Wesley Pegden

Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice

Highly nonrepetitive sequences: Winning strategies from the local lemma

\mathbb{Z}^d

Assessing significance in a Markov chain without mixing

, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of

The lefthanded local lemma characterizes chordal dependency graphs

n

Walker-Breaker Games

chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as

Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

n\to \infty

Online purchasing under uncertainty

. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as

Separating subadditive euclidean functionals

n \to \infty

Critical graphs without triangles: An optimum density construction

. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

A note on the localization number of random graphs: Diameter two case

We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Lovasz Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the significant dependencies of events. This appears to represent the first successful application of a Local Lemma to games.