Abstract: for (transient) one dimensional random walk in random environment, conditions are known that ensure an annealed CLT. One then also have a quenched CLT, with a different (environment dependent) centering.

In higher dimensions, annealed CLT's have been derived in the ballistic case by Sznitman. We prove that in dimension 4 or more, annealed CLT's together with a mild integrability condition imply a quenched CLT. The proof is based on controlling the intersections of two RWRE paths in the same environment.

(joint work with N. Berger)

We will discuss a strong law of large numbers, an annealed CLT, and
the limit law of the ``environment viewed from the particle" for transient
random walks on a strip (product of Z with a finite set). The model was
introduced by Bolthausen and Goldsheid and includes in particular RWRE
with bounded jumps on the line as well as some one-dimensional RWRE with a
memory.

We consider a nonlinear Schr\"odinger equation (NLS) with random
coefficients, in a regime of separation of scales corresponding to
diffusion approximation. The primary goal is to propose and
study an efficient numerical scheme in this framework. We use a
pseudo-spectral splitting scheme and we establish the order of the
global error. In particular we show that we can take an integration step
larger than the smallest scale of the problem, here the correlation
length of the random medium. We study
the asymptotic behavior of the numerical solution in the diffusion
approximation regime.

Given the measure on random walk paths $P_0$ and a Hamiltonian $H$ the Gibbs perturbation of $H$ defined by
$$\frac{dP_{\beta,t}}{dP_0}=Z^{-1}_{\beta,t}\exp\{\beta H(x)\}$$
with
$$Z_{\beta,t}=\int \exp\{-\beta H(x)\}dP_0(x)$$
gives a new measure on paths $x$ which can be viewed as polymers.
In the case $H(x)=\int_0^t\delta_0(x_{s})ds(\int_0^t\delta_0(x_{s})dW_s)$ we say the resulting measure is concentrated on "homopolymers" ("heteropolymers") and are interested in the influence of dimension and $\beta$ on their behavior.