Spectrum of Markov generators on sparse random graphs

We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random n× n matrix L defined by Ljk = Xjk if k 6= j and Ljj = − ∑ k 6=j Ljk , where (Xjk)j 6=k are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n → ∞ of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n) → 0 as long as np(n) ≫ (log(n)). The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.