### Rutgers Discrete Mathematics Seminar

Title: The Frog Model on Trees

Speaker: **Toby Johnson**, New York University

Date: Monday, September 19, 2016 2:00 pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:
Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at one vertex wakes up and begins a simple random walk. When it moves to a new vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model, and many basic questions about it remain open. I'll talk about the frog model on infinite trees, where the process exhibits some interesting phase transitions. In particular, I'll (mostly) answer a question posed by Serguei Popov in 2003 by showing that on a binary tree, all frogs wake up with probability one, while on a 5-ary or higher tree, some frogs remain asleep with probability one. This work is joint with Christopher Hoffman and Matthew Junge.