ELSEVIER Physica A 263 (1999) 180-186 PHYSICA Metastable states: smooth continuations through the critical point Senya Shlosman CPT/CNRS, Luminy, Marseille 1 Introduction In this paper we describe the recent results concerning the understanding of the metasta- bility phenomenon as a dynamical feature of the infinite-volume systems. The results pre- sented are obtained in collaboration with Professor Roberto Schonmann and are subject of the paper [1]. The topic of metastability has a long history, and is discussed in [1] and in the references therein. As in [1], the only model which will be considered, is the two dimensional Ising model. It seems however that the approach proposed is much more general and enables one to think rigorously about the metastable behavior of any model of statistical mechanics in the proximity of the first order phase transition. In what follows we will study the 2D Ising model at an arbitrary subcritical temperature T and under a small positive external magnetic field h. We will construct the metastable states of this model. These states are close to the (-)-phase of the Ising model, in spite of the presence of the positive external field. Due to the positivity of the field the metastable states are "higher" than the Gibbs states with the same T under any negative external magnetic field. Of course, these metastable states are not the Gibbs states of our model; in the region of parameters we are talking about there is a unique Gibbs state, and this state is quite far from the (-)-phase. What we show is the following picture. Let us take a typical configuration of the (-)- phase for the initial configuration, and let us run the Glauber dynamics for times of order exp(A/h) with A below a certain critical value Ac. Then the system would arrive into a sort of metastable state, close to the (-)-phase. On the other hand, for a time of order exp(A/h) with A > Ac the system would already relax and so would be close to the (+)-phase. The critical value A~ is given by the expression w(T)2 (1) A~ = A~(T) - 12Tm*(T) 1 On leave from the Department of Mathematics, University of California at Irvine. The work is partially supported by the NSF through the grant DMS-9800860 and by the Russian Fund for Fundamental Research through the grant 930101470. 0378-4371/99/$ - see front matter © 1999 Published by Elsevier Science B.V. All rights reserved. PIh S0378-437 1(98)00531 -7