Stochastic homogenization and first-passage percolation

First-passage percolation is a random growth model on the cubic lattice Z^d. It models, for example, the spread of fluid in a random porous medium. Quantitatively describing the ``average time'' required for the fluid to percolate through the medium ---known as the time-constant of first-passage percolation--- is a classical, but unsolved problem in probability. We view first-passage percolation as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman equation, and derive an exact variational formula for the time-constant. The random fluctuations of the model are (conjecturally) universal; we will present some new results in this direction.