Variational Formula for the Time Constant of First‐Passage Percolation
Arjun S. Krishnan · Communications on Pure and Applied Mathematics · 2016
We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ℤd. LetT(x) be the first‐passage time from the origin to a pointxin ℤd. The convergence of the scaled first‐passage timeT([nx])/nto the time constant asn → ∞ 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). We explicitly identify when the iteration produces correctors.© 2016 Wiley Periodicals, Inc.