Asymptotic expansion of the partition function for ${\hat{I}}^2$-ensembles with complex potentials

Abstract

In this work we establish under certain hypotheses the $N\to +\infty$ asymptotic expansion of integrals of the form $${\mathcal{Z}}_{N,\hat{I}“}[V]={\int }_{\hat{I}{“}^N}\prod _{a<b}^N(z_a-z_b{)}^2\prod _{k=1}^N{\mathrm{e}}^{-N{\hat{I}}^{2}V(z_k)}\mathrm{d}\mathbf{z}$$ where $V\in \mathbb{C}[X]$, ${\hat{I}}^2\in 2{\mathbb{N}}^{\ast }$ is an even integer and $\hat{I}“\subset \mathbb{C}$ is an unbounded contour such that the integral converges. For even degree, real valued $V$s and when $\hat{I}“=\mathbb{R}$, it is well known that the large-$N$ expansion is characterised by an equilibrium measure corresponding to the minimiser of an appropriate energy functional. This method bears a structural resemblance with the Laplace method. By contrast, in the complex valued setting we are considering, the analysis structurally resembles the classical steepest-descent method, and involves finding a critical point \textit{and} a steepest descent curve, the latter being a deformation of the original integration contour. More precisely, one minimises a curve-dependent energy functional with respect to measures on the curve and then maximises the energy over an appropriate space of curves. Our analysis deals with the one-cut regime of the associated equilibrium measure. We establish the existence of an all order asymptotic expansion for $\ln {\mathcal{Z}}_{N,\hat{I}“}[V]$ and explicitly identify the first few terms.

64 pages