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.

For a class of stationary Markov-dependent sequences
(A_n,B_n)
in R^2, we consider the random linear recursion S_n=A_n+B_n
S_{n-1}, n \in \zz, and show that the distribution tail of its
stationary solution has a power law decay.