Metastability of Exponentially Perturbed Markov Chains

Motivated by studies of the stochastic Ising models at very low temperature,neural networks and simulating annealing, we consider a family of Markov chains asa perturbation of another Markov chain. The main theme of this paper is to demon-strate that the metastable behavior happens universally for this wide class of process.Assume the state space X = {ξ,η,…} is finite, and the transition matrices P~β =