Eigenvalue distribution of some nonlinear models of random matrices

Large Random Matrices: Eigenvalue Distribution

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is universal, is recovered and the three and four-point functions are given explicitly. One observes that higher order correlation functions are linear combinations of u...

Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secul...

Universality of some models of random matrices and random processes

Many limit theorems in probability theory are universal in the sense that the limiting distribution of a sequence of random variables does not depend on the particular distribution of these random variables. This universality phenomenon motivates many theorems and conjectures in probability theory. For example the limiting distribution in the central limit theorem for the suitably normalized su...

Random matrices and communication systems: WEEK 6 1 Wishart random matrices: joint eigenvalue distribution

Again, this can be viewed as a change of variables; on the left-hand side, there are n diagonal free parameters wjj and n(n−1) 2 off-diagonal free parameters wjk, j < k in the matrix W (the remaining offdiagonal parameters are fixed, as W is symmetric); on the right-hand side, there are n free parameters in the matrix Λ and (n − 1) + (n − 2) + . . . + 1 + 0 = n(n−1) 2 free parameters in the mat...

Distribution of Eigenvalues for Some Sets of Random Matrices