Kinetically Constrained Models

Abstract

Kinetically constrained spin models (KCSM) are interacting particle systems which are intensively studied in physics literature as models for systems undergoing glass or jamming transitions. KCSM leave on discrete lattices and evolve via a Glauber-like dynamics which is reversible w.r.t. a simple product measure. The key feature is that the creation/destruction of a particle at a given site can occur only if the current configuration satisfies proper local constraints. Due to the fact that creation/destruction rates can be zero, the whole analysis of the long time behavior becomes quite delicate. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density remained open for most KCSM (with the exception of the East model in d = 1 Aldous and P. Diaconis, J. Stat. Phys. 107(5-6):945-975 2002). Here we review a novel multi-scale approach which we have developed in Cancrini et al. (Probab. Theory Relat. Fields 140:459-504, 2008; Lecture Notes in Mathematics, vol. 1970, pp. 307-340, Springer, 2009) trough which we: (i) prove positivity of the spectral gap in the whole ergodic region for a wide class of KCSM on Z(d), (ii) establish (sometimes optimal) bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iii) prove pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations. In particular (i) above establishes exponential decay of auto-correlation functions disproving the stretched exponential decay which had been conjecture for some KCSM and (ii) disproves some of the scalings which had been extrapolated from numerical simulations for the relaxation times (inverse of the spectral gap). NR - 27 PU - SPRINGER-VERLAG BERLIN PI - BERLIN PA - HEIDELBERGER PLATZ 3, D-14197 BERLIN, GERMANY