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Resolvent of large random graphs
 Random Structures and Algorithms
"... We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieltjes transform of the spectral measure of such graphs. We illustrate our results on the unif ..."
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Cited by 16 (5 self)
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We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieltjes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, ErdösRényi graphs and graphs with a given degree sequence. We give examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. MSCclass: 05C80, 15A52 (primary), 47A10 (secondary). 1
Circular law theorem for random Markov matrices
 PROBAB THEORY RELATED FIELDS
, 2010
"... Let (Xjk)jk�1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ 2. Let M be the n × n random Markov matrix with i.i.d. rows defined by Mjk = Xjk/(Xj1+···+Xjn). In particular, when X11 follows an exponential law, the random matrix M belongs to the D ..."
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Cited by 14 (1 self)
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Let (Xjk)jk�1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ 2. Let M be the n × n random Markov matrix with i.i.d. rows defined by Mjk = Xjk/(Xj1+···+Xjn). In particular, when X11 follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ1,...,λn be the eigenvalues of √ nM i.e. the roots in C of its characteristic polynomial. Our main result states that with probability one, the counting probability measure 1 1 δλ1 n nδλn converges weakly as n → ∞ to the uniform law on the disk {z ∈ C: z  � m −1 σ}. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.
CENTRAL LIMIT THEOREMS FOR LINEAR STATISTICS OF HEAVY TAILED RANDOM MATRICES
"... Abstract. We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of αstable laws and entries with moments exploding with the dimension, as in the adjacency matrices of ErdösRényi grap ..."
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Cited by 4 (2 self)
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Abstract. We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of αstable laws and entries with moments exploding with the dimension, as in the adjacency matrices of ErdösRényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables. hal00769741, version 3 6 Sep 2013
Spectrum of nonHermitian heavy tailed random matrices
 Comm. Math. Phys
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SPECTRAL ANALYSIS OF 1D NEAREST–NEIGHBOR RANDOM WALKS WITH APPLICATIONS TO SUBDIFFUSIVE RANDOM TRAP AND BARRIER MODELS
, 905
"... Abstract. Given a family X (n) (t) of continuous–time nearest–neighbor random walks on the one dimensional lattice Z, parameterized by n ∈ N+, we show that the spectral analysis of the Markov generator of X (n) with Dirichlet conditions outside (0, n) reduces to the analysis of the eigenvalues and e ..."
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Abstract. Given a family X (n) (t) of continuous–time nearest–neighbor random walks on the one dimensional lattice Z, parameterized by n ∈ N+, we show that the spectral analysis of the Markov generator of X (n) with Dirichlet conditions outside (0, n) reduces to the analysis of the eigenvalues and eigenfunctions of a suitable generalized second order differential operator −DmnDx with Dirichlet conditions outside (0, 1). If in addition the measures dmn weakly converge to some measure dm, similarly to Krein’s correspondence we prove a limit theorem of the eigenvalues and eigenfunctions of −DmnDx to the corresponding spectral quantities of −DmDx. Applying the above result together with the Dirichlet–Neumann bracketing, we investigate the limiting behavior of the small eigenvalues of subdiffusive random trap and barrier models and establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions. Key words: random walk, generalized differential operator, Sturm–Liouville theory, random trap model, random barrier model, self–similarity. MSCclass: 60K37, 82C44, 34B24. 1.
0 SPECTRUM OF LARGE RANDOM REVERSIBLE MARKOV CHAINS: HEAVY TAILED WEIGHTS ON THE COMPLETE GRAPH
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"... limiting distributions of large heavy Wigner and arbitrary random matrices ..."
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limiting distributions of large heavy Wigner and arbitrary random matrices