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

Understand

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 $n \to \infty$ can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation.
• We derive an exact variational formula for the time-constant, and construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption).