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Cutoff for the Ising model on the lattice

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Abstract

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L 1 on a system of size n is O(logn). Whether in this regime there is cutoff, i.e. a sharp transition in the L 1-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem.

We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ℤ2 this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (ℤ/nℤ)d with periodic boundary conditions has cutoff at (d/2λ ∞)logn, where λ ∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited.

The proof hinges on a new technique for translating L 1-mixing to L 2-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e.g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

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Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA, 98052-6399, USA

    Eyal Lubetzky

  2. Department of Statistics, UC Berkeley, Berkeley, CA, 94720, USA

    Allan Sly

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  1. Eyal Lubetzky
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  2. Allan Sly
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Correspondence to Eyal Lubetzky.

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Lubetzky, E., Sly, A. Cutoff for the Ising model on the lattice. Invent. math. 191, 719–755 (2013). https://doi.org/10.1007/s00222-012-0404-5

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