2009 Probability Seminar - 10

The aim of this talk is to highlight the usefulness of continuous time branching process theory in understanding refined asymptotics about various random network models. We shall exhibit their usefulness in two different contexts: (1) First passage percolation: Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory. In the modern context these problems take an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. In the context of the configuration model of random networks we shall show how branching processes allow us to find the limiting distribution of the minimal weight path as well as establishing a general central limit theorem for the hopcount with matching means and variances. (2) Spectral distribution of random trees: Many models of random trees (including general models embedded in continuous time branching processes) satisfy a general form of convergence locally to limiting infinite objects. In this context we find via soft arguments, the convergence of the spectral distribution of the adjacency matrix to a limiting (model dependent) non random distribution. For any $\gamma$ we also find a sufficient condition for there to be a positive mass at $\gamma$ in the limit. Joint work with Remco van der Hoftsad, Gerard Hooghiemstra, Steve Evans and Arnab Sen.