"Limit shapes in first-passage percolation."

Abstract: First-passage percolation was introduced by Hammersley and Welsh in 1965 as a model of fluid flow through a random medium. Mathematically, on the integer lattice, one places i.i.d. nonnegative random variables at the nearest-neighbor edges on $\mathbb{Z}^d$, and study the induced (pseudo)metric space on $\mathbb{Z}^d$. As $t$ grows, the random ball of radius $t$ centered at the origin (also known as the wet region), scaled by a factor of $t$, converges to a deterministic limit shape $\mathcal{B}$. Although the model has been introduced for more than 50 years, many properties of $\mathcal{B}$ are yet to be verified. In this talk, I will cover some known results and conjectured properties of $\mathcal{B}$. If time allowed, I will briefly mention some results on the ``surface area'' of the wet region. This talk will be introductory and will not assume any knowledge of percolation.