Abstract

Properties of first passage percolation above the (hypothetical) critical dimension

Kenneth Alexander
University of Southern California (USC)
Mathematics

It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\ZZ^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $dWe show that at least one of three fundamental properties of FPP known or believed to hold when $\chi>0$ must be false if $\chi=0$. A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false. These consequences support the idea that when $\chi=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.