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

TL;DR

This paper derives an exact variational formula for the time-constant in first-passage percolation on lattices, connecting it to homogenization of HJB equations and providing a method to compute it explicitly.

Contribution

It introduces a novel variational formula for the time-constant in first-passage percolation, linking discrete homogenization with continuum HJB theory and offering an explicit iterative computation method.

Findings

Derived an exact variational formula for the time-constant.

Constructed an explicit iteration to compute the time-constant.

Discussed duality aspects of the variational formula.

Abstract

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. We then construct an explicit iteration that produces the minimizer of the variational formula (under a symmetry assumption), thereby computing the time-constant. The variational formula may also be seen as a duality principle, and we discuss some aspects of this duality.