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

Abstract

We prove the topological expansion for the cubic log–gas partition function ZN(t)=∫Γ⋯∫Γ ∏1≤j<k≤N (zj−zk)2 N ∏ k=1 e−N(− z3 3 +tz)dz1⋯dzN, where t is a complex parameter and Γ is an unbounded contour on the complex plane extending from eπi∞ to eπi/3∞. 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 logZN(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.