\( \mu(p)\) 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
History First written here March 2009, but mentioned in conversation for several years earlier.