Spatial mixing and nonlocal Markov chains

Antonio Blanca Pimentel, Pietro Caputo, Alistair Sinclair, Eric Vigoda

Research output: Contribution to journalArticle

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 (ie, 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.

Original languageEnglish (US)
JournalRandom Structures and Algorithms
DOIs
StatePublished - Jan 1 2019

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Mixing Conditions
Markov processes
Markov chain
Spin Systems
Relaxation Time
Relaxation time
Monotone Systems
Imply
Convergence to Equilibrium
Mixing Time
Potts model
Spectral Gap
Functional Analysis
Potts Model
Ising model
Functional analysis
Linear algebra
Ising Model
Regular hexahedron
Nearest Neighbor

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

Cite this

Pimentel, Antonio Blanca ; Caputo, Pietro ; Sinclair, Alistair ; Vigoda, Eric. / Spatial mixing and nonlocal Markov chains. In: Random Structures and Algorithms. 2019.
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Spatial mixing and nonlocal Markov chains. / Pimentel, Antonio Blanca; Caputo, Pietro; Sinclair, Alistair; Vigoda, Eric.

In: Random Structures and Algorithms, 01.01.2019.

Research output: Contribution to journalArticle

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AU - Sinclair, Alistair

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