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