Cutoff for General Spin Systems with Arbitrary Boundary Conditions

Eyal Lubetzky, Allan Sly

Research output: Contribution to journalArticle

Abstract

The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the d-dimensional torus (ℤ/nℤ)d for any d ≥ ;1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions, and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for the stochastic Ising model on any sequence of bounded-degree graphs with subexponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures, including the gas hard-core model, the Potts model, the antiferromagnetic Potts model, and the coloring model.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - 2013

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Stochastic Ising Model
Potts model
Ising model
Spin Systems
Potts Model
External Field
Torus
Hard-core Model
Boundary conditions
Glauber Dynamics
Spectral Gap
Arbitrary
Coloring
Half-plane
Markov processes
Colouring
Monotonicity
Markov chain
Ball
Intersection

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

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title = "Cutoff for General Spin Systems with Arbitrary Boundary Conditions",
abstract = "The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the d-dimensional torus (ℤ/nℤ)d for any d ≥ ;1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions, and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for the stochastic Ising model on any sequence of bounded-degree graphs with subexponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures, including the gas hard-core model, the Potts model, the antiferromagnetic Potts model, and the coloring model.",
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AU - Sly, Allan

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N2 - The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the d-dimensional torus (ℤ/nℤ)d for any d ≥ ;1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions, and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for the stochastic Ising model on any sequence of bounded-degree graphs with subexponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures, including the gas hard-core model, the Potts model, the antiferromagnetic Potts model, and the coloring model.

AB - The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the d-dimensional torus (ℤ/nℤ)d for any d ≥ ;1. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions, and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for the stochastic Ising model on any sequence of bounded-degree graphs with subexponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures, including the gas hard-core model, the Potts model, the antiferromagnetic Potts model, and the coloring model.

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