Let $(\xi(s))_{s\geq 0}$ be a standard Brownian motion in $d\geq 1$
dimensions and let $(D_s)_{s \geq 0}$ be a collection of open sets in $\R^d$.
For each $s$, let $B_s$ be a ball centered at 0 with $\vol(B_s) = \vol(D_s)$.
We show that $\E[\vol(\cup_{s \leq t}(\xi(s) + D_s))] \geq \E[\vol(\cup_{s \leq
t}(\xi(s) + B_s))]$, for all $t$. In particular, this implies that the expected
volume of the Wiener sausage increases when a drift is added to the Brownian
motion.

We consider the following dynamic Boolean model introduced by van den Berg,
Meester and White (1997). At time 0, let the nodes of the graph be a Poisson
point process in R^d with constant intensity and let each node move
independently according to Brownian motion. At any time t, we put an edge
between every pair of nodes if their distance is at most r.

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any
$\alpha < 1/d$ we construct an $\alpha$-H\"{o}lder continuous function $f
\colon [0,1] \to \mathbb{R}^d$ so that the range of $B-f$ covers an open set.
This strengthens a result of Graversen (1982) and answers a question of Le Gall
(1988).

Let $U \subseteq \C$ be a bounded domain with smooth boundary and let $F$ be
an instance of the continuum Gaussian free field on $U$ with respect to the
Dirichlet inner product $\int_U \nabla f(x) \cdot \nabla g(x) dx$. The set
$T(a;U)$ of $a$-thick points of $F$ consists of those $z \in U$ such that the
average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi}
\log \tfrac{1}{r}$ as $r \to 0$.

Consider Glauber dynamics for the Ising model on a graph of $n$ vertices.
Hayes and Sinclair showed that the mixing time for this dynamics is at least
$n\log n/f(\Delta)$, where $\Delta$ is the maximum degree and $f(\Delta) =
\Theta(\Delta \log^2 \Delta)$. Their result applies to more general spin
systems, and in that generality, they showed that some dependence on $\Delta$
is necessary. In this paper, we focus on the ferromagnetic Ising model and
prove that the mixing time of Glauber dynamics on any $n$-vertex graph is at
least $(1/4+o(1))n \log n$.

Let $C_1$ be the largest component of the Erd\H{o}s-R\'enyi random graph
$G(n,p)$. The mixing time of random walk on $C_1$ in the strictly supercritical
regime, $p=c/n$ with fixed $c>1$, was shown to have order $\log^2 n$ by
Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In
the critical window, $p=(1+\epsilon)/n$ where $\lambda=\epsilon^3 n$ is
bounded, Nachmias and Peres proved that the mixing time on $C_1$ is of order
$n$.