The hitting and cover times of Metropolis walks

Given a finite graph G = (V, E) and a probability distribution π = (π_v)_{v ∈ V} on V, Metropolis walks, i.e., random walks on G building on the Metropolis-Hastings algorithm, obey a transition probability matrix P = (p_{u v})_{u, v ∈ V} defined by, for any u, v ∈ V, p_{u v} = {(frac(1, d_u) min {frac(d_u π_v, d_v π_u), 1}, if v ∈ N (u),; 1 - under(∑, w ≠ u) p_{u w}, if u = v,; 0, otherwise,) and are guaranteed to have π as the stationary distribution, where N (u) is the set of adjacent vertices of u ∈ V and d_u = | N (u) | is the degree of u. This paper shows that the hitting and the cover times of Metropolis walks are O (f n²) and O (f n² log n), respectively, for any graph G of order n and any probability distribution π such that f = max_{u, v ∈ V} π_u / π_v < ∞. We also show that there are a graph G and a stationary distribution π such that any random walk on G realizing π attains Ω (f n²) hitting and Ω (f n² log n) cover times. It follows that the hitting and the cover times of Metropolis walks are Θ (f n²) and Θ (f n² log n), respectively.