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

Abstract

We study the asymptotic edge statistics of the Gaussian β-ensemble, a collection of n particles, as the inverse temperature β tends to zero as n tends to infinity. In a certain decay regime of β, the associated extreme point process is proved to converge in distribution to a Poisson point process as n → +∞. 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.

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

“…For local statistics, in [7], the bulk eigenvalues are shown to exhibit Poissonian statistics by inspecting the correlation functions under the assumption nβ = γ ≥ 0. The corresponding result for the edge in the regime nβ 1 is treated by same means in [14]. Later, the bulk result for nβ = γ ≥ 0 is retrieved by Minami's method in [13] and an extension of the CLT to this new regime is introduced.…”

Large deviations for Ablowitz-Ladik lattice, and the Schur flow

“…for some constant λ P α , see [35,Lemma 3.2] for example. The fluctuations of the eigenvalues in the bulk and at the edge of a configuration were studied for example in [8,47,57,58,61]. These fluctuations were shown to be described by Poisson statistics in this regime.…”

Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $β$-ensemble and double confluent Heun equation

“…On the other hand, in [19,59,60] the authors reconstruct the densities from the moment generating functions. Several authors [12,38,44,45,48,61] investigated the local fluctuations of the eigenvalues, they observed that in this regime they are described by a Poisson process. In particular, in [38] Lambert studied the local fluctuations for general Gibbs ensembles on N -dimensional manifolds, moreover he also studied the asymptotic behaviour of the maximum eigenvalue for the classical beta ensemble at high-temperature.…”