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.

A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$
of the Gibbs measure of the form
$$
P(\textd\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}\textd\phi,
$$
where $V$ is a potential, $\beta$ is the inverse temperature and $\textd\phi$ is the product
Lebesgue measure. The simplest example is the (lattice) Gaussian free field
$V(\eta)=\frac12\kappa\eta^2$. A well known result of Funaki and Spohn (and Sheffield)
asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are
characterized by the \emph{tilt}, which is a vector $u\in\R^d$ such that
$E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example
for which this result fails once $V$ is sufficiently non-convex thus showing that
the conditions of Funaki-Spohn's theory are generally optimal. The underlying
mechanism is an order-disorder phase transition known, e.g., from the context
of the $q$-state Potts model with sufficiently large $q$. Based on joint work
with Roman Koteck\'y.