Vertex-removal stability and the least positive value of harmonic measures
报告题目(Title)：Vertex-removal stability and the least positive value of harmonic measures
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 work, despite vertex- removal stability no longer holding. Joint work(s) with Z. Cai, G. Kozma, and E.B. Procaccia.