On fluctuations of eigenvalues of random permutation matrices

Arous, Gérard Ben; Dang, Kim

doi:10.1214/13-aihp569

Cited by 11 publications

(20 citation statements)

References 41 publications

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“…Note that the speed of the central limit theorem is N −1/2 as for independent integrable random variables, but differently from what happens for standard Wigner's matrices. This phenomenon has also already been observed for adjacency matrix of random graphs [9,24] and we will see below that it also holds for Lévy matrices. It suggests that the repulsive interactions exhibited by the eigenvalues of most models of random matrices with lighter tails than heavy tailed matrices no longer work here.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 74%

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Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

Benaych-Georges

Guionnet

Male

2014

Commun. Math. Phys.

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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 α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi 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.

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Section: Introduction and Statement Of Resultssupporting

confidence: 74%

“…(3) About recentering with respect to the limit instead of the expectation, it depends on the rate of convergence in (9) or in (6). For instance, if NE(a 2k 11 )−C k = o(N −1/2 ) for any k ≥ 1, then…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

Benaych-Georges

Guionnet

Male

2014

Commun. Math. Phys.

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“…A limiting Gaussian wave character of eigenvectors have also been conjectured [14][15][16]. Some fine properties of eigenvalues and eigenvectors can indeed be proved for a single permutation matrix; see [33] and [4].…”

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confidence: 99%

Cycles and eigenvalues of sequentially growing random regular graphs

Johnson¹,

Pal²

2014

Ann. Probab.

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Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge.Comment: Published in at http://dx.doi.org/10.1214/13-AOP864 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…Therefore 1 2 S has the same eigenbasis as M (σ, 1) but the eigenvalues are projected to the real axis. If σ is a cycle of length n, then the eigenvalues of the corresponding permutation matrix are exp(2πim/n) with 0 ≤ m < n (see [4] or [26]). Thus the eigenvalues of S(σ) are 2 cos(0), 2 cos 2πi n , .…”

Section: Introductionmentioning

confidence: 99%

The characteristic polynomial of a random permutation matrix at different points

Dang

Zeindler

2014

Stochastic Processes and their Applications

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Abstract. We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study also the wreath product of permutation matrices and diagonal matrices with iid entries and more general class functions on the symmetric group with a multiplicative structure..

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On fluctuations of eigenvalues of random permutation matrices

Cited by 11 publications

References 41 publications

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

Cycles and eigenvalues of sequentially growing random regular graphs

The characteristic polynomial of a random permutation matrix at different points

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