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The distribution of eigenvalues of randomized permutation matrices

Joseph Najnudel, Ashkan Nikeghbali (2013)

Annales de l’institut Fourier

In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter θ > 0 ) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations....

The local relaxation flow approach to universality of the local statistics for random matrices

László Erdős, Benjamin Schlein, Horng-Tzer Yau, Jun Yin (2012)

Annales de l'I.H.P. Probabilités et statistiques

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues {xj}j=1N are close to their classical location {γj}j=1N determined by the limiting density of eigenvalues. Under...

The random paving property for uniformly bounded matrices

Joel A. Tropp (2008)

Studia Mathematica

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of...

The rate of convergence for spectra of GUE and LUE matrix ensembles

Friedrich Götze, Alexander Tikhomirov (2005)

Open Mathematics

We obtain optimal bounds of order O(n −1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

The Wigner semi-circle law and the Heisenberg group

Jacques Faraut, Linda Saal (2007)

Banach Center Publications

The Wigner Theorem states that the statistical distribution of the eigenvalues of a random Hermitian matrix converges to the semi-circular law as the dimension goes to infinity. It is possible to establish this result by using harmonic analysis on the Heisenberg group. In fact this convergence corresponds to the topology of the set of spherical functions associated to the action of the unitary group on the Heisenberg group.

Transforming stochastic matrices for stochastic comparison with the st-order

Tuğrul Dayar, Jean-Michel Fourneau, Nihal Pekergin (2003)

RAIRO - Operations Research - Recherche Opérationnelle

We present a transformation for stochastic matrices and analyze the effects of using it in stochastic comparison with the strong stochastic (st) order. We show that unless the given stochastic matrix is row diagonally dominant, the transformed matrix provides better st bounds on the steady state probability distribution.

Transforming stochastic matrices for stochastic comparison with the st-order

Tuğrul Dayar, Jean-Michel Fourneau, Nihal Pekergin (2010)

RAIRO - Operations Research

We present a transformation for stochastic matrices and analyze the effects of using it in stochastic comparison with the strong stochastic (st) order. We show that unless the given stochastic matrix is row diagonally dominant, the transformed matrix provides better st bounds on the steady state probability distribution.

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