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

Michael Damron, Julian Gold, Wai-Kit Lam, Xiao Shen
2022 arXiv   pre-print
First-passage percolation is a random growth model defined on ℤ^d using i.i.d. nonnegative weights (τ_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 ∈ℤ^d : T(0,x) ≤ t}. It is known that for all such τ_e, the number of vertices (volume) of B(t) is at least order t^d, and under mild conditions on τ_e, this volume grows like a deterministic constant times t^d. Defining a hole in
more » ... B(t) to be a bounded component of the complement B(t)^c, we prove that if τ_e is not deterministic, then a.s., for all large t, B(t) has at least ct^d-1 many holes, and the maximal volume of any hole is at least clog 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.
arXiv:2205.09733v1 fatcat:jhob6t5izbcqllfjdkxrj25ihe