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

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.