Convergence to equilibrium of biased plane Partitions

Caputo, Pietro; Martinelli, Fabio; Toninelli, Fabio Lucio

doi:10.1002/rsa.20339

Cited by 4 publications

(4 citation statements)

References 34 publications

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“…For one thing, macroscopic shapes in this case minimize not the free energy but the free energy with a volume constraint, and the mixing time turns out to scale like δ −1+o(1) rather than δ −2+o(1) . We refer to [LL19] for the ASEP in an interval (the authors prove the sharp estimate T mix ∼ cδ −1 , as well as the occurrence of the cut-off phenomenon) and to [CMT11,GPR09] for a dynamics of biased plane partitions, which is a lozenge tiling dynamics where the two updates of Figure 1.2 have different transition rates p and q. Also in the latter case, the result is that T mix = δ −1+o(1) (see [GPR09], where this is proven for small enough bias log(p/q), and [CMT11] for the general case of arbitrary non-zero bias).…”

Section: The Broader Contextmentioning

confidence: 99%

The mixing time of the lozenge tiling Glauber dynamics

Laslier,

Toninelli

2023

Annales Henri Lebesgue

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The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time T mix . In the (d + 1)-dimensional setting, d ⩾ 2, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [Hen97, Spo93] and simulations (see [Des02] and the references in [Hen97, Wil04]), one expects convergence to equilibrium to occur on time-scales of order ≈ δ −2 in any dimension, with δ → 0 the lattice mesh.We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as (2 + 1)-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [CKP01], the height function concentrates as δ → 0 around a deterministic profile ϕ, the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the

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Section: The Broader Contextmentioning

confidence: 99%

The mixing time of the lozenge tiling Glauber dynamics

Laslier,

Toninelli

2023

Annales Henri Lebesgue

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“…There is a well-studied height function that maps hexagonal lozenge tilings bijectively to plane partitions lying in an n × n × n box (see, e.g., [25]), and it follows that lozenge tilings with a fixed average height of k are precisely the plane partitions with volume k. The Markov chain that adds or removes single cubes on the surface of the plane partition (corresponding to rotating three nested lozenges 180 degrees) is known to mix rapidly in the unbiased case. Caputo et al [7] studied the biased version of this chain with a preference toward removing cubes, and showed that this chain converges in O(n 3 ) time.…”

Section: Lozenge Tilings With Fixed Average Heightmentioning

confidence: 99%

Approximately Sampling Elements with Fixed Rank in Graded Posets

Bhakta

Cousins

Fahrbach

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often #P-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call balanced bias.A noteworthy application of our method is sampling restricted classes of integer partitions of n. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of n from general restricted classes. Several observations allow us to improve the efficiency of this chain to require O(n 1/2 log(n)) space, and for unrestricted integer partitions, expected O(n 9/4 ) time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.

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“…In this paper we prove that the mixing time is O(n log n) as long as the spectral gap of the process is Ω(1). Our technique bares some similarities to those employed in [6] to analyse the Glauber dynamics of biased plane partitions.…”

Section: Introductionmentioning

confidence: 99%

Mixing time bounds for oriented kinetically constrained spin models

Chleboun

Martinelli

2013

Electron. Commun. Probab.

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We analyze the mixing time of a class of oriented kinetically constrained spin models (KCMs) on a d-dimensional lattice of n d sites. A typical example is the North-East model, a 0-1 spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin 0, with rate one it resets its own spin by tossing a p-coin, at all other times its spin remains frozen. Such models are very popular in statistical physics because, in spite of their simplicity, they display some of the key features of the dynamics of real glasses. We prove that the mixing time is O(n log n) whenever the relaxation time is O(1). Our study was motivated by the "shape" conjecture put forward by G. Kordzakhia and S.P. Lalley.

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Convergence to equilibrium of biased plane Partitions

Cited by 4 publications

References 34 publications

The mixing time of the lozenge tiling Glauber dynamics

The mixing time of the lozenge tiling Glauber dynamics

Approximately Sampling Elements with Fixed Rank in Graded Posets

Mixing time bounds for oriented kinetically constrained spin models

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