Largest eigenvalue of large random block matrices: A combinatorial approach

Abstract

We study the largest eigenvalue of certain block matrices where the number of blocks and the block size both increase with suitable conditions on their relative growth. In one of them, we employ a symmetric block structure with large independent Wigner blocks and in the other we have the Wigner block structure with large independent symmetric blocks. The entries are assumed to be independent and identically distributed with mean [Formula: see text] variance [Formula: see text] with an appropriate growth condition on the moments. Under our conditions the limit spectral distribution of these matrices is the standard semi-circle law. It is natural to ask if the extreme eigenvalues converge to the extreme points of its support, namely [Formula: see text]. We exhibit models where this indeed happens as well as models where the spectral norm converges to [Formula: see text]. Our proofs are based on combinatorial analysis of the behavior of the trace of large powers of the matrix.