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We prove the topological expansion for the cubic log-gas partition function\[ Z_N(t)= \int_\Gamma\cdots\int_\Gamma\prod_{1\leq j<k\leq N}(z_j-z_k)^2\prod_{k=1}^Ne^{-N\left(-\frac{z^3}{3}+tz\right)}\mathrm dz_1\cdots \mathrmdz_N, \] where $t$ is a complex parameter and $\Gamma$ is an unbounded contouron 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 regionson the complex $t$-plane, with one cut and two cuts, separated by analyticcritical arcs of the two types of phase transition: split of a cut and birth ofa cut. The common point of the critical arcs is a tricritical point of thePainlev\'e 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 theRiemann--Hilbert approach to semiclassical asymptotic expansions for theassociated orthogonal polynomials and the theory of $S$-curves and quadraticdifferentials.

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