## Metropolis on Cayley graphs

Define a one-parameter family $$\mu(p), 0 < p < 1$$ of probability distributions on a finite Cayley graph by:
$$\mu(p)$$ is the distribution of $$X(T_p-1)$$ , where $$X(i)$$ is random walk started at the identity, and $$T_p$$ has Geometric(p) distribution.
Take $$\mu(0)$$ to be the uniform distribution. Now consider the reversible Markov chain defined by:
the Metropolis chain, based on random walk, with stationary distribution $$\mu(p)$$.
This has some relaxation time $$\tau (p)$$. Can one prove anything about τ(p) in this general setting? For instance, can one prove
(i) $$\tau (p)$$ is monotone decreasing in $$p$$?
(ii) $$\tau (p)\leq C \tau (\infty )$$ for some universal C ?
(ii) Some decreasing bound on $$\tau (p)$$ in terms of $$\tau (\infty ), p$$, number of states and familiar parameters of the graph?

History First written here March 2009, but mentioned in conversation for several years earlier.