Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Charlie Dworaczek Guera

TL;DR

This work analyzes the real $β$-ensemble in the high-temperature regime with $Nβ=2P$, proving an all-orders large-$N$ expansion for the partition function $Z_N[V]$ when $V(x)=x^2+φ(x)$ and $φ$ is bounded. The authors develop a loop-equation framework built around the master operator $\Xi$ and its inverse, establish sharp controls on $\Xi^{-1}$ in Sobolev and $L^∞$ norms, and prove continuity of the thermal equilibrium density with respect to potential perturbations via a Banach fixed-point approach. They obtain the linear-statistics expansions, show the density’s potential-dependence is continuous, and, crucially, interpolate between Gaussian and general potentials to deduce the $1/N$ expansion of $\log \mathcal{Z}_N$; the leading term equals the negative of the equilibrium energy plus an entropy term, with the next-order correction provided by the inverse master operator. The results illuminate how entropy competes with energy at high temperature and establish a robust method that yields precise asymptotics for a broad class of partition-function integrals, with potential extensions to more general confining potentials and interactions.

Abstract

We consider the real $β$-ensemble (or 1D log-gas) of dimension $N$ in the high-temperature regime, \textit{i.e.} where the inverse temperature $β$ scales as $Nβ=2P$ 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 $V(x)=x^2+φ(x)$ with $φ$ 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-$β$ 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-$β$ 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.

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

TL;DR

This work analyzes the real -ensemble in the high-temperature regime with , proving an all-orders large- expansion for the partition function when and is bounded. The authors develop a loop-equation framework built around the master operator and its inverse, establish sharp controls on in Sobolev and norms, and prove continuity of the thermal equilibrium density with respect to potential perturbations via a Banach fixed-point approach. They obtain the linear-statistics expansions, show the density’s potential-dependence is continuous, and, crucially, interpolate between Gaussian and general potentials to deduce the expansion of ; the leading term equals the negative of the equilibrium energy plus an entropy term, with the next-order correction provided by the inverse master operator. The results illuminate how entropy competes with energy at high temperature and establish a robust method that yields precise asymptotics for a broad class of partition-function integrals, with potential extensions to more general confining potentials and interactions.

Abstract

We consider the real -ensemble (or 1D log-gas) of dimension in the high-temperature regime, \textit{i.e.} where the inverse temperature scales as with a fixed positive parameter. We establish the large- 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 with 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- case, but rather scales at the same order in . Consequently, at large , 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- 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.
Paper Structure (42 sections, 47 theorems, 341 equations)

This paper contains 42 sections, 47 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Assumptions
  4. Main results
  5. Connection with the literature and motivations
  6. Comparison with the fixed-temperature regime
  7. The high-temperature regime
  8. Asymptotic analysis of multiple integrals
  9. Link with integrable systems
  10. The thermal equilibrium measure
  11. Outline of the proof
  12. Expansion of linear statistics
  13. Controls on $\Xi^{-1}$
  14. Potential dependence of the equilibrium density
  15. Notations and conventions
  16. ...and 27 more sections

Key Result

Theorem 1.2

Assume $V$ satisfies assumptions assumptions, then for all smooth functions $\phi\in H^{r}(\mathbb{R}^k)$ for $r>0$ (depending on $K$) big enough, there exists a unique sequence $(b_i)_{i\geq\lceil k/2\rceil}$ depending on $V$, $\phi$ and $P$ such that for all $K\in\mathbb{N}$:

Theorems & Definitions (101)

  • Theorem 1.2: Asymptotic expansion of linear statistics
  • Theorem 1.3
  • Theorem 1.4: Asymptotic expansion of the partition function
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4: A priori bound on linear statistics
  • Proof
  • Definition 3.1: Extension of operators
  • Definition 3.2
  • ...and 91 more