Metastability and Multiscale Extinction Time on a Finite System of Interacting Stochastic Chains

We studied metastability and extinction time of a finite system with a large number of interacting components in discrete time by means of analytical and numerical investigation. The system is markovian with respect to the potential profile of the components, which are subject to leakage and gain effects simultaneously. We show that the only invariant measure is the null configuration, that the system ceases activity almost surely in a finite time and that extinction time presents a cutoff behavior. Moreover, there is a critical parameter determined by leakage and gain below which the extinction time does not depend on the system size. Above such critical ratio, the extinction time depends on the number of components and the system tends to stabilize around a unique metastable state. Furthermore, the extinction time presents infinitely many scales with respect to the system size.