EXACT ASYMPTOTICS FOR DUARTE AND SUPERCRITICAL ROOTED KINETICALLY CONSTRAINED MODELS

Kinetically constrained models (KCM) are reversible interacting particle systems on Z^d with continuous-time Markov dynamics of Glaubertype, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as U-bootstrap percolation. Furthermore, KCM have an interest in their own since the display some of the most striking features of the liquid-glass transition, a major and longstanding open problem in condensed matter physics. A key issue for KCM is to identify the divergence of the characteristic time scales when the equilibrium density of empty sites,q, goes to zero. In [19, 20] a general scheme was devised to determine a sharp upper bound for these time scales. Our paper is devoted to developing a (very different) technique which allows proving matching lower bounds. We analyze the class of two-dimensional supercritical rooted KCMand the Duarte KCM, the most studied critical1-rooted model. We prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM aseΘ((log)2)and for Duarte KCM aseΘ((log)4/q2)when q↓0. These results prove the conjectures put forward in [20, 22], and establish that the time scales for these KCM diverge much faster than for the corresponding-bootstrap processes, the main reason being the occurrence of energy barriers which determine the dominant behavior for KCM, but which do not matter for the bootstrap dynamics.