Empirical distributions, geodesic lengths, and a variational formula in first-passage percolation
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We consider the standard first-passage percolation model on $\mathbb{Z}^d$, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to $n \mathbf{e}_1$. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as $n \to \infty$. These families include arbitrarily small $L^\infty$-perturