Front progression in the East model

Blondel, Oriane

doi:10.1016/j.spa.2013.04.014

Cited by 22 publications

(47 citation statements)

References 16 publications

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“…Remark. The proof strategy of the equivalent theorem in [Blo13] combined with Lemma 4.4 and Corollary 3.3 would also allow us to give a result of the type "in the box [L, L+M ] seen from the front, the distribution is within e −c √ L∧t of µ in total variation distance", under suitable hypotheses on the initial configuration (depending on the respective regimes of L, t, M ). A proper statement would however be too technical for the purpose of the present paper and we restrict to the above.…”

Section: Relaxation Far From the Frontmentioning

confidence: 99%

“…The motion of the front has been analyzed in [Blo13,GLM15] for another one dimensional KCM, the East model, for which the constraint requires the site at the right of x to be empty: ergodicity of the measure seen from the front, law of large numbers, central limit theorem and cutoff results have been established. A key tool for the East model introduced in [AD02] and used in [Blo13,GLM15] for the study of the front, is the construction of a distinguished zero, a sort of moving boundary which induces local relaxation to equilibrium. This construction relies heavily on the oriented nature of the East constraint and cannot be extended to FA-1f and to generic KCM.…”

Section: Introductionmentioning

confidence: 99%

“…Afterward, we apply the techniques of [BCM+13] (Corollary 3.3) to prove relaxation to equilibrium using these zeros in Section 5. Once this result is established, we construct a coupling inspired by [Blo13,GLM15] to prove ergodicity of the process behind the front, namely convergence to the unique invariant measure for the process seen from the front (Section 6). This convergence result allows us to analyze the increments of the front (Section 7.1) and to deduce a law of large numbers.…”

Section: Introductionmentioning

confidence: 99%

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Front evolution of the Fredrickson-Andersen one spin facilitated model

Blondel¹,

Deshayes²,

Toninelli³

2019

Electron. J. Probab.

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The Fredrickson-Andersen one spin facilitated model (FA-1f) on Z belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability q (respectively p = 1 − q), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for q larger than a thresholdq < 1, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.Let Ω = {0, 1} Z be the space of configurations andbe the subspace of configurations with a leftmost zero at the origin. For a configuration σ for which there exists x ∈ Z such that for every y < x, σ(y) = 1 and σ(x) = 0, we call x the front of configuration σ and we denote it by X(σ). For Λ ⊂ Z and σ ∈ Ω, we denote by σ Λ the restriction of σ to the set Λ. For σ ∈ Ω and x ∈ Z, let σ x be the configuration σ

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Section: Relaxation Far From the Frontmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Front evolution of the Fredrickson-Andersen one spin facilitated model

Blondel¹,

Deshayes²,

Toninelli³

2019

Electron. J. Probab.

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“…If the East dynamics tries to move (by ±1) the front away from the origin, then the whole configuration is shifted by ∓1 in order to keep to restore the front at the origin. Blondel showed process seen from the front has an invariant measure ν on

false(- \infty, - 1 false] \cap double-struckZ

whose marginal on

false(- \infty, - N false] \cap double-struckZ

approaches exponentially fast (in N ) the same marginal of the Bernoulli(1/2) product measure μ . She also proved that

\frac{1}{t} X false(η false(t false) false)

converges in probability to v ∗ , where v ∗ was defined right after Theorem .…”

Section: Preliminary Resultsmentioning

confidence: 99%

Upper triangular matrix walk: Cutoff for finitely many columns

Ganguly

Martinelli

2019

Random Struct Algorithms

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We consider random walk on the space of all upper triangular matrices with entries in double-struckF2 which forms an important example of a nilpotent group. Peres and Sly proved tight bounds on the mixing time of this walk up to constants. It is well known that the column projection of this chain is the one dimensional East process. In this article we complement the Peres‐Sly result by proving a cutoff result for the mixing of finitely many columns in the upper triangular matrix walk at the same location as the East process of the same dimension. The proof is based on a recursive argument which uses a local version of a dual process appearing in a previous study, various combinatorial consequences of mixing and concentration results for the movement of the front in the one dimensional East process.

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“…We first observe that Assumption 3 trivially implies j (ε) (η) = −j (−ε) (η). This identity alone is not enough to prove the antisymmetry relation (12)…”

Section: A Class Of Rw With An Antisymmetry Propertymentioning

confidence: 99%

A Class of Random Walks in Reversible Dynamic Environments: Antisymmetry and Applications to the East Model

2016

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Abstract. We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in [2]. As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical interfaces between empty and occupied regions. We study the asymptotic velocity and density profile for the environment seen from the walker. In particular, we determine the sign of the velocity when the density of the underlying East process is not 1/2, and we discuss the appearance of a drift in the balanced setting given by density 1/2.

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Front progression in the East model

Cited by 22 publications

References 16 publications

Front evolution of the Fredrickson-Andersen one spin facilitated model

Front evolution of the Fredrickson-Andersen one spin facilitated model

Upper triangular matrix walk: Cutoff for finitely many columns

A Class of Random Walks in Reversible Dynamic Environments: Antisymmetry and Applications to the East Model

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