Abstract. We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form − + ∇F (·)∇ on R d or subsets of R d , where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as ↓ 0, to the capacities of suitably constructed sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring-Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.

We study a large class of reversible Markov chains with discrete state space and transition matrix P N . We define the notion of a set of metastable points as a subset of the state space Γ N such that (i) this set is reached from any point x ∈ Γ N without return tox with probability at least b N , while (ii) for any two point x, y in the metastable set, the probability T −1x,y to reach y from x without return to x is smaller than a −1 N ≪ b N . Under some additional non-degeneracy assumption, we show that in such a situation:(i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely.(ii) To each metastable point corresponds one simple eigenvalue of 1 − P N which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.1 6 E.g. the lack of precision in the relation T M = O(1/(1 − (1 − λ) t )) in [GS] is partly due to this fact. 7 There is no difficulty in applying our results to continuous time chains by using suitable embeddings. 8 The case of irreversible Markov chains will be studied in a forthcoming publication [EK].

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form − + ∇F (•)∇ on R d or subsets of R d , where F is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius centered at the positions of the local minima of F. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring-Kramers formula.

This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.