Energy of Flows on Percolation Clusters |
| |
Authors: | Hoffman Christopher Mossel Elchanan |
| |
Institution: | (1) Department of Mathematics, University of Maryland, College Park, MD, 20742, U.S.A;(2) Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel |
| |
Abstract: | It is well known for which gauge functions H there exists a flow in Z
d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z
d we let our (random) graph cal
C
cal
(Z
d,p) be the graph obtained from Z
d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p
c(Z
d), simple random walk on cal
C
cal
(Z
d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x
2 energy e
f(e)2 is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p
c(Z
d), then cal
C
cal
(Z
d,p) supports a nonzero flow f such that the x
q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z
d. In this paper we close the gap by showing that if d3 and Z
d supports a flow of finite H energy then the infinite percolation cluster on Z
d also support flows of finite H energy. This disproves a conjecture of Levin and Peres. |
| |
Keywords: | percolation energy electrical networks nonlinear potential theory |
本文献已被 SpringerLink 等数据库收录! |
|