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.

Let x and y be points chosen uniformly at random
from the four-dimensional discrete torus with side length n.
We show that the length of the loop-erased random walk from
x to y is of order n^2 (log n)^{1/6}, resolving a conjecture
of Benjamini and Kozma. We also show that the scaling limit
of the uniform spanning tree on the four-dimensional discrete
torus is the Brownian continuum random tree of Aldous. Our
proofs use the techniques developed by Peres and Revelle,
who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the
d-dimensional discrete torus for d >= 5, in combination with
results of Lawler concerning intersections of
four-dimensional random walks.

Given a set $K$ in a metric space $M$ one may ask when is $K$ contained in
$\Gamma$, a connected set of finite 1-dimensional Hausdorff length, and
for estimates on the minimal length of such a $\Gamma$ (up to
multiplicative constants). This was first answered for $M$ = the
Euclidean plane by P. Jones and extended to $M=R^d$ by K. Okikiolu.
Recently, there have been several new relevant results (by several people)
for $M$ being: a Hilbert space, the Heisenberg group, and a general metric
space. for some of these one restricts the discussion to $K$ and $\Gamma$
in specific categories. In some of these categories which we will discuss
the anwswer is in IFF form, whereas in others it is not. The answer to
this question usually comes together with a multiresolutional analysis of
the set $K$ and a construciton of a $\Gamma$ containing $K$ which is not
'too long'. Essentially no prior knowledge in analysis is needed to
understand this talk.