Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics

¹ The Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, 34100 Trieste, Italy
² Service de Physique Théorique, CEA/Saclay Orme des Merisiers, F-91191 Gif-sur-Yvette Cedex, France
³ Laboratoire de Physique Théorique, École Normale Supérieure 24 rue Lhomond, Paris, France

Received: 17 September 2004
Accepted: 2 December 2004

Abstract

We show that facilitated spin models of cooperative dynamics introduced by Fredrickson and Andersen display on Bethe lattices a glassy behaviour similar to the one predicted by the mode-coupling theory of supercooled liquids and the dynamical theory of mean-field disordered systems. At low temperature such cooperative models show a two-step relaxation and their equilibration time diverges at a finite temperature according to a power law. The geometric nature of the dynamical arrest corresponds to a bootstrap percolation process which leads to a phase space organization similar to the one of mean-field disordered systems. The relaxation dynamics after a subcritical quench exhibits aging and converges asymptotically to the threshold states that appear at the bootstrap percolation transition.