The second largest eigenvalue of a transition matrix @math has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when @math is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a generalization of the well-known Friedman theorem, for undirected regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015).
We finally mention some related questions and conjectures. What is the link between @math and the cutoff phenomenon for the Markov chain ? Do graphs in our model having @math exibit cutoff ? Is the upper bound optimal ? In the Friedman theorem, the difficult part was to prove the upper bound while the lower bound had been proven very early ( @cite_26 ) using the full strength of the symmetric nature of @math . We have proven an upper bound for our model, but no lower bound is known yet. This paper deals with the second eigenvalue of random digraphs in general. In the specific case of @math -regular digraphs, it is conjectured in [Section 7] bordenave_chafai_survey that the whole empirical spectral measure of the adjacency matrix of a @math -regular digraph converges almost surely in distribution to @math , a complex version of the Kesten-McKay distribution, namely [ ( d z) = ^ -1 d^2(d-1) (d^2 - |z|^2)^2 1 _ |z| d d z. ]
@cite_26
2036878392
Abstract It is shown that the second largest eigenvalue of the adjacency matrix of any d-regular graph G containing two edges the distance between which is at least 2k + 2 is at least 2 d − 1 − (2 d − 1 − 1) (k+1) .