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

We prove the topological expansion for the cubic log–gas partition function Z N ( t ) = ∫ Γ ⋯ ∫ Γ ∏ 1 ≤ j < k ≤ N ( z j - z k ) 2 ∏ k = 1 N e - N - z 3 3 + t z dz 1 ⋯ dz N , where t is a complex parameter and Γ is an unbounded contour on the complex plane extending from e π i ∞ to e π i / 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 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.