Variational Formula for the Time Constant of First‐Passage Percolation |
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| Authors: | Arjun Krishnan |
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| Affiliation: | University of Utah, Salt Lake City, UT, USA |
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| Abstract: | We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ?d. Let T(x) be the first‐passage time from the origin to a point x in ?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. |
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