 1) + aτ  x  �  = −  1  f(x)  (1 + τ)(1 + 2c)  2  (aτx − 1)(x − aτ)  x  = (aτx − 1)(x − a)  (ax − 1)(x − aτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore if x < 1/(aτ),  d  �  aτx − (a2τ 2 + 1) + aτ  x  �  < τ|f(x)| < |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  From this, we notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='55) does not hold for wζ∗ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Furthermore, this implies that the right-hand side  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='55) is real-valued if and only if wζ∗ ∈ ∂D, equivalently, ζ∗ ∈ ∂ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' This contradicts with the assumption  that ζ∗ ∈ �Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Now the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  14  SUNG-SOO BYUN  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Conformal mapping method: the pre-critical case  In this appendix, we present the conformal mapping method, which is helpful to derive the candidate of the  droplet given in terms of the rational function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let τ ∈ (τc, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Suppose that �S in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18) is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let f be a unique conformal  map (¯Dc, ∞) → (�Sc, ∞), which satisfies  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1)  f(z) = r1 z + r2 + O  �1  z  �  ,  z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (i) The conformal map f is a rational function of the form  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2)  f(z) = r1z + r2 + r3  z +  r4  z − a,  a ∈ (−1, 0),  which satisfies  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3)  f(1/a) = r1  a + r2 + r3a +  ar4  1 − a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (ii) The parameters rj (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' , 4) are given by  r1 = 1 + τ  2  �  1 + 2c  τ  ,  r2 = 1 + τ  2τ  (τ(1 + 2c) + 2τ − 1),  r3 = 1 + τ  2  τ  �  τ(1 + 2c),  r4 = (1 − τ)2(1 + τ)(1 − (1 + 2c)τ)  2τ  �  τ(1 + 2c)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4)  and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5)  a = −  1  �  τ(1 + 2c)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that the rational function f with the choice of parameters (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4) corresponds to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Therefore  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 gives rise to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 (ii) under the assumption that �S is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' However, there  is no general theory characterising the connectivity of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (Nevertheless, we refer the reader to [49, 48]  for sharp connectivity bounds of the droplets associated with a class of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=') Thus we need to directly  verify the variational principles as in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' By differentiating the variational equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29), we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6)  ∂ζ �Q(ζ) = C(ζ) :=  � d�µ(z)  ζ − z ,  ζ ∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14), this can be rewritten as  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7)  ¯ζ = ζ  �  (1 − τ 2)  �  C(ζ) + c  ζ  �  + τ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore the Schwarz function F associated with the droplet �S exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Furthermore, it is expressed in terms  of C as  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8)  F(ζ) = ζ  �  (1 − τ 2)  �  C(ζ) + c  ζ  �  + τ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that for z ∈ ∂D,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9)  f(1/¯z) = f(z) = f(z)  �  (1 − τ 2)  �  C(f(z)) +  c  f(z)  �  + τ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Using this, we define f : ¯D\\{0} → C by analytic continuation as  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10)  f(z) := f(1/¯z)  �  (1 − τ 2)  �  C(f(1/¯z)) +  c  f(1/¯z)  �  + τ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore f has simple poles only at 0, ∞ and the point a ∈ R such that f(1/a) = 0, which leads to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  15  Next, we need to specify the constants rj and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For this, we shall find interrelations among the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11)  r3 = r1τ 2  and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12)  r4 = a(1 − τ 2)  �  r2 − 2τ(1 + c)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Furthermore, we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13)  r2 = r1  1 + a2τ 2  1 − a2τ 2  a2 − 1  a  + 2a2(1 − τ 2)τ(1 + c)  1 − a2τ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Note that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14)  f(1/¯z) = r1  z + r2 + r3z +  r4z  1 − az .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Therefore, we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='15)  1  f(1/¯z)  = 1  r1  z − r2  r2  1  z2 + r2  2 − r1r3 − r1r4  r3  1  z3 + O(z4),  z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Since the Cauchy transform C satisfies the asymptotic behaviour  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='16)  C(ζ) = 1  ζ + O( 1  ζ2 ),  ζ → ∞,  we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='17)  C(f(1/¯z)) = 1  r1  z + O(z2),  z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining these equations with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10), we obtain  f(z) = r1τ 2  z  +  �  r2τ 2 + 2τ(1 − τ 2)(1 + c)  �  + O(z),  z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18)  On the other hand, by using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2), we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='19)  f(z) = r3  z +  �  r2 − r4  a  �  + O(z),  z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then by comparing the coefficients in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='18) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='19), we