Three Problems in Discrete Probability

Abstract

In this thesis we present three problems. The first problem is to find a good description of the number of fixed points of a 231-avoiding permutation. We use a bijection from Dyck paths to 231-avoiding permutations that allows us to compute the scaled distribution of the number of fixed points of a 231-avoiding permutation chosen uniformly at random. We also show a strong connection with a these permutations and Brownian excursion. The second problem is a study of bootstrap percolation on the Hamming torus. We give a thorough description of the behavior of this model for finite lattices of all dimensions when the percolation threshold is 2. Lastly we present a problem on jigsaw percolation as a model for collaborative problem solving. This process considers a pair of graphs on a shared set of vertices and forms clusters of vertices based on the edges of the two underlying graphs. We consider the process where both graphs are Erdos-Renyi random graphs.