Cutoff phenomena for random walks on random regular graphs
  The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\geq 3$ is an expander, and even essentially Ramanujan, implying a mixing-time of $O(\log n)$. According to a conjecture of Peres, the simple random walk on $\G(n,d)$ for such $d$ should then exhibit cutoff with high probability. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is w.h.p. $(6+o(1))\log_2 n$.   In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d\geq3$, the simple random walk on $\G(n,d)$ w.h.p. has cutoff at $\frac{d}{d-2}\log_{d-1} n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random walk on $\G(n,d)$ w.h.p. has cutoff already at $\log_{d-1} n$ with constant window order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers. 
Eyal Lubetzky, Allan Sly