Title: Sharp bounds on the least positive harmonic measure in lattices

Abstract: We discuss the least positive harmonic measure for sets in Z^d and planar lattices. For planar lattices we present the optimal asymptotic rate and we conjecture the sets achieving it are *-connected spirals. For Z^d, d>2, it remains open to prove that optimal sets are *-connected, thus we consider scattered sets with some minimal distance between points of the set. We prove sharp bounds that present a phase transition in the minimal distance parameter. Various open problems will be presented. 

Based on joint work with Zhenhao Cai, Yam Berent, Gady Kozma and Yuan Zhang.