We consider a gas of N particles with a general two-body interaction and confined by an external potential in the mean field or high temperature regime, that is when the inverse temperature satisfies $\beta N \to \kappa \ge 0$ as $N\to+\infty$. We show that under general conditions on the interaction and the potential, the local fluctuations are described by a Poisson point process in the large N limit. We present applications to Coulomb and Riesz gases on $\mathbb{R}^n$ for any $n\ge 1$, as well as to the edge behavior of $\beta$-ensembles on $\mathbb{R}$.

Poisson statistics for Gibbs measures at high temperature

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $\beta N \to const \in (0, \infty)$, with $N$ the system size and $\beta$ the inverse temperature. In this regime, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle by Liu and Wu (Stochastic Processes and their Applications (2019)). This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.

Poisson statistics for beta ensembles on the real line at high temperature

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $$\beta N \rightarrow const \in (0, \infty )$$, with N the system size and $$\beta $$ the inverse temperature. For the global behavior, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle. This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.

It was shown in [J. A. Ramirez, B. Rider and B. Virag. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $\beta$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/\beta$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R}$ of intensity $e^x dx$ and that the eigenfunctions converge to Dirac masses centered at IID points with exponential laws. Furthermore, we obtain a precise description of the microscopic behavior of the eigenfunctions near their localization centers.

The stochastic Airy operator at large temperature

We consider the Gaussian $$\beta $$
 -ensemble when $$\beta $$
 scales with n (the number of particles) such that $$\displaystyle {{n}^{-1}\ll \beta \ll 1}$$
 . Under a certain regime for $$\beta $$
 , we show that the largest particle satisfies a large deviations principle in $$\mathbb {R}$$
 with speed $$n\beta $$
 and explicit rate function. As a consequence, the largest particle converges in probability to 2, the rightmost point of the semicircle law.

Large Deviations Principle for the Largest Eigenvalue of the Gaussian $$\beta $$β-Ensemble at High Temperature

We study the asymptotic edge statistics of the Gaussian $\beta$-ensemble, a collection of $n$ particles, as the inverse temperature $\beta$ tends to zero as $n$ tends to infinity. In a certain decay regime of $\beta$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.

/pdf/poisson-statistics-at-the-edge-of-gaussian-beta-ensembles-at-29vblsln65.pdf

Cambyse Pakzad

Papers

Poisson statistics at the edge of Gaussian beta-ensembles at high temperature