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

We consider a model for 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{ie} where the inverse temperature $\beta_N$ scales as $N\beta_N\rightarrow2P>0$. 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 an extension of [Ben-Arous, Dembo, Guionnet 2001] and [Pakzad 2020 where such estimates were shown for the largest particle of the $\beta$-ensemble respectively at fixed $\beta_N=\beta>0$ and $\beta_N\gg N^{-1}$. 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.