| Speaker: Jesse Goodman Affiliation: University of Auckland Time: 2 pm Thursday, 19 August, 2021 Location: 303-310 |
| First passage percolation is a dynamic exploration process used for finding shortest paths. To visualise it, imagine fluid flowing through the edges of a graph at constant speed starting from a source vertex. Then the time it takes for fluid to reach a given target vertex measures the graph distance from the source. First passage percolation adds randomness: choose the length of each edge randomly and independently according to a specified length distribution. When the underlying graph is tree-like, each newly wetted vertex begins sending fluid towards (mostly) new neighbours, and so the exploration process resembles a branching process. Specifically, because fluid flows through edges only gradually, we obtain an age-dependent branching process, where children are born at different rates depending on the ages of their parents. This talk will outline some of the ways of creating and analysing age-dependent branching processes. Particularly, we will discuss the relationship between the intensity measure for children and the average growth rate of the population size, including connections to the renewal equation, Malthusian parameters, and random walks. |