Abstract. We propose a new definition of metastability of Markov processes on countable state spaces. We obtain sufficient conditions for a sequence of processes to be metastable. In the reversible case these conditions are expressed in terms of the capacity and of the stationary measure of the metastable states.

Abstract. Let r : S × S → R + be the jump rates of an irreducible random walk on a finite set S, reversible with respect to some probability measure m.Consider a zero range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). Let N stand for the total number of particles. In the stationary state, as N ↑ ∞, all particles but a finite number accumulate on one single site. We show in this article that in the time scale N 1+α the site which concentrates almost all particles evolves as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.

We presented in [1,5] an approach to derive the metastable behavior of continuous-time Markov chains. We assumed in these articles that the Markov chains visit points in the time scale in which it jumps among the metastable sets. We replace this condition here by assumtpions on the mixing times and on the relaxation times of the chains reflected at the boundary of the metastable sets.

We proposed in [1] a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet principle proved in [11].

We prove the metastable behavior of reversible Markov processes on finite state spaces under minimal conditions on the jump rates. To illustrate the result we deduce the metastable behavior of the Ising model with a small magnetic field at very low temperature.

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature β on a two dimensional torus Λ L = {0, . . . , L − 1} 2 . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are n 2 particles, n < L/2, and that the initial state is the configuration in which all sites of the square {0, . . . , n − 1} 2 are occupied. We show that in the time scale e 2β the process evolves as a Markov process on Λ L which jumps from any site x to any other site y = x at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.

We prove that the metric space associated with a uniformly distributed planar quadrangulation with n faces and no pendant vertices converges modulo a suitable rescaling to the Brownian map. This is a first step towards the extension of recent convergence results for random planar maps to the case of graphs satisfying local constraints.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP13 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

We prove uniqueness of a martingale problem with boundary conditions on a simplex associated to a differential operator with an unbounded drift. We show that the solution of the martingale problem remains absorbed at the boundary once it attains it, and that, after hitting the boundary, it performs a diffusion on a lower dimensional simplex, similar to the original one. We also prove that in the diffusive time scale condensing zerorange processes evolve as this absorbed diffusion.