# On heavy-tail phenomena in some large deviations problems

## TL;DR

This paper investigates heavy-tail phenomena in large deviations, showing they can be explained by translation mechanisms, and applies these results to spectral measures, eigenvalues, traces, and last-passage times with heavy-tailed distributions.

## Contribution

It establishes a general large deviations principle for functionals under heavy-tailed measures, revealing translation as the key mechanism behind observed deviations.

## Key findings

- Heavy-tail phenomena explained by translation mechanisms.
- Large deviations principles for spectral measures and eigenvalues.
- Application to last-passage times with heavy-tailed weights.

## Abstract

In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borel, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Y_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the Wigner matrices without Gaussian tails, of the empirical spectral measure by Bordenave and Caputo, the largest eigenvalue and traces of polynomials by the author. We also apply our large deviations result to the last-passage time, which yields a large deviations principle when the weights have the density $Z_{\alpha}^{-1} e^{-x^{\alpha}}$ with respect to Lebesgue measure on $\mathbb{R}_+$, with $\alpha \in (0,1)$.