Efficient sampling and counting algorithms for the Potts model on ℤᵈ at all temperatures

Borgs, Christian; Chayes, Jennifer; Helmuth, Tyler; Perkins, Will; Tetali, Prasad

doi:10.1145/3357713.3384271

Cited by 32 publications

(51 citation statements)

References 37 publications

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“…This began with the work [23], which gave polynomial-time sampling algorithms for the Potts model on pZ{nZq d at sufficiently low temperatures. Further work [8] derives similar results at all temperatures, provided the parameter q is sufficiently large as a function of the dimension d. In a related direction, [12,16] obtain rapid mixing results for the so-called "polymer dynamics" at sufficiently low temperatures on expander graphs, and deduce polynomial mixing for Glauber dynamics restricted to a small region around the ground states (measured in terms of the size of its largest polymer). It should be noted that such approaches are ultimately based on convergence of the cluster expansion and are unlikely to cover the full low-temperature regime for small q.…”

Section: Related Workmentioning

confidence: 75%

Low-temperature Ising dynamics with random initializations

Gheissari¹,

Sinclair²

2021

Preprint

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It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be polynomially fast. However, such phenomena have largely evaded rigorous analysis, as most tools in the analysis of Markov chain mixing times are tailored to worst-case initializations.In this paper, we establish this conjectured behavior for the classical case of the Ising model on an N -vertex torus in Z d . Namely, we show that the mixing time for the Glauber dynamics, initialized in a 1 2 -1 2 mixture of the all-plus and all-minus configurations, is OpN log N q in all dimensions, at all temperatures below the critical one. We also give a matching lower bound, showing that this is optimal. A key innovation in our analysis is the introduction of the notion of "weak spatial mixing within a phase", an adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to imply rapid mixing of the Glauber dynamics from the above random initialization, and that it holds for the Ising model at all low temperatures and in all dimensions. As byproducts of our analysis, we also prove rapid mixing of the Glauber dynamics restricted to the plus phase, and the Glauber dynamics on a box in Z d with plus boundary conditions, when initialized from the all-plus configuration.

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Section: Related Workmentioning

confidence: 75%

Low-temperature Ising dynamics with random initializations

Gheissari¹,

Sinclair²

2021

Preprint

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“…For discrete classical spin systems, expansion methods have also been used at low temperatures, i.e., when β ≫ 1. 4,[26][27][28][29] It would be interesting to adapt these methods to quantum systems, e.g., by developing algorithms based on Pirogov-Sinai methods for quantum perturbations of classical systems. 30 We remark, however, that it seems difficult to use this approach for low-temperature quantum systems with an infinite degeneracy of ground states, for example, when the set of ground states possesses a continuous symmetry.…”

Section: Discussionmentioning

confidence: 99%

Efficient algorithms for approximating quantum partition functions

Mann

Helmuth

2021

Journal of Mathematical Physics

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We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.

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“…Polymer models have recently been used to obtain algorithms for spin systems in regimes where standard algorithmic tools (such as correlation-decay algorithms or Gibbs sampling/Glauber dynamics) are inefficient. The prototypical class of graphs where polymer models have been applied to are classes of expander and random regular graphs [21,6,18,1,22,8,14], see also [19,4,20] for applications on the grid.…”

Section: Introductionmentioning

confidence: 99%

Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

Chen¹,

Galanis²,

Štefankovič³

et al. 2021

Preprint

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For spin systems, such as the q-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an FPRAS for counting q-colorings for even q = O ∆ log ∆ on almost every ∆-regular bipartite graph. This is within a factor O(log ∆) of the sampling algorithm for general graphs in the uniqueness region and improves significantly upon the previous best bound of q = O √ ∆ (log ∆) 2

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Efficient sampling and counting algorithms for the Potts model on ℤᵈ at all temperatures

Cited by 32 publications

References 37 publications

Low-temperature Ising dynamics with random initializations

Low-temperature Ising dynamics with random initializations

Efficient algorithms for approximating quantum partition functions

Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

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