Spatial mixing and non-local Markov chains

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Citations (Scopus)

Abstract

We consider spin systems with nearest-neighbor interactions on an n-vertex d-dimensional cube of the integer lattice graph Zd. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium of non-local Markov chains. We prove that SSM implies O(log n) mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an O(1) bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of Z2 is O(1) throughout the subcritical regime of the q-state Potts model, for all q ≥ 2. 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). Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

Original languageEnglish (US)
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Pages1965-1980
Number of pages16
ISBN (Electronic)9781611975031
DOIs
StatePublished - Jan 1 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: Jan 7 2018Jan 10 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
CountryUnited States
CityNew Orleans
Period1/7/181/10/18

Fingerprint

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

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

Cite this

Pimentel, A. B., Caputo, P., Sinclair, A., & Vigoda, E. (2018). Spatial mixing and non-local Markov chains. In A. Czumaj (Ed.), 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 (pp. 1965-1980). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611975031.128
Pimentel, Antonio Blanca ; Caputo, Pietro ; Sinclair, Alistair ; Vigoda, Eric. / Spatial mixing and non-local Markov chains. 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. editor / Artur Czumaj. Association for Computing Machinery, 2018. pp. 1965-1980 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).
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Pimentel, AB, Caputo, P, Sinclair, A & Vigoda, E 2018, Spatial mixing and non-local Markov chains. in A Czumaj (ed.), 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, pp. 1965-1980, 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, United States, 1/7/18. https://doi.org/10.1137/1.9781611975031.128

Spatial mixing and non-local Markov chains. / Pimentel, Antonio Blanca; Caputo, Pietro; Sinclair, Alistair; Vigoda, Eric.

29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. ed. / Artur Czumaj. Association for Computing Machinery, 2018. p. 1965-1980 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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N2 - We consider spin systems with nearest-neighbor interactions on an n-vertex d-dimensional cube of the integer lattice graph Zd. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium of non-local Markov chains. We prove that SSM implies O(log n) mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an O(1) bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of Z2 is O(1) throughout the subcritical regime of the q-state Potts model, for all q ≥ 2. 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). Systematic scan dynamics are widely employed in practice but have proved hard to analyze. 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 Zd. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium of non-local Markov chains. We prove that SSM implies O(log n) mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an O(1) bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of Z2 is O(1) throughout the subcritical regime of the q-state Potts model, for all q ≥ 2. 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). Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

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Pimentel AB, Caputo P, Sinclair A, Vigoda E. Spatial mixing and non-local Markov chains. In Czumaj A, editor, 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. Association for Computing Machinery. 2018. p. 1965-1980. (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). https://doi.org/10.1137/1.9781611975031.128