We explore planar random walk conditioned to avoid its past convex hull. We prove that it escapes at a positive lim sup speed. Experimental results show that fluctuations from a limiting direction are on the order of $n^{3/4}$. This behavior…

Sufficient conditions are given for the strict convexity of the limit shape in standard first-passage percolation. These conditions involve (1) asymptotic ``straightness'' of the geodesics, and (2) existence of mean-zero limit distributions for the first-passage times.

We define the first passage bridge from 0 to $\lambda$ as the Brownian motion on the time interval $[0,1]$ conditioned to first hit $\lambda$ at time 1. We show that this process may be related to the Brownian bridge, the…

We disprove a conjecture of Russ Lyons---that for every locally finite, connected graph $G$, the critical probability for (Bernoulli bond) percolation on $G$ is equal to the "first moment method" lower bound on this probability---and propose a possible alternative.

A probability measure $\mu$ on the lattice $2^{[n]}$ is said to be positively associated if any two increasing functions on the lattice are positively correlated with respect to $\mu$. Pemantle asked whether, in order to establish positive association for a…

We show that, for one generating set, the on-diagonal decay of the heat kernel on the lamplighter group is asymptotic to $c_1 n^{1/6}\exp[-c_2 n^{1/3}]$. We also make off-diagonal estimates which show that there is a sharp threshold for which elements…

We prove the existence of a class $A$ of subsets of $\mathbb{R}^d$ of VC dimension 1 such that the symmetric convex hull $F$ of the class of characteristic functions of sets in $A$ is rich in the following sense. For…

Let $C_n$ be the origin-containing cluster in subcritical percolation on the lattice $\frac{1}{n} \mathbb Z^d$, viewed as a random variable in the space $\Omega$ of compact, connected, origin-containing subsets of $\mathbb R^d$, endowed with the Hausdorff metric $\delta$. When $d…