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November 2014 A large deviation principle for Wigner matrices without Gaussian tails
Charles Bordenave, Pietro Caputo
Ann. Probab. 42(6): 2454-2496 (November 2014). DOI: 10.1214/13-AOP866


We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb{P} (|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha }}$ for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+\alpha /2}$ with a good rate function $J(\mu)$ that is finite only if $\mu$ is of the form $\mu=\mu_{\mathrm{sc}}\boxplus\nu$ for some probability measure $\nu$ on $\mathbb{R} $, where $\boxplus$ denotes the free convolution and $\mu_{\mathrm{sc}}$ is Wigner’s semicircle law. We obtain explicit expressions for $J(\mu_{\mathrm{sc}}\boxplus\nu)$ in terms of the $\alpha $th moment of $\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.


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Charles Bordenave. Pietro Caputo. "A large deviation principle for Wigner matrices without Gaussian tails." Ann. Probab. 42 (6) 2454 - 2496, November 2014.


Published: November 2014
First available in Project Euclid: 30 September 2014

zbMATH: 1330.60012
MathSciNet: MR3265172
Digital Object Identifier: 10.1214/13-AOP866

Primary: 15A18‎ , 47A10 , 60B20

Keywords: Free convolution , large deviations , Local weak convergence , random matrices , random networks , spectral measure

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 6 • November 2014
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