Probabilistic Inequalities and Bootstrap Percolation

Download or read book Probabilistic Inequalities and Bootstrap Percolation written by Tomas Juskevicius and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation focuses on two topics. In the first part of we address a number of extremal probabilistic questions:The Littlewood-Offord problem: we provide an alternative and very elementary proof of a classical result by Erdos that avoids using Sperner's Theorem. We also give a new simple proof of Sperner's Theorem itself. Upper bounds for the concentration function: answering a question of Leader and Radcliffe we obtain optimal upper bounds for the concentration function of a sum of real random variables when individual concentration information about the summands is given. The result can be viewed as the optimal form of a well-known Kolmogorov-Rogozin inequality. Small ball probabilities for sums of random vectors with bounded density: we provide optimal upper bounds the probability that a sum of random vectors lies inside a small ball and derive an upper bound for the maximum density of this sum. In particular, our work extends a result of Rogozin who proved the best possible result in one dimension and improves some recent results proved by Bobkov and Chystiakov. Two extremal questions of bounded symmetric random walks: we obtain optimal upper bounds for the probability that a sum of independent bounded symmetric random variables exceeds a given value or hits it. The second part of the dissertation is concerned with a problem in Bootstrap Percolation. Let G be a graph and let I be a set of initially infected vertices. The set of infected vertices is updated as follows: if a healthy vertex has the majority of its neighbours infected it itself becomes infected. In the description of the bootstrap process above the superscripts of the sets correspond to the time steps when infections occur. If the process ends up infecting all of the vertices we say that percolation occurs. In this dissertation we shall investigate this process on the Erdos-Renyi random graph G(n, p). In this graph on n vertices each edge is included independently with probability p. We shall be interested in the smallest cardinality, say m=m(n), of a uniformly chosen initially infected set of vertices I, such that the probability of percolation at least 1/2. We call this quantity the critical size of the initially infected set. In the regime p>clog (n)/n (the connectivity threshhold) we prove sharp upper and lower bounds for m that match in the first two terms of the asymptotic expansion.