Circular law for the sum of random permutation matrices

Abstract

Let $P_n^1,\dots , P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum _{\ell =1}^d P_n^\ell $. We show that if $\log ^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d} $ converges weakly to the circular law in probability as $n \to \infty $.