Variational formula for the time-constant of first-passage percolation

Arjun Krishnan

Published 2013-11-01, updated 2014-11-02Version 2

We consider first-passage percolation with positive, stationary-ergodic weights on the square lattice $\mathbb{Z}^d$. Let $T(x)$ be the first-passage time from the origin to a point $x$ in $\mathbb{Z}^d$. The convergence of the scaled first-passage time $T([nx])/n$ to the time-constant as $n$ tends to infinity can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula for the time-constant. As an application, we construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption), thereby computing the time-constant. In certain situations, the iteration produces correctors.