Escaping scattered sets in Z^d, d>=3

We consider the smallest escape probability (or harmonic measure) among subsets A of Z^d, d>= 3, |A|=n, where the smallest distance between any two vertices of A is at least n^\kappa. We prove a phase transition occurs at \kappa_c=\frac{2}{d(d-2)}, at which the harmonic measure turns from stretch exponential to polynomial.

Joint work with Zhenhao Cai and Eviatar Procaccia.

This talk is Yam’s master’s seminar.