scholarly journals Semicircle law for random matrices of long-range percolation model

2009 ◽  
Vol 17 (1) ◽  
pp. 1-35 ◽  
Author(s):  
S. Ayadi
Keyword(s):  
Random Matrices ◽  
Long Range ◽  
Percolation Model ◽  
Semicircle Law

Scaling properties of a percolation model with long-range correlations

1996 ◽  
Vol 54 (4) ◽  
pp. 3870-3880 ◽  
Author(s):  
Muhammad Sahimi ◽  
Sumit Mukhopadhyay
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Scaling Properties ◽  
Long Range Correlations

scholarly journals EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250007 ◽  
Author(s):  
S. DALLAPORTA
Keyword(s):  
Random Matrices ◽  
Spectral Measure ◽  
Wasserstein Distance ◽  
Covariance Matrices ◽  
Finite Range ◽  
Variance Bounds ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Wigner Random Matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

scholarly journals Percolation of Words on Zd with Long-Range Connections

2011 ◽  
Vol 48 (4) ◽  
pp. 1152-1162 ◽  
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model on Zd, with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1}N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

scholarly journals Random Matrices From Linear Codes and Wigner’s Semicircle Law

2019 ◽  
Vol 65 (10) ◽  
pp. 6001-6009
Author(s):  
Chin Hei Chan ◽  
Enoch Kung ◽  
Maosheng Xiong
Keyword(s):  
Random Matrices ◽  
Linear Codes ◽  
Semicircle Law

Wigner matrices

2018 ◽  
Author(s):  
Peter Forrester
Keyword(s):  
Simple Model ◽  
Random Matrices ◽  
Symmetric Matrices ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Global Properties ◽  
Wigner Random Matrices ◽  
Local Properties ◽  
Concentration Properties ◽  
Adjoint Operators

This article reviews some of the important results in the study of the eigenvalues and the eigenvectors of Wigner random matrices, that is. random Hermitian (or real symmetric) matrices with iid entries. It first provides an overview of the Wigner matrices, introduced in the 1950s by Wigner as a very simple model of random matrices to approximate generic self-adjoint operators. It then considers the global properties of the spectrum of Wigner matrices, focusing on convergence to the semicircle law, fluctuations around the semicircle law, deviations and concentration properties, and the delocalization of the eigenvectors. It also describes local properties in the bulk and at the edge before concluding with a brief analysis of the known universality results showing how much the behaviour of the spectrum is insensitive to the distribution of the entries.

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