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arXiv · 1606.04303

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

Pavel M. Bleher·Alfredo Deaño·Maxim Yattselev
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Abstract

We prove the topological expansion for the cubic log-gas partition function \[ Z_N(t)= \int_Γ\cdots\int_Γ\prod_{1\leq j<k\leq N}(z_j-z_k)^2 \prod_{k=1}^Ne^{-N\left(-\frac{z^3}{3}+tz\right)}\mathrm dz_1\cdots \mathrm dz_N, \] where $t$ is a complex parameter and $Γ$ is an unbounded contour on the complex plane extending from $e^{π\mathrm i}\infty$ to $e^{π\mathrm i/3}\infty$. 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.

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Pavel M. Bleher, Alfredo Deaño, Maxim Yattselev. 2016-06-14. Topological Expansion in the Complex Cubic Log-Gas Model. One-Cut Case. https://doi.org/10.1007/s10955-016-1621-x

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