Cutoff Phenomenon for Random Walks on Kneser Graphs

The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers n and k, the Kneser graph K(2n+ k, n) is defined as the graph with vertex set being all subsets of {1, . . . , 2n+k} of size n and two vertices A and B being connected by an edge if A∩B = ∅. We show that for any k = O(n), the random walk on K(2n+ k, n) exhibits a cutoff at 1 2 log1+k/n (2n+ k) with a window of size O( n k ).