On the number and size of holes in the growing ball of first-passage percolation
Journal
Transactions of the American Mathematical Society
ISSN
0002-9947
1088-6850
Date Issued
2023-12-22
Author(s)
Abstract
First-passage percolation is a random growth model defined on Zd using i.i.d. nonnegative weights (τe) 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 ∈ Zd : T(0, x) ≤ t}. It is known that for all such τe, the number of vertices (volume) of B(t) is at least order td, and under mild conditions on τe, this volume grows like a deterministic constant times td. Defining a hole in B(t) to be a bounded component of the complement B(t)c, we prove that if τe is not deterministic, then a.s., for all large t, B(t) has at least ctd−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)Ctd−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.
Publisher
American Mathematical Society (AMS)
Type
journal article