Jesse Goodman

Scaling limit of the invasion percolation cluster on a regular tree

We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.

Short Paths for First Passage Percolation on the Complete Graph

We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight Wn and the number of edges Hn of the minimal weight path between two distinct vertices in the weak disorder regime. We establish novel and simple first and second moment methods using path counting to derive first order asymptotics for the considered quantities. Our results are stated in terms of a sequence of parameters

Exponential Growth of Ponds in Invasion Percolation on Regular Trees

(s_{n})_{n\in\mathbb{N}}

Degree distribution of shortest path trees and bias of network sampling algorithms

that quantifies the extreme-value behaviour of the edge weights, and that describes different universality classes for first passage percolation on the complete graph. These classes contain both n-independent and n-dependent edge weight distributions. The method is most effective for the universality class containing the edge weights

Lectures on self-avoiding walks

E^{s_{n}}

Extremal geometry of a Brownian porous medium

, where E is an exponential random variable and snlogn→∞,

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

s_{n}^{2} \log n \to0

Long paths in first passage percolation on the complete graph II. Global branching dynamics

. We discuss two types of examples from this class in detail. In addition, the class where snlogn stays finite is studied. This article is a contribution to the program initiated by Bhamidi and van der Hofstad (Ann. Appl. Probab. 22(1):29–69, 2012).

Invasion percolation on regular trees

In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.