Markov chain decomposition for convergence rate analysis

Abstract

In this paper we develop tools for analyzing the rate at which a reversible Markov chain converges to stationarity. Our techniques are useful when the Markov chain can be decomposed into pieces which are themselves easier to analyze. The main theorems relate the spectral gap of the original Markov chains to the spectral gaps of the pieces. In the first case the pieces are restrictions of the Markov chain to subsets of the state space; the second case treats a Metropolis-Hastings chain whose equilibrium distribution is a weighted average of equilibrium distributions of other Metropolis-Hastings chains on the same state space.

On the swapping algorithm

“…The key result for this is a theorem due to Caracciolo, Pelissetto, and Sokal [2]. For more detailed discussion and a proof of this result, see Madras and Randall [20].…”

Dichotomy for Holant Problems of Boolean Domain

“…Great progress has been made on the complexity of partition functions, giving classification theorems [16,4,19,37,6,15,5] in terms of polynomial time tractability or #P-hardness. A major further research direction is when a #P-hard partition function can be approximated [22,14,12,23,30,33,18]. Now consider the problem of counting perfect matchings.…”

Random sampling of 3‐colorings in ℤ2

“…The free-boundary problem was considered by Madras and Randall [11] as a possible application of their new decomposition method, but the application contained an error [12]. In [12] they stated that the free-boundary problem was still open.…”