Topological Expansion in the Complex Cubic Log–Gas Model: One-Cut Case

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• 作者：Pavel Bleher ; Alfredo Deaño ; Maxim Yattselev
• 关键词：Log–gas model ; Partition function ; Topological expansion ; Equilibrium measure ; S ; curve ; Quadratic differential ; Orthogonal polynomials ; Non ; Hermitian orthogonality ; Riemann–Hilbert problem ; Nonlinear steepest descent method
• 刊名：Journal of Statistical Physics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：166
• 期：3-4
• 页码：784-827
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：166

文摘

We prove the topological expansion for the cubic log–gas partition function $$\begin{aligned} Z_N(t)= \int _\Gamma \cdots \int _\Gamma \prod _{1\le j<k\le N}(z_j-z_k)^2 \prod _{k=1}^Ne^{-N\left( -\frac{z^3}{3}+tz\right) }\mathrm{dz}_1\cdots \mathrm{dz}_N, \end{aligned}$$where t is a complex parameter and \(\Gamma \) is an unbounded contour on the complex plane extending from \(e^{\pi \mathrm{i}}\infty \) to \(e^{\pi \mathrm{i}/3}\infty \). The complex cubic log–gas model exhibits two phase regions on the complex t-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painlevé I type. In the present paper we prove the topological expansion for \(\log Z_N(t)\) in the one-cut phase region. The proof is based on the Riemann–Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of S-curves and quadratic differentials.KeywordsLog–gas modelPartition functionTopological expansionEquilibrium measureS-curveQuadratic differentialOrthogonal polynomialsNon-Hermitian orthogonalityRiemann–Hilbert problemNonlinear steepest descent methodDedicated to David Ruelle and Yakov Sinai.