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Random incidence matrices: moments of the spectral density
 J. Stat. Phys
, 2001
"... We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large ..."
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We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of “small ” eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.
Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices, Journal of Theoretical Probability 20
 The Modulo 1 Central Limit Theorem and Benford’s Law for Products, to appear in the International Journal of Algebra.http://arxiv.org/abs/math/0607686 MN2
, 2007
"... Abstract. Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of nor ..."
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Cited by 7 (0 self)
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Abstract. Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides
A.: The distribution of the largest nontrivial eigenvalues in families of random regular graphs
 Exper. Math
, 2008
"... Keywords: Ramanujan graphs, random graphs, largest nontrivial eigenvalues, TracyWidom distribution Recently Friedman proved Alon’s conjecture for many families of dregular graphs, namely that given any ǫ> 0 “most ” graphs have their largest nontrivial eigenvalue at most 2 √ d − 1+ ǫ in absolute ..."
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Cited by 6 (0 self)
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Keywords: Ramanujan graphs, random graphs, largest nontrivial eigenvalues, TracyWidom distribution Recently Friedman proved Alon’s conjecture for many families of dregular graphs, namely that given any ǫ> 0 “most ” graphs have their largest nontrivial eigenvalue at most 2 √ d − 1+ ǫ in absolute value; if the absolute value of the largest nontrivial eigenvalue is at most 2 √ d − 1 then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding theory and cryptography. As many of these applications depend on the size of the largest nontrivial positive and negative eigenvalues, it is natural to investigate their distributions. We show these are wellmodeled by the β = 1 TracyWidom distribution for several families. If the observed growth rates of the mean and standard deviation as a function of the number of vertices holds in the limit, then in the limit approximately 52% of dregular graphs from bipartite families should be Ramanujan, and about 27 % from nonbipartite families (assuming the largest positive and negative eigenvalues are independent).
Sparse regular random graphs: Spectral density and eigenvectors
"... Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to ..."
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Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized. 1.
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs
"... Abstract. Trace formulae for dregular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit cont ..."
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Cited by 3 (0 self)
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Abstract. Trace formulae for dregular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only periodic orbits which contribute are the non back scattering orbits, and the smooth part in the trace formula coincides with the KestenMcKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with backscatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of dregular graphs and the theory of random matrices. 1. Introduction and
Eigenvalue Spacings for Quantized Cat Maps
"... According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions ..."
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According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms { \cat maps". Our numerical experiments indicate that for \generic" choices of cat maps, the unfolded consecutive spacings distribution in the irreducible components of the Nth quantization (given by the Ndimensional Weil representation) approaches the GOE/GSE law of Random Matrix Theory. For certain special \arithmetic " transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacings distribution follows Poisson statistics; we provide a sharp estimate in that direction.
DISTRIBUTION OF EIGENVALUES FOR THE ENSEMBLE OF REAL SYMMETRIC PALINDROMIC TOEPLITZ MATRICES
, 2005
"... Abstract. Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of nor ..."
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Abstract. Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges (weakly and almost surely), independent of p, to a distribution which is almost the Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices (matrices where the first row is a palindrome), and the resulting spectral measures converge (weakly and almost surely) to the Gaussian. 1.