A large deviation principle for Wigner matrices without Gaussian tails

By Charles Bordenave, Pietro Caputo. 2012

47 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/06be256acdff721ca60a40eadf0014c002bd362c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Large deviations for the largest eigenvalue of Rademacher matrices (2018, background): “Caputo [11] tackled the case of matrices with heavy tails, that is Wigner matrices with entries with stretched exponential tails, going to zero at infinity more slowly than a Gaussian tail.”
Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails (2015, methodology): “But this event has an infinite cost at the exponential scale Nα/2 since the empirical spectral measure follows a large deviation principle with speed N1+α/2 according to [9].”
Large deviations of empirical neighborhood distribution in sparse random graphs (2013, background): “ut that our Corollary 1.10 gives a corrected version of [18, Corollary 2.2]. Under a stronger sparsity assumption, large deviations of neighborhood distributions for random networks have been used in [10] to study the large deviations of the spectral measure of certain random matrices. 2. Local weak”