Abstract:

We prove that for Z^d (d>1), the vertex-removal stability of harmonic measures (i.e. it is feasible to remove some vertex while changing the harmonic measure by a bounded factor) holds if and only if d=2. The proof mainly relies on geometric arguments, with a surprising use of the discrete Klein bottle. Moreover, a direct application of this stability verifies a conjecture of Calvert, Ganguly and Hammond, for the exponential decay of the least positive value of harmonic measures on Z^2. Furthermore, the analogue of this conjecture for Z^d with d> 2 is also proved in this paper, despite vertex-removal stability no longer holding.