Exponential convergence in supercritical kinetically constrained models

L. Mareche

2020, v.26, Issue 5, 861-884


Kinetically constrained models (KCMs) were introduced by physicists to
model the liquid-glass transition. They are interacting particle systems on $\mathds{Z}^d$ in which
each element of $\mathds{Z}^d$ can be in state 0 or 1 and tries to update its state to 0 at rate $q$ and to 1 at rate $1-q$,
provided that a constraint is satisfied.
In this article, we prove the first non-perturbative result of convergence to
equilibrium for KCMs with general constraints: for any KCM in the class
termed ``supercritical'' in dimension 1 and 2, when the initial configuration has product $\mathrm{Bernoulli}(1-q')$
law with $q' \neq q$, the dynamics converges to equilibrium with exponential speed when $q$ is close enough to 1,
which corresponds to the high temperature regime.

Keywords: interacting particle systems; Glauber dynamics; kinetically constrained models; bootstrap percolation; convergence to equilibrium


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