Matrizes circulantes aleatórias.

Abstract

The Random Matrix Theory (RMT) is a very important theory known by Wigner’s works in 1955 that solved several numerous problems of many areas of science. A wellknown problem is that of matrix integrals. Brezin et al (1) were the first to note the relation between matrix integrals and map enumeration. It has been found that integrals on gaussian hermitian ensemble of the 2k matrix elements product can be calculated by summing the genus g maps with k edges. We discuss about the sprectrum of stochastic circulant random matrices. It was verified that the spectrum of circulant Hankel matrices has Rayleigh distribution, while circulant Toeplitz matrices is the set of all convex combinations of the roots of the unit. The second largest eigenvalue has Tracy-Widom distribution for Hankel matrices and a Gumbel distribution for Toeplitz matrices. The repulsion of eigenvalues of Hankel matrices has the same distribution of matrices with independent entries. The product between circulant matrices has an interesting algebra: the Hankel matrices product is a Toeplitz matrix, but the Toeplitz matrix product remains a Toeplitz matrix. As we multiply more matrices, the combinations of the permutation matrices are no longer convex.