The hitting and cover times of Metropolis walks

Given a finite graph G = (V, E) and a probability distribution pi = (pi(v))(v is an element of V) on V. Metropolis walks, i.e., random walks on G building on the Metropolis-Hastings algorithm, obey a transition probability matrix P = (p(uv))(u,v is an element of V) defined by, for any u, v is an element of V. p(uv) = {1/du min{du pi v/dv pi u, 1} if v is an element of N(u), 1 - Sigma(w not equal u) p(uw) if u = v, otherwise, and are guaranteed to have pi as the stationary distribution, where N(u) is the set of adjacent vertices of u is an element of V and d(u) = vertical bar N(u)vertical bar is the degree of u. This paper shows that the hitting and the cover times of Metropolis walks are O(fn(2)) and O(fn(2) log n), respectively, for any graph G of order n and any probability distribution it such that f = max(u,v is an element of V) pi(u)/pi(v) < infinity. We also show that there are a graph G and a stationary distribution pi such that any random walk on G realizing pi attains Omega(fn(2)) hitting and Omega(fn(2) log n) cover times. It follows that the hitting and the cover times of Metropolis walks are Theta(fn(2)) and Theta(fn(2) log n), respectively. (C) 2010 Elsevier B.V. All rights reserved.