REM universality for linear random energy

TL;DR

This paper proves that for a class of linear random Hamiltonians, the distribution of energy levels converges to a Poisson process, demonstrating a form of universality known as REM universality, even with exponentially many sampled configurations.

Contribution

The work extends REM universality to models with exponentially many configurations and characterizes fluctuations and Gibbs weights.

Findings

Energy levels converge to a Poisson process with exponential intensity.

Universality holds for exponentially many sampled configurations.

Asymptotic distribution of Gibbs weights is derived.

Abstract

We consider a sequence of random Hamiltonians $H_n(h,\sigma)=\sum^n_{i=1}h_i(\sigma_i-m)$, and study the asymptotic ($n\to \infty$) distribution of the energy levels $(H_n(h,\sigma))_{\sigma\in \{-1,1\}^n}$, where $h_1,h_2,\cdots$ are i.i.d. random variables. We show that, when $e^{O(n)}$ configurations are sampled at random, the corresponding collection of energy levels converges in distribution to a Poisson point process with exponential intensity measure. This establishes the Random Energy Model (REM) universality for the present model. Our results strengthen earlier works on local REM universality by characterizing the distribution of $O(1)-$order fluctuations of $H_n$. In addition, we improve upon the REM universality by dilution studied by Ben Arous, Gayrard, Kuptsov by allowing an exponentially large number $e^{O(n)}$ of sampled configurations, instead of $e^{o(\sqrt{n})}$. Finally, we derive the asymptotic distribution of the Gibbs weight.