Random reversible Markov matrices with tunable extremal eigenvalues

Abstract

Random sampling of large Markov matrices with a tunable spectral gap, a nonuniform stationary distribution and a nondegenerate limiting empirical spectral distribution (ESD) is useful. Fix $c>0$ and $p>0$. Let $A_{n}$ be the adjacency matrix of a random graph following $\mathrm{G}(n,p/n)$, known as the Erdős–Rényi distribution. Add $c/n$ to each entry of $A_{n}$ and then normalize its rows. It is shown that the resulting Markov matrix has the desired properties. Its ESD weakly converges in probability to a symmetric nondegenerate distribution, and its extremal eigenvalues, other than 1, fall in $[-1/\sqrt{1+c/k},-b]\cup[b,1/\sqrt{1+c/k}]$ for any $0<b<1/\sqrt{1+c}$, where $k=\lfloor p\rfloor+1$. Thus, for $p\in(0,1)$, the spectral gap tends to $1-1/\sqrt{1+c}$.