Blanket times and the Gaussian Free Field

The blanket time of a random walk on a graph G is the expected time until the proportion of time spent at each vertex approximates the stationary distribution. In 1996, Winkler and Zuckerman conjectured that the blanket time is of the same order as the cover time. Ding, Lee and Peres proved this conjecture in 2010 by relating both terms to the maximum of the Gaussian free field on G. We outline the connection between the blanket time and the Gaussian free field. 1. Covert Times and Blanket Times We begin by defining the blanket time of a random walk, discussing an application, historical aspects, and its relationship to the cover time. Before doing so, we must review several important concepts. Let G = (V,E) be an undirected graph with n vertices and m edges, and let {Xt} be a random walk on G. For τ v cov = min{t : V ⊂ {Xs}s=0} where X0 = v, we define the cover time tcov = max v E(τ v cov). Here, τ v cov is the (random) exact time it takes for {Xt} to visit every state when starting at v, while tcov is the expected time to visit every state, starting from the worst possible v. Recall the hitting time Hu(v) is the expected time for the random walk to travel from u to v. We state the best bound for the cover time: Theorem 1.1 (Matthews’ Theorem). For any G with |V | = n, min u,v Hu(v)n log n ≤ tcov ≤ max u,v Hu(v)n log n. The cover time can be viewed as a measurement of certain types of connectivity for the random walk. Familiar concepts in this vein include the hitting time and the commute time κuv = Hu(v) + Hv(u), which measures the expected time to go from u to v and back. The blanket time time is a similar quantity, measuring the expected time it takes to visit every state approximately proportional to the stationary distribution. In order to define the blanket time, we must quantify the amount of time spent at each state. For the remainder of the paper, it is assumed that X0 has some known initial distribution ν. The local time at v is Lt = E( ∑t s=0 1Xs=v) πv . This quantity represents the proportion of time spent at v up to time t. Note