Asymptotic expansion of the partition function for $\beta$-ensembles with complex potentials

TL;DR

This paper derives the large-N asymptotic expansion of partition functions for complex potential beta-ensembles, extending classical methods to complex contours and establishing explicit terms of the expansion.

Contribution

It introduces a novel steepest-descent inspired method for complex potentials, providing the first all-order asymptotic expansion for the partition function.

Findings

Established existence of all-order asymptotic expansion

Identified first few terms explicitly

Extended analysis to complex-valued potentials and contours

Abstract

In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N \beta V(z_k) } \, \mathrm{d}\mathbf{z}$$ where $V \in \mathbb{C}[X]$, $\beta \in 2 \mathbb{N}^*$ is an even integer and $\Gamma \subset \mathbb{C}$ is an unbounded contour such that the integral converges. For even degree, real valued $V$s and when $\Gamma = \mathbb{R}$, 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 $\ln \mathcal{Z}_{N,\Gamma}[V]$ and explicitly identify the first few terms.