# Xiao Shen: On the number and size of holes in the growing ball of first-passage percolation

Time: Tue 2022-09-27 15.15 - 16.15

Location: Zoom

Video link: Meeting ID: 698 3346 0369

Lecturer: Xiao Shen (University of Utah)

### Abstract

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 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 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)^Ct^{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. (Joint work with Michael Damron, Julian Gold, Wai-Kit Lam).