Exponential convergence to equilibrium for the d-dimensional East model

Laure Marêché

doi:10.1214/19-ecp261

Home/Archive/arXiv

arXiv · 1905.00765

Exponential convergence to equilibrium for the d-dimensional East model

Laure Marêché

View original source ↗Read source document ↗Export metadata ↓

Abstract

Kinetically constrained models (KCMs) are interacting particle systems on $Z^d$ with a continuous-time constrained Glauber dynamics, which were introduced by physicists to model the liquid-glass transition. One of the most well-known KCMs is the one-dimensional East model. Its generalization to higher dimension, the d-dimensional East model, is much less understood. Prior to this paper, convergence to equilibrium in the d-dimensional East model was proven to be at least stretched exponential, by Chleboun, Faggionato and Martinelli in 2015. We show that the d-dimensional East model exhibits exponential convergence to equilibrium in all settings for which convergence is possible.

Explore related subjects

math.PR

Keep this discovery

BibTeX RIS

Laure Marêché. 2019-05-02. Exponential convergence to equilibrium for the d-dimensional East model. https://doi.org/10.1214/19-ecp261

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