On the number and size of holes in the growing ball of first-passage percolation

Abstract

First-passage percolation is a random growth model defined on ${\mathbb{Z}}^d$ using i.i.d. nonnegative weights $({\tau }_e)$ on the edges. Letting $T(x,y)$ be the distance between vertices $x$ and $y$ induced by the weights, we study the random ball of radius $t$ centered at the origin, $B(t)=\{x\in {\mathbb{Z}}^d:T(0,x)\le t\}$. It is known that for all such ${\tau }_e$, the number of vertices (volume) of $B(t)$ is at least order $t^d$, and under mild conditions on ${\tau }_e$, this volume grows like a deterministic constant times $t^d$. Defining a hole in $B(t)$ to be a bounded component of the complement $B(t{)}^c$, we prove that if ${\tau }_e$ is not deterministic, then a.s., for all large $t$, $B(t)$ has at least $c{t}^{d-1}$ many holes, and the maximal volume of any hole is at least $c\log t$. Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large $t$, the number of holes is at most $(\log t{)}^C{t}^{d-1}$, and for $d=2$, no hole in $B(t)$ has volume larger than $(\log t{)}^C$. Without curvature, we show that no hole has volume larger than $Ct\log t$.

28 pages, 8 figures