Spectrum of Markov generators on sparse random graphs

Understand

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 nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d.
• random weights.
• Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling.