RANDOM REVERSIBLE MARKOV MATRICES WITH TUNABLE EXTREMAL EIGENVALUES

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 An be the adjacency matrix of a random graph following G(n, p/n), known as the Erdős–Rényi distribution. Add c/n to each entry of An 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 $[-1/\sqrt{1+\mathrm{c}/\mathrm{k}},-\mathrm{b}]\cup [\mathrm{b},1/\sqrt{1+\mathrm{c}/\mathrm{k}}]$ for any $0