Variational Formula for the Time Constant of First‐Passage Percolation

Source abstract

We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ℤ d . Let T ( x ) be the first‐passage time from the origin to a point x in ℤ d . The convergence of the scaled first‐passage time T ([ nx ])/ n to the time constant as n → ∞ 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.