obtain (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Note that by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='20)  r4 = a2 − 1  a  �r1  a + r2 + r3a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11), we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='21)  r4 = a2 − 1  a  �r1  a + r2 + r1aτ 2�  = r1  (1 + a2τ 2)(a2 − 1)  a2  + r2  a2 − 1  a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining this identity with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12), we obtain  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='22)  a(1 − τ 2)r2 − 2a(1 − τ 2)τ(1 + c) = r1  (1 + a2τ 2)(a2 − 1)  a2  + r2  a2 − 1  a  ,  which leads to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' We have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='23)  �  (2 − a2 + a4τ 2)r1 + ar2  ��  r2 − 2τ(1 + c)  �  = (1 − τ 2)c2a(a2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  16  SUNG-SOO BYUN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='24)  1  f(1/¯z)  =  a2(a2 − 1)  (2 − a2)r1 + ar2 + a4r3  1  z − a + O(1),  z → a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Then by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11), we obtain  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='25)  r4 = (1 − τ 2)2c2 a2(a2 − 1)  (2 − a2)r1 + ar2 + a4r3  =  (1 − τ 2)2c2 a2(1 − a2)2  r1(a2τ 2 − 1)(1 − a2)2 + r4a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Now lemma follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Since �µ is a probability measure, we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='26)  1 =  �  �S  1  2(1 − τ 2)  1  |z| dA(z) =  1  2πi  �  ∂ �S  1  1 − τ 2  �  ¯z  z dz,  where we have used Green’s formula for the second identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Using the change of variable z = f(w), where f is  of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2), this can be rewritten as  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27)  1  2πi  �  ∂D  �  f(1/ ¯w)f(w) f ′(w)  f(w) dw = 1 − τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2), we have  f(z) = 1 − az  z(z − a)  �  − r1  a z2 +  �a2 − 1  a2  r1 − r2  a  �  z − aτ 2r1  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='28)  f(1/¯z) =  z − a  z(1 − az)  �  − aτ 2r1z2 +  �a2 − 1  a2  r1 − r2  a  �  z − r1  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='29)  Note here that by construction, two zeros of f other than 1/a are contained in the unit disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Using these  together with straightforward residue calculus, we obtain  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30)  Resw=0  ��  f(1/ ¯w)f(w) f ′(w)  f(w)  �  = (1 + c)(1 − τ 2)  and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='31)  Resw=a  ��  f(1/ ¯w)f(w) f ′(w)  f(w)  �  = −1  a  ��1 + a2τ 2  a  r1 + r2  ��a4τ 2 − a2 + 2  a  r1 + r2  ��1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Furthermore, it follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3 that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='32)  Resw=a  ��  f(1/ ¯w)f(w) f ′(w)  f(w)  �  = −c(1 − τ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='27), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='30) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='32), one can notice that the function f has a double zero, which implies that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='33)  a2 − 1  a2  r1 − r2  a = 2r1τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By solving the system of equations given in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='33), the desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  □  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4 (The use of higher moments of the equilibrium measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In a more complicated case, for instance  for the case with multiple point charges such as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7), the mass-one condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='26) may not be enough to  characterise the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' In this case, one can further use the higher order asymptotic expansions appearing  in the above lemmas, which involve the k-th moments of the equilibrium measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Thus in principle,  one can always find enough (algebraic) interrelations to characterise the parameters appearing in the conformal  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' For the case τ = 0 and p > 0, it was shown in [12] that if  c > (1 − p2)2  4p2  ,  EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE  17  the droplet associated with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) is a simply connected domain whose boundary is given by the image of the  conformal map  f(z) = R z −  κ  z − q − κ  q ,  R = 1 + p2q2  2pq  ,  κ = (1 − q2)(1 − p2q2)  2pq  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Here, q is given by the unique solution of P(q2) = 0, where  P(x) := x3 −  �p2 + 4c + 2  2p2  �  x2 +  1  2p4  such that 0 < q < 1 and κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Beyond the case τ = 0, the conformal mapping method described above also works for the potential (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1) with  general τ ∈ [0, 1), c ∈ R and p ∈ C under the assumption that the associated droplet is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Under  this assumption, one can show that the boundary of the droplet is given by the image of the rational conformal  map f of the form  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='34)  f(z) = R1 z + R2 + R3  z +  R4  z − q ,  q ∈ D,  which satisfies f(1/q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Furthermore, following the strategy above, one can characterise the coefficients Rj  (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' , 4) of this rational map as well as the position of the pole q ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  However, as previously mentioned, it is far from being obvious to characterise a condition for which the  droplet is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Nevertheless, since