Spectrum of Non-Hermitian Heavy Tailed Random Matrices

By Charles Bordenave, Pietro Caputo, Djalil CHAFAÏ. 2010

66 citations recorded by the source corpus.

Let (Xjk)j,k ≥ 1 be i.i.d. complex random variables such that |Xjk| is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko’s circular law. Namely, under some additional smoothness assumptions on the law of Xjk, we prove that there exist a deterministic sequence an ~ n1/α and a probability measure μα on $${\mathbb{C}}$$ depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix $${(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}$$ converges weakly to μα as n → ∞. Our approach combines Aldous & Steele’s objective method with Girko’s Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous’ Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of μα. In contrast with the Hermitian case, we find that μα is not heavy tailed.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/35c1680da8e51ebcba339269bea5bb5067eee248

How researchers see this paper

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

In Computational Theory and Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Spectrum of Markov Generators on Sparse Random Graphs (2012, methodology): “This approach was introduced in the mathematical physics literature [21, 24, 34] for the analysis of non-Hermitian random matrices; see also [11] and [13, sec.”
Around the circular law (2011, methodology): “The approach taken from [26] involves many ingredients including the Hermitization of section 4.”
Localization and delocalization of eigenvectors for heavy-tailed random matrices (2012, methodology): “1 (Concentration for spectral measures [11]) Let A be an n × n random Hermitian matrix.”
Circular law theorem for random Markov matrices (2008, unknown): “This program is addressed in a separate paper [13].”
Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph (2009, background): “This follows from an a priori concentration estimate (see [12]).”