Metastability on the hierarchical lattice

Abstract

We study metastability for Glauber spin-flip dynamics on the N-dimensional hierarchical lattice with n hierarchical levels. Each vertex carries an Ising spin that can take the values or . Spins interact with an external magnetic field . Pairs of spins interact with each other according to a ferromagnetic pair potential , where is the strength of the interaction between spins at hierarchical distance i. Spins flip according to a Metropolis dynamics at inverse temperature β. In the limit as , we analyse the crossover time from the metastable state (all spins ) to the stable state (all spins ). Under the assumption that is non-increasing, we identify the mean transition time up to a multiplicative factor . On the scale of its mean, the transition time is exponentially distributed. We also identify the set of configurations representing the gate for the transition. For the special case where , , with the relevant formulas simplify considerably. Also the hierarchical mean-field limit can be analysed in detail.