### 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 language | English (US) |
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Journal | Random Structures and Algorithms |

DOIs | |

State | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Random Structures and Algorithms*. https://doi.org/10.1002/rsa.20844

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*Random Structures and Algorithms*. https://doi.org/10.1002/rsa.20844

**Spatial mixing and nonlocal Markov chains.** / Pimentel, Antonio Blanca; Caputo, Pietro; Sinclair, Alistair; Vigoda, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Spatial mixing and nonlocal Markov chains

AU - Pimentel, Antonio Blanca

AU - Caputo, Pietro

AU - Sinclair, Alistair

AU - Vigoda, Eric

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85062515281&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062515281&partnerID=8YFLogxK

U2 - 10.1002/rsa.20844

DO - 10.1002/rsa.20844

M3 - Article

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

ER -