Informal Probability Workshop – 22nd January 2018

22 Jan 18

Speaker:  Eleonora Kreacic, University of Oxford

Date:         Monday 22nd January 2018, The Mathematical Institute, Andrew Wiles Building, L5

Title:         The spread of fire on a random multigraph

Abstract: We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge-lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability 1/2. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having n vertices of degree 3 and alpha(n) vertices of degree 4, where alpha(n)=o(n), with i.i.d. standard exponential edge-lengths. Depending on whether alpha(n) >> \sqrt{n} or alpha(n)=O(\sqrt{n}), we prove that as n tends to infinity these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. This is joint work with Christina Goldschmidt.