On the Essential Features of Metastability: Tunnelling Time and Critical Configurations

We consider Metropolis Markov chains with finite state space and transition probabilities of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$P(\eta ,\eta ')=q(\eta ,\eta ')e^{- \beta [H(\eta ') - H(\eta)]_+}$$ \end{document} for given energy function H and symmetric Markov kernel q. We propose a simple approach to determine the asymptotic behavior, for large β, of the first hitting time to the ground state starting from a particular class of local minima for H called metastable states. We separate the asymptotic behavior of the transition time from the determination of the tube of typical paths realizing the transition. This approach turns out to be useful when the determination of the tube of typical paths is too difficult, as for instance in the case of conservative dynamics. We analyze the structure of the saddles introducing the notion of “essentiality” and describing essential saddles in terms of “gates.” As an example we discuss the case of the 2D Ising Model in the degenerate case of integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\tfrac{{2j}}{h}}$$ \end{document}.