Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

Abstract We consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in low temperature limit. generalize previous model-independent results several directions. First, we give an estimate mixing time terms maximal stability level. Second, assuming is reversible, associated spectral gap. Third, precise asymptotics expected from any metastable state to stable using potential-theoretic techniques. do this a general reversible setting where two or more states are allowed and some them may even be degenerate. This generalizes that hold series only states. then focus on specific Probabilistic Cellular Automata (PCA) configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + } ? $$\varLambda \subset {\mathbb {Z}}^2$$ ? Z 2 finite box periodic boundary conditions. apply our find sharp estimates $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ ? c o e $$\underline{+1}$$ . Here $${\underline{c}}^o,{\underline{c}}^e$$ denote odd chessboard respectively. To this, identify rigorously by giving explicit upper bounds level every other configuration. rely these prove recurrence property dynamics, which cornerstone pathwise approach metastability.