Entropic Repulsion for the Free Field:¶Pathwise Characterization in d≥ 3

We study concentration properties of the lattice free field , i.e. the centered Gaussian field with covariance given by the Green function of the (discrete) Laplacian, when constrained to be positive in a region of volume O(N ^d ) (hard–wall condition). It has been shown in [3] that, as N→∞, the conditioned field is pushed to infinity: more precisely the typical value of the ϕ-variable to leading order is , and the exact value of c was found. It was moreover conjectured that the conditioned field, once this diverging height is subtracted, converges weakly to the lattice free field. Here we prove this conjecture, along with other explicit bounds, always in the direction of clarifying the intuitive idea that the free field with hard–wall conditioning merely translates away from the hard wall. We give also a proof, alternative to the one presented in [3], of the lower bound on the probability that the free field is everywhere positive in a region of volume N ^d . Received: 26 October 1998 / Accepted: 5 April 1999     

Aging for interacting diffusion processes

We study the aging phenomenon for a class of interacting diffusion processes {Xt(i),i∈Zd}. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f(Xt) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.     

Entropic repulsion of the lattice free field,II. The 0-boundary case

This paper is a continuation of [5]. We consider the Euclidean massless free field on a boxV _N of volumeN ^d with O-boundary condition; that is the centered Gaussian field with covariances given by the Green function of the simple random walk on ^d ,d3, killed as it exitsV _N . We show that the probability, that all the spins are positive in the boxV _N decays exponentially at a surface rateN ^d–1 . This is in contrast with the rateN ^d–2 logN for the infinite field of [5].     

Non-Gaussian surface pinned by a weak potential

We consider a model of a two-dimensional interface of the (continuous) SOS type, with finite-range, strictly convex interactions. We prove that, under an arbitrarily weak pinning potential, the interface is localized. We consider the cases of both square well and δ potentials. Our results extend and generalize previous results for the case of nearest-neighbours Gaussian interactions in [7] and [11]. We also obtain the tail behaviour of the height distribution, which is not Gaussian. Received: 3 November 1998 / Revised version: 14 June 1999