On the Geometry of the Growing Ball of First-Passage Percolation

In first-passage percolation, one puts i.i.d. nonnegative weights on the nearest-neighbor edges of the integer lattice $\mathbb{Z}^d$, and studies the induced (pseudo)metric. It is well-known that a metric ball of radius $t$, after rescaled by $1 / t$, converges to a deterministic limit shape as $t \to \infty$. However, not too much is known about the geometry of a ball for finite $t$. In this talk, we will talk about the geometry of a ball of radius $t$, by looking at its boundary. We will discuss the size of the boundary of a ball (based on an earlier joint work with M. Damron and J. Hanson), and then we will talk about holes in a ball (based on a recent joint work with M. Damron, J. Gold and X. Shen).