ARM EXPONENTS IN HIGH DIMENSIONAL PERCOLATION

摘要

It is widely believed that there is no infinite component almost surely in critical percolation on any d-dimensional lattice for any d > 1. Proving this is considered to be one of the most challenging problems in probability. This was proved for d = 2 by Harris [21] and Kesten [26] and in high dimensions by Hara and Slade [19]. By high dimensions we mean one of the two underlying graphs: (i) Z~d with d > 19 or, (ii) the graph with vertex set Z~d such that x and y are neighbors if y < L for sufficiently large L and d > 6 (see further definitions below).Having no infinite component almost surely is equivalent to the assertion that the probability that the origin is connected by an open path to 0Q,, the boundary of the cube {-r, , r}d tends to 0 as r co. Physicists' lore (see for example [1], page 31) maintains that not only is there no infinite component for any d > 2, but also that these probabilities decay according to some power law in r; that is, Pp (0 H 0Q,) = r-1/P+o(1) for some critical exponent p > 0 which depends only on the dimension d, and not on the local structure of the lattice. In this paper we prove that p = 1/2 in high dimensions.