Speaker: Arjun Krishnan, Fields Institute, University of Toronto
ABSTRACT
Consider first-passage percolation with positive, stationary-ergodic weights on the square lattice in d-dimensions. Let T(x) be the first-passage time from the origin to x in Z^d. The convergence of T([nx])/n to the time constant as n tends to infinity is a consequence of the subadditive ergodic theorem. This convergence 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 will derive an exact variational formula (duality principle) for the time-constant. Under a symmetry assumption, we will use the variational formula to construct an explicit iteration that produces the limit shape.