Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

TL;DR

This paper characterizes the perfect freezing transition in Gaussian beta ensembles, showing that almost sure equidistribution occurs if and only if the inverse temperature grows at least logarithmically with the number of particles, using Beurling-Landau densities.

Contribution

It establishes a precise growth condition on inverse temperature for equidistribution in Gaussian beta ensembles, extending techniques from Coulomb gas models to one dimension.

Findings

Almost sure equidistribution occurs if eta_nollows log n growth.

Fekete configurations are uniformly spread with respect to a regularized semicircle law.

Growth rate log n is necessary and sufficient for perfect freezing in these ensembles.

Abstract

The Gaussian $\beta$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$ picked randomly with respect to the Boltzmann factor $e^{-\frac\beta 2H_n}$, $H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The point process $\{x_j\}_1^n$ tends to follow the semicircle law $\sigma(x)=\tfrac 1{2\pi}\sqrt{(4-x^2)_+}$ in certain average senses. A Fekete configuration (minimizer of $H_n$) is spread out in a much more uniform way in the interval $[-2,2]$ with respect to the regularization $\sigma_n(x)=\max\{\sigma(x),n^{-\frac 1 3}\}$ of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to $\sigma_n(x)$, in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences $\beta_n$ of inverse temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We find that a necessary and sufficient condition is that $\beta_n$ grows at least logarithmically in $n$: $$\beta_n\gtrsim \log n.$$ We call this growth rate the perfect freezing regime. We give several further results on the distribution of particles when $\beta_n\gtrsim\log n$, for example on minimal spacing, discrepancies, and sampling and interpolation for weighted polynomials. The condition $\beta_n\gtrsim\log n$ was introduced by some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.