The influence of dimension on the relaxation process of East-like models: Rigorous results

¹ Mathematics Institute, University of Warwick - Coventry CV4 7AL, UK
² Department of Mathematics, La Sapienza University - P.le Aldo Moro 2, 00185 Rome, Italy
³ Department of Mathematics and Physics, Roma Tre University - Largo S.L. Murialdo 1, 00146 Rome, Italy

^(a) paul@chleboun.co.uk
^(b) faggiona@mat.uniroma1.it
^(c) martin@mat.uniroma3.it

Received: 8 July 2014
Accepted: 11 July 2014

Abstract

We study facilitated models which extend to arbitrary dimensions the one-dimensional East process and which are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low-temperature regime (small density of the facilitating sites). In the literature the relaxation process has been assumed to be quasi–one-dimensional and the equilibration time has been computed using the relaxation time of the East model on the equilibrium length scale in d-dimension. This led to a super-Arrhenius scaling for the relaxation time of the form . In a companion paper, using renormalization group ideas and electrical networks methods, we rigorously establish that instead , contradicting the quasi–one-dimensional assumption. The above scaling confirms previous MCAMC simulations. Next we compute the relaxation time at finite and mesoscopic length scales, and show a dramatic dependence on the boundary conditions. Our final result is related to the out-of-equilibrium dynamics. Starting with a single facilitating site at the origin we show that, up to length scales , its influence propagates much faster (on a logarithmic scale) along the diagonal direction than along the axes directions.