Asymptotics of the partition function for $\beta$-ensembles at high temperature

TL;DR

This paper derives the large-$N$ asymptotic expansion of the partition function for high-temperature $eta$-ensembles, revealing the role of the thermal equilibrium measure and advancing the loop equations method.

Contribution

It provides the first successful implementation of loop equations at high temperature, analyzing the thermal equilibrium measure and the associated master operator.

Findings

Established all-order asymptotic expansion of the partition function.

Identified the first two terms of the expansion.

Developed new analytical tools for the inverse of the master operator.

Abstract

We consider the real $\beta$-ensemble (or 1D log-gas) of dimension $N$ in the high-temperature regime, \textit{i.e.} where the inverse temperature $\beta$ scales as $N\beta=2P$ with $P$ a fixed positive parameter. We establish the large-$N$ asymptotic expansion at all orders of the partition function: \begin{equation*} Z_N[V]=\int_{\mathbb{R}^N}\prod_{i<j}^{N}\left |x_i-x_j\right|^{\frac{2P}{N}}\cdot\prod_{i=1}^{N}e^{-V(x_i)} \mathrm{d}x_i \end{equation*} for $V(x)=x^2+\phi(x)$ with $\phi$ a bounded smooth function, and identify the first two terms of this expansion. In this regime, the energy no longer dominates the entropy, as in the fixed-$\beta$ case, but rather scales at the same order in $N$. Consequently, at large $N$, the system is macroscopically described by the so-called\textit{ thermal equilibrium measure} which is supported on the entire real line. Our proof relies on the loop equations method, previously applied in the fixed-$\beta$ setting in \cite{BoG1,BoG2}, and provides the first example in which this approach can be successfully implemented using the thermal equilibrium measure. This requires a detailed understanding of both the thermal equilibrium measure and the associated master operator, an unbounded differential operator, leading to several new analytical challenges. In this setting, we carry out a technically involved analysis to obtain precise estimates for the inverse of the master operator in suitable functional norms. In addition we establish, through subtle operator arguments, a crucial continuity property of the equilibrium density with respect to the potential dependence. These two results constitute the main novelties of the paper and allow us to exhibit a new class of multiple integrals for which such an expansion can be obtained, while providing a deeper understanding of the thermal equilibrium measure and its properties.