In the context of a tight-binding approximation of the Gross-Pitaevskii energy functional with a random background potential we want to discuss in dependence on the interaction coupling constant a criteria when the Gross-Pitaevskii ground state and the single particle ground state coincide.

Random interface models have been an important object of study during the last decades. Next to the widely studied gradient model, the Laplacian model is of importance in modelling for example semiflexible membranes. In this talk, we present the Gaussian Laplacian model, that is, the Gausian field on the d-dimensional integer lattice with covariances given by the Green's function of the discrete bilaplacian. We compare it with the gradient model, and explain why crucial techniques like the random walk representation and the FKG inequalities fail. We show the methods we use to avoid some of these problems, and prove entropic repulsion for the Laplacian model in sufficiently high dimensions.

In this talk, we consider the asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of traps is given by independent displacements of the lattice points, which gives a model of the Frenkel defects in a crystal. We determined both annealed and quenched asymptotics for the logarithm of the survival probability up to multiplicative constant. As applications, we show the Lifshitz tail effect of the density of states of associated random Schrödinger operator and intermittency in the parabolic problem.

An NxN square Young tableau can be thought of as a set of instructions for building an NxN brick wall by laying bricks sequentially in such a way that the heights of the brick columns, read from left to right, form a monotone (weakly) decreasing sequence at any point during the construction. I will talk about the problem of the "typical" limit shape, or growth profile, of such a wall - namely, if a square Young tableau is chosen at random uniformly from all tableaux of this size, what can one say about the growth profile of the wall at various times during its construction when N is large? The analysis leads to a problem in the calculus of variations which has a similar structure to the problem solved by Vershik-Kerov and Logan-Shepp in 1977 in their solution of the famous Ulam problem on the length of the longest increasing subsequences of a random permutation. At the end of the talk, if time permits I will tell about some recent applications of the limit shape result. The talk is based on joint work with Boris Pittel.

We consider a random walk in random environment in dimension greater than or equal to three satisfying any of the standard ballisticity conditions (either $T$, or $T'$ or $T_\gamma$). We consider the event $A(n)$ that at time $n$ the distance of the walker from he origin is less than half of its expected value. We show that for every $\alpha

We investigate the recurrence/transience property of a particular class of self interacting random walks called "cookie random walks". These processes have been given particular attention in the lattice cases Z and Z^d. We here consider a similar model when the state space is a regular tree. We show that such a walk can be recurrent or transient depending on the underlying cookie environment and that the limiting behaviour of the walk depends strongly on the order of the cookies in the pile (which contrast with the one-dimensional setting). The main ingredient for this study is a construction of a branching Markov chain closely related to the local time process of the walk.

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cram\'er's condition. We prove moderate deviation principles in dimensions d> 1, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case d>3 we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In d>2, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in d=2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.
(joint work with K.Fleischmann and P. Morters)

We show a quenched invariance principle for ballistic random walk in i.i.d. elliptic random environment in dimension greater than or equal to 4 under mild integrability conditions for the regeneration times. I will then show a new and simpler proof of Rassoul-Agha and Seppalainen's quenched invariance principle in dimensions two and three.
Based on joint work with Ofer Zeitouni.

We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first order perturbation theory for quantum many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation.
REF: Reviews in Mathematical Physics, Vol. 18, No. 3 (April 2006)

A fundamental question of statistics first formulated and answered by Fischer is what must one know a priori about an unknown functional dependence in order to estimate it on the basis of observations.
Until the 1980's the answer to this was : "Almost everything", namely one must know the functional dependence up to values of a number of finite parameters. The research of V. Vapnik et.al. replaced this answer by a new paradigm that overcame the restrictions.
In this talk I will introduce the general learning problem as the minimization of a functional that depends on an unkown probability measure and describe the results of Vapniks research. In particular I will discuss the principle of empirical risk minimization, probabilistic bounds on the risk and how the generalization ability of a learning machine can be controled to optimize learning from small sample sizes.

The decay of correlation mentioned in the title is an essential information needed in order to control the phases of some Kac-systems. In the work of A. Mazel, J.Lebowitz and E. Presutti it was shown rigorously the existence of a phase transition with the density as order parameter for these systems. Although, one has not a unique state we were nevertheless able to modify Drobushin's uniqueness method in such a way, that it becomes applicable. The Dobrushin condition cannot hold for all configurations, because otherwise it would imply global uniqueness. However, the configurations which violate the condition have small probability in the phase under consideration. This can be formalized using the notion of restricted ensemble. The decay of correlation for the restricted ensemble can be used to control the finite volume corrections of the pressure. The latter one are essential to obtain information about the initially considered systems from the restricted ensemble.

In this talk we present a first attempt to derive deterministic mesoscopic theories for continuous spin lattice systems with applications to micromagnetics. We will also discuss the difficulty that arises in the treatment of the dipolar energy term.