The size of the boundary in the Eden model

Stochastics Seminar
Thursday, September 15, 2016 - 15:05
1 hour (actually 50 minutes)
Skiles 006
School of Mathematics, Georgia Tech
The Eden model, a special case of first-passage percolation, is a stochastic growth model in which an infection that initially occupies the origin of Z^d spreads to neighboring sites at rate 1. Infected sites are colonized permanently; that is, an infected site never heals. It is known that at time t, the infection occupies a set B(t) of vertices with volume of order t^d, and the rescaled set B(t)/t converges to a convex, compact limiting shape. In joint work with J. Hanson and W.-K. Lam, we partially answer a question of K. Burdzy, concerning the order of the size of the boundary of B(t). We show that, in various senses, the boundary is relatively smooth, being typically of order t^{d-1}. This is in contrast to the fractal behavior of interfaces characteristic of percolation models.