The circular law for random regular digraphs with random edge weights

We consider random n × n matrices of the form Yn = 1 dAn Xn, where An is the adjacency matrix of a uniform random d-regular directed graph on n vertices, with d = ?pn? for some fixed p (0, 1), and Xn is an n × n matrix of i.i.d. centered random variables with unit variance and finite (4 + ?)th moment (here denotes the matrix Hadamard product). We show that as n →∞, the empirical spectral distribution of Yn converges weakly in probability to the normalized Lebesgue measure on the unit disk.

Duke Scholars

Author Nicholas A Cook Mathematics