Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

Abstract

We consider a model of a gas of $N$ confined particles subject to a two-body repulsive interaction, namely the one-dimensional log or Riesz gas. We are interested in the so-called \textit{high temperature} regime, \textit{i.e.} when the inverse temperature is given by $\beta_N=2\alpha/N$ for some $\alpha>0$. We establish, in the log case, a large deviation (LD) principle and moderate deviations estimates for the largest particle $x_\mathrm{max}$ when appropriately rescaled. Our result is in the continuity of [Ben Arous Dembo Guionnet 01', Pakzad 20'] where such estimates were shown for the largest particle of the $\beta$-ensemble at fixed $\beta_N=\beta>0$ and $\beta_N\gg N^{-1}$ respectively. We show that the corresponding rate function is the same as in the case of iid particles. We also provide LD estimates in the Riesz case. Additionally, we consider related models of symmetric tridiagonal random matrices with independent entries having Gaussian tails; for which we establish the LD principle for the top eigenvalue. In a certain specialization of the entries, we recover the result for the largest particle of the log-gas. We show that LD are created by a few entries taking abnormally large values.