Summary of Lectures 16 and 17 ( Nov . 5 and Nov . 7 , 2008 ) Scribe : Srinath Puducheri

1 Executive Summary We started with an introduction to the notion of percolation over random graphs with a 2-D lattice as an example. We then studied branching processes over random trees and proved conditions under which the such a process continues forever with positive probability. Finally, we defined the probability space of a random graph over a 2-D lattice. 2 Introduction to Percolation Consider the graph formed by a 2-D lattice L , G(V,E) with vertex set V = Z and edge set E = {(x, y) ∈ V 2 : ||x− y|| = 1}. We derive a random graph as follows: each edge in L exists with probability p. Such edges are said to be open. This gives rise to the bond percolation model. Alternatively, if we randomly retain vertices instead of edges, we have the site percolation model.