The empirical spectral distribution of symmetric random matrices with correlated entries - An asymptotic analysis employing the method of moments
Michael Fleermann, UT-EWI-SOR
A random matrix is nothing other than a matrix containing random variables. Although contemporary analysis of random matrices is of rather theoretical interest, its origins were in the applications, ranging from data analysis to energy levels of heavy nuclei. From the beginning of the investigations the focus was on the empirical distribution of eigenvalues when matrix dimensions reach infinity. For a large class of random matrices – especially for suitable models with independent entries - this empirical spectral distribution converges weakly almost surely against the famous semicircle distribution. This result is called Wigner’s Semicircle law and can be regarded as the central limit theorem for random matrices. Ever since its establishment by Wigner, great efforts have been made to analyze the universality of the semicircle law. For example, in which way can one allow for correlation within the random matrices without jeopardizing the semicircle law?
In this presentation we will embark on a journey into the investigation of random matrices. To prepare a survival kit, we will first establish results about weak convergence, stochastic weak convergence, the method of moments and the stochastic method of moments. On the way, we will learn about the semicircle distribution and its mysterious connection to an ubiquitous sequence of numbers – the Catalan numbers. Having reached the top the hill of universality, we will obtain the semicircle law for random matrices with weakly correlated entries.