Spectrum of non Hermitian random Markov matrices with heavy tailed weights

Abstract

This thesis concerns the convergence of the empirical spectral distribution of random matrices, that is the probability measure concentrated on the spectrum f_1(M); :::; _n(M)g of a complex n_n matrix M. Namely we de_ne the empirical spectral distribution _M as _M := 1/n Xn i=1 __i(M): We will consider matrices M whose entries will be random variables, as a consequence _M will be a random probability measure. The study of the spectrum of random matrices goes back to the 050 when the Hungarian physicist Eugene Wigner, proved that the empirical spectral distribution (ESD) of a sequence of Hermitian random matrices, whose entries are independent random variables with unitary variance, up to rescaling weakly converges to the probability measure _sc(dx) = 1 2_ p 4 􀀀 x2 1jxj<2dx: The law _sc is the so called Wigner semi-circular law. In the same years Wigner conjectured that the limit spectral distribution of a sequence of non Hermitian matrices with unitary variance independent random entries, un to rescaling, is the uniform law on the unitary disc of the complex plane. The proof of the conjecture has a long history, it indeed has been proved by Tao and Vu in 2009, after more than 40 years of partial results. We just mention very few of them, for a more detailed sequence of the results we refer to [9, Section 2]. The _rst piece of proof for the circular law theorem is by Metha in [12], who in 1967, using the work of Ginibre [11], proved the result for the expected empirical distribution for complex Gaussian entries. In 1997 Bai, in [2], is the _rst to obtain results in the universal case, using the work of Vyacheslav Girko, but assuming stronger hypothesis on the law of the entries, such as bounded density and _nite sixth moment. Finally in 2009, Tao and Vu proved the original conjecture, in [14]. The proof is based on the Girko Hermitization method, a trick to pull back the non-Hermitian problem to Hermitian matrices. We give more details in chapter 1, but the core of the method is to work with the spectrum of the singular values of the non- Hermitian matrix and to take advantage of their logarithmic bond with the spectrum of the eigenvalues of M. In 2008 Ben Arous and Guionnet, proved a heavy tailed counterpart of the Wigner theorem, see also Zakharevich [15]. They indeed proved an existence result for the limiting spectral distribution of a sequence of Hermitian matrices whose entries are independent with common law in the domain of attraction of the _-stable law, with _ 2 (0; 2). As in the _nite second moment scenario, the limiting spectral measure does not depend on the law of the entries of the matrix, but only on the parameter _. Remarkable work in this regime is also by Belinschi, Dembo and Guionnet [3]. These works established rigorously a number of prediction made by physicists Bouchaudand and Fizeau [10] In 2010 Bordenave, Caputo, and Chafa _ [5], with a new and independent approach based on the objective method introduced by Aldous and Steele in [1], give an alternative proof of the convergence in the heavy tailed setting. They prove that the heavy tailed matrix, suitably rescaled, locally converges to an in_nite poissonian weighted tree called PWIT. The heavy tails regime is in many ways more di_cult than the bounded variance regime. Indeed we do not have an explicit expression for the limiting distribution. Nevertheless the approach of [5], is powerful enough to give some properties of the limiting spectral measure, by means of recursive analysis on the limiting tree. In 2012 the same authors in [6], prove an analogous of the Circular law, for non- Hermitian matrices with i.i.d. heavy tailed entries, using the same approach from [5] combined with the Hermitization techniques by Tao and Vu. Very little is known if the entries of the matrix are not independent. An interesting problem with non-independent entries is obtained by considering Markov matrices, i.e. matrices with non-negative entries and row sum equal to 1. In this case is natural to associate the random matrix with the corresponding weighted random graph, and to interpret the elements of the matrix as the transition probabilities of the random walk on the graph. Suppose Ui;j is a collection of i.i.d. random variables, and de_ne the Markov matrix Xn = (Xi;j)n i;j=1, Xi;j = Ui;j _i _i = Xn j=1 Ui;j : By construction, the spectrum of Xn is a subset of fz 2 C : jzj _ 1g. The convergence of ESD of P has been investigated in recent works by Bordenave, Caputo, and Chafa _. When the variables fUi;jg have _nite variance _2 2 (0;+1) and unitary mean, _p n _ Xn behaves as the ESD of pUn n_2 where Un = (Ui;j) is the non normalized matrix. Namely it converges to the circular law when Un is non-Hermitian, see [7], and to the Wigner semi-circular law when Un is Hermitian, see [4]. This analysis can be extended to other models with non-independent entries such as random Markov generators and zero sum matrices, see the work of Bordenave, Caputo, and Chafa _ [8] and Tao [13]. In [5], Bordenave, Caputo, and Chafa _, study the markovian case when Un is a symmetric heavy tailed random matrix. Of remarkable interest is the case _ 2 (0; 1). In this regime the ESD of the matrix Xn, without any scaling factor, converges to a non trivial measure concentrated on the unitary disc of the complex plane. In this work we study the non-Hermitian version of this heavy tailed matrix, that is the case where Un has i.i.d. heavy tailed entries, with _ 2 (0; 1). Our main result concerns the convergence of the ESD of the associated Markov matrix Xn, to a non trivial measure supported in the unitary disc of the complex plane, depending only on the parameter _. In contrast with the Hermitian case, this limiting distribution should have a remarkable concentration on a disc with radius r < 1.