Ágnes Backhausz: On the eigenvectors of random sign matrices and their connection to action convergence

Speaker: Ágnes Backhausz

Title: On the eigenvectors of random sign matrices and their connection to action convergence

Abstract: Random sign matrices (in which all entries are independent and  equal to  +1 or -1 with equal probabilities) are much less understood than their symmetric counterparts (which are special Wigner matrices). In a recent work, by defining the notion of action convergence (which unifies dense graph convergence and local-global convergence), we could use methods coming from graph limit theory to understand some of their spectral properties. Namely, we could prove that the empirical distribution of the eigenvectors of random sign matrices (normalized with n^{1/2}) cannot be too far from the Gaussian distribution. The upper bound is the strongest for eigenvalues with absolute value 1; in this case the limit is the Gaussian distribution itself. This result will be presented in the talk, together with the notion of action convergence, entropy inequalities for typical measures related to random sign matrices and other tools that are used in the proofs. Joint work with Balázs Szegedy.