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September 2020 Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős–Rényi graphs
Fanny Augeri
Ann. Probab. 48(5): 2404-2448 (September 2020). DOI: 10.1214/20-AOP1427

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 Erdős–Rényi graphs down to the sparsity threshold $n^{-1/2}$.

Citation

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Fanny Augeri. "Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős–Rényi graphs." Ann. Probab. 48 (5) 2404 - 2448, September 2020. https://doi.org/10.1214/20-AOP1427

Information

Received: 1 October 2018; Revised: 1 December 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152647
Digital Object Identifier: 10.1214/20-AOP1427

Subjects:
Primary: 05C80, 52A40, 60B20, 60F10

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 5 • September 2020
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