Bisection for kinetically constrained models revisited

Ivailo Hartarsky

doi:10.1214/21-ecp434

Abstract

The bisection method for kinetically constrained models (KCM) of Cancrini, Martinelli, Roberto and Toninelli is a vital technique applied also beyond KCM. In this note we present a new way of performing it, based on a novel two-block dynamics with a probabilistic proof instead of the original spectral one. We illustrate the method by very directly proving an upper bound on the relaxation time of KCM like the one for the East model in a strikingly general setting. Namely, we treat KCM on finite or infinite one-dimensional volumes, with any boundary condition, conditioned on any of the irreducible components of the state space, with arbitrary site-dependent state spaces and, most importantly, arbitrary inhomogeneous rules.

Highlights

The bisection method is one of the fundamental techniques in the rigorous theory of kinetically constrained models (KCM), introduced by Cancrini, Martinelli, Roberto and Toninelli [4, Section 4] and inspired by [17, Proposition 3.5] for the Glauber dynamics of the Ising model
In this note we present a new way of performing it, based on a novel two-block dynamics with a probabilistic proof instead of the original spectral one
An exposition of the original bisection method can be found in the upcoming monograph on KCM by Toninelli [23]

Summary

Introduction

The bisection method ( halving or two-block) is one of the fundamental techniques in the rigorous theory of kinetically constrained models (KCM), introduced by Cancrini, Martinelli, Roberto and Toninelli [4, Section 4] and inspired by [17, Proposition 3.5] for the Glauber dynamics of the Ising model. It is our hope that this new approach itself will be of independent interest and, in particular, our substitute for the two-block dynamics, Proposition 4, and its proof We apply it in the following setting of unprecedented generality. Let us note that for such general KCM there are usually many irreducible components (there are always at least two, save for trivialities) and their combinatorial structure can be very intricate They have proved hard to deal with due to the long-range dependencies they introduce, like those present in conservative KCM. As we will see, non-interval domains, boundary conditions and irreducible components other than the ergodic one can be absorbed in the inhomogeneity of the rules, but such arbitrarily inhomogeneous KCM have not been considered previously

Definition of the models and notation

Result

MR2344312

MR3134727

MR4108128