For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p → 0 and L → ∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p) → 1 if lim inf p log L> λ, and I(L,p) → 0 if lim sup p log L < λ, where λ = π 2 /18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ ′ = π 2 /6. The existence of the thresholds λ,λ ′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2]. 1

. We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analiticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the assoc...

. We consider the kinetic Ising models (Glauber dynamics) corresponding to the infinite volume Ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h. Minimal conditions on the flip rates are assumed, so that all the common choices are being considered. We study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature T and the evolution is started from a distribution which is stochastically lower than the (\Gamma)-phase. We show that as h & 0 the relaxation time blows up as exp( c (T )=h), with c (T ) = w(T ) 2 =(12Tm (T )). Here m (T ) is the spontaneous magnetization and w(T ) is the integrated surface tension of the Wulff body of unit volume. Moreover, for 0 ! ! c , the state of the process at time exp(=h) is shown to be close, when h is small, to the (\Gamma)-phase. The difference between this state and the (\Gamma)-phase can be described in terms of an asymptotic expa...

. We consider a Glauber dynamics reversible with respect to the two dimensional Ising model in a finite square of side L, in the absence of an external field and at large inverse temperature fi. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary condition. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of \Gamma 1 fiL log(gap) as L ! 1 and show that it coincides with the surface tension along one of the coordinat axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin-spin time correlation in the infinite volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by...

In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let > 0 be the inverse temperature and let Z 2 be two nite boxes. Particles perform independent random walks on n and inside feel exclusion as well as a binding energy U > 0 with particles at neighboring sites, i.e., inside the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial conguration is chosen such that is empty, while a total of j j particles is distributed randomly over n with no exclusion. That is to say, initially the system is in equilibrium with particle density conditioned on being empty. For large , the system in equilibrium has fully occupied because of the binding energy. We consider the case where = e for some 2 (U; 2U) and investigate how the transition from empty to full takes place under the dy...

In this paper we review and discuss results on metastability. We consider ergodic aperiodic Markov chains with exponentially small transition probabilities and we give a complete description of the typical tube of trajectories during the first excursion outside a general domain Q. 25=aprile=1997 [1] 0:1 Section 1. Introduction. In this review we want to describe some recent results in the theory of metastability from the point of view of mathematical physics and probability theory. Mathematically we will deal with the study of some large deviation problems for families of Markov chains with finite state space and transition probabilities decaying exponentially fast in a large external parameter fi: the so called Freidlin-Wentzell regime. From physical point of view we analyze a central problem arising in the dynamical description of phase transitions: the decay from metastable to stable equilibrium. Phase transitions (like, for instance, liquid-vapour for a fluid or positive-negativ...

Abstract. We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N ×N box, with boundary conditions which are “plus ” except for small regions at the corners which are either free or “minus. ” The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order log N. This means that removing as few as O(log N) plus spins from the corners produces a spectral gap far smaller than the order N −2 gap believed to hold under the all-plus boundary condition. Our results are valid at all subcritical temperatures.

We continue a study of Schonmann (1994), Schonmann and Shlosman(1996) and Greenwood and Sun (1997) regarding the competing influences of boundary conditions and external field for the Ising model. We find a critical point B 0 in the competing influences for low temperature in dimension d 2.

We study, at infinite volume and very low temperature, the relaxation mechanisms towards stable equilibrium in presence of two competing metastable states. Following Dehghanpour and Schonmann we introduce a simplified nucleation-growth irreversible model as an approximation for the stochastic Blume-Capel model, a ferromagnetic lattice system with spins taking three possible values: \Gamma1; 0; 1. Starting from the less stable state \Gamma1 (all minuses) we look at a local observable. We find that, when crossing a special line in the space of the parameters, there is a change in the mechanism of transition towards the stable state +1: we pass from a situation: 1) Where the intermediate phase 0 is really observable before the final transition with a permanence in 0 typically much longer than the first hitting time to 0; to the situation: 2) Where 0 is not observable since the typical permanence in 0 is much shorter than the first hitting time to 0 and, moreover, large growing 0-droplets...