Oct. 6, 3:30pm, Wean 8220

Yury Sokolov, University of Pittsburgh

A modified bootstrap percolation on a random graph coupled with a lattice

Yury Sokolov, University of Pittsburgh

A modified bootstrap percolation on a random graph coupled with a lattice

Abstract:

Bootstrap percolation is an interactive particle system defined on a graph with a homogeneous local update rule. In
*k*-neighbor bootstrap percolation, every vertex is in one of two possible states, active or inactive. Some
vertices are initially activated. Active vertices stay so forever, and an inactive vertex becomes active whenever
it has at least *k* active neighbors. During the last decades, bootstrap percolation has been extensively
investigated on square lattices.

We introduce a random graph model *G*_{ℤ2N,pd}, which is a combination of fixed torus grid edges in
(ℤ/*N*ℤ)^{2} and some additional random ones. The random edges are called long, and the
probability of having a long edge between vertices *u*,*v*∈(ℤ/*N*ℤ)^{2} with graph distance *d* on
the torus grid is *p _{d}*=

Based on joint work with Svante Janson, Robert Kozma and Miklós Ruszinkó.