Variational formula for the timeconstant of firstpassage percolation
Abstract
We consider firstpassage percolation with positive, stationaryergodic weights on the square lattice $\mathbb{Z}^d$. Let $T(x)$ be the firstpassage time from the origin to a point $x$ in $\mathbb{Z}^d$. The convergence of the scaled firstpassage time $T([nx])/n$ to the timeconstant as $n$ tends to infinity can be viewed as a problem of homogenization for a discrete HamiltonJacobiBellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula for the timeconstant. We then construct an explicit iteration that produces the minimizer of the variational formula (under a symmetry assumption), thereby computing the timeconstant. The variational formula may also be seen as a duality principle, and we discuss some aspects of this duality.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.1108
 Bibcode:
 2014arXiv1406.1108K
 Keywords:

 Mathematics  Probability;
 60K35;
 82B43
 EPrint:
 112 pages, double spaced, 2 figures. PhD Thesis, Courant Institute, New York University