Consider the directed process (i S i ) i 0 where the second component is simple random walk on Z(S 0 = 0). De ne a transformed path measure by w eighting each n-step path with a factor exp P 1 i n (! i +h)sign(S i )]. Here, (! i ) i 1 is an i.i.d. sequence of random variables taking values 1 with probability 1 =2 (acting as a random medium), while 2 0 1) a n d h 2 0 1) are parameters. The weight factor has a tendency to pull the path towards the horizontal, because it favors the combinations S i > 0 ! i = +1 and S i < 0 ! i = ;1. The transformed path measure describes a`heteropolymer', consisting of hydrophylic and hydrophobic monomers, near an oil-water interface.We study the free energy of this model as n ! 1 , and show that there is a critical curve ! h c ( ) where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive s e v eral properties of this curve, in particular, its behavior for # 0. To obtain this behavior, we prove that as h # 0 the free energy scales to its Brownian motion analogue.AMS 1991 subject classi cations. 60F10, 60J15, 82B26.

No abstract

No abstract

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 ␤Ͼ0 be the inverse temperature and let ⌳ ʚ⌳ ␤ ʚZ 2 be two finite boxes. Particles perform independent random walks on ⌳ ␤ ‫⌳گ‬ and inside ⌳ feel exclusion as well as a binding energy UϾ0 with particles at neighboring sites, i.e., inside ⌳ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that ⌳ is empty, while a total of ͉⌳ ␤ ͉ particles is distributed randomly over ⌳ ␤ ‫⌳گ‬ 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 ϭe Ϫ⌬␤ for some ⌬ (U,2U) and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit as ␤→ϱ for fixed ⌳ and lim ␤→ϱ (1/␤) log͉⌳ ␤ ͉ϭϱ. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice ⌬ (U,2U) corresponds to the situation where the critical droplet has side length l c (1,ϱ), i.e., the system is metastable. The side length of ⌳ must be much larger than l c and independent of ␤, but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in ⌳ ␤ ‫⌳گ‬ can be treated as a reservoir that creates particles with rate at sites on the interior boundary of ⌳ and annihilates particles with rate 1 at sites on the exterior boundary of ⌳ . Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside ⌳ in the presence of a nonconservative boundary. The dynamics inside ⌳ is still conservative and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For instance, rectangular droplets tend to become square through a movement of particles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions ͑at least for not too long a time͒ without being attracted by a square.

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gärtner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical measures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition.MSC2010: Primary 60F10, 60G60, 60K35; Secondary 82B26, 82C22.