Nonlinear large deviation bounds with applications to traces of Wigner matrices and cycles counts in Erd\"os-Renyi graphs

TL;DR

This paper develops broad nonlinear large deviation bounds applicable to various distributions, and demonstrates their use in analyzing traces of Wigner matrices, cycles in Erdös-Renyi graphs, and the Ising model.

Contribution

It introduces a new approach to nonlinear large deviation estimates that do not require second order smoothness, expanding applicability to diverse problems.

Findings

Large deviation bounds for Wigner matrix traces and Erdös-Renyi cycle counts.

Application of bounds to Ising model partition function approximation.

Effective analysis down to the connectivity threshold in sparse graphs.

Abstract

We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries, and the upper tail of cycles counts in sparse Erd\"os-Renyi graphs down to the connectivity threshold $n^{-1/2}$.