Spatial mixing and nonlocal Markov chains*. Issue 3 (6th March 2019)

• Record Type: Journal Article
• Authors: Blanca, Antonio; Caputo, Pietro; Sinclair, Alistair; Vigoda, Eric
• Is Part Of: Random structures & algorithms. Volume 55:Issue 3(2019)
• Page Start: 584
• Page End: 614
• Publication Date: 2019-03-06
• Subjects: block dynamics -- Markov chains -- spin systems -- strong spatial mixing -- Swendsen‐Wang dynamics; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20844
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : We consider spin systems with nearest‐neighbor interactions on an n ‐vertex d ‐dimensional cube of the integer lattice graph Z d . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O ( r ), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O (1) bound for the relaxation time of the Swendsen‐Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is O ( log n ( log log n ) 2 ) . Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.