the radius of curvature of the ellipse (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3) at the point (1 +  τ)√1 + c is given by  (1 − τ)2  1 + τ  √  1 + c,  one can expect that if  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='35)  p > max  � 4τ  1 + τ  √  1 + c , (1 + τ)  √  1 + c −  �  1 − τ 2√c  �  then the droplet is a simply connected domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' One-dimensional equilibrium measure problem in the Hermitian limit  In this appendix, we present a proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Let us write  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='1)  V (z) ≡ Vp(z) = z2  2 − 2c log |z − p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Recall that µV ≡ µVp is the equilibrium measure associated with Vp(x) (x ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  We define  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2)  R(z) :=  �V ′(z)  2  �2  −  �  R  V ′(z) − V ′(s)  z − s  dµV (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By applying Schiffer variations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [35, Section 3]), we have  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3)  R(z) =  � � dµV (s)  z − s − V ′(z)  2  �2  ,  z ∈ C \\ supp(µV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining the asymptotic behaviour  � dµV (s)  z − s  ∼ 1  z ,  z → ∞,  with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), we obtain  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4)  R(z) = 1  4z2 − (c + 1) − cp  z + O  � 1  z2  �  ,  z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  On the other hand, since  V ′(z) = z −  2c  z − p,  V ′(z) − V ′(s)  z − s  = 1 +  2c  z − p  1  s − p,  18  SUNG-SOO BYUN  we have  R(z) = 1  4  �  z −  2c  z − p  �2  − 1 −  2c  z − p  �  R  dµV (s)  s − p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5)  Thus we obtain  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6)  R(z) =  c2  (z − p)2 + O  �  1  z − p  �  ,  z → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  In the expression (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='5), one can observe that R is a rational function with a double pole at z = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Therefore  it is of the form  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7)  R(z) = 1  4z2 + Az2 + Bz + C  (z − p)2  for some constants A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' As in the previous subsection, we need to specify these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By direct computations, we have  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8)  R(z) = 1  4z2 + A + 2Ap + B  z  + O  � 1  z2  �  ,  z → ∞,  and  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9)  R(z) = Ap2 + Bp + C  (z − p)2  + O  �  1  z − p  �  ,  z → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By comparing coefficients in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='8), we have  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10)  A = −c − 1,  −cp = 2Ap + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Similarly, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='6) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='9),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11)  Ap2 + Bp + C = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  By solving these algebraic equations, we obtain  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12)  B = p(c + 2),  C = c2 − p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Combining all of the above with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='7), we have shown that  R(z) = 1  4z2 + −(c + 1)z2 + p(c + 2)z + (c2 − p2)  (z − p)2  = ((z − p)(z − 2) − 2c)((z − p)(z + 2) − 2c)  4(z − p)2  =  �4  j=1(z − λj)  4(z − p)2  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='13)  where λj’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Therefore by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='3), the Stieltjes transform of µV is given by  � dµV (s)  z − s  = V ′(z)  2  − R(z)1/2 = z  2 −  c  z − p − 1  2  ��4  j=1(z − λj)  (z − p)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='14)  Letting z = x + iε → x ∈ R, we find  lim  ε→0+ Im  �  dµV (s)  (x + iε) − s =  �  �  �  �  �  �  �  �  − �4  j=1(x − λj)  2|x − p|  if x ∈ [λ1, λ2] ∪ [λ3, λ4],  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Now the desired identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='10) follows from the Sokhotski-Plemelj inversion formula, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' [39, Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The author is greatly indebted to Yongwoo Lee for the figures and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Scaling limits of planar symplectic ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' SIGMA Symmetry Integrability  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' 007, 40, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Akemann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Duits, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Molag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' The elliptic Ginibre ensemble: A unifying approach to local and global statistics for  higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Ebke, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content='  [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Ameur and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Byun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Almost-Hermitian random matrices and bandlimited point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' From the Ginzburg-Landau model to vortex lattice problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Microscopic description of Log and Coulomb gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
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+page_content='  Center for Mathematical Challenges, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu,  Seoul 02455, Republic of Korea  Email address: sungsoobyun@kias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}
+page_content='kr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfefjV/content/2301.00324v1.pdf'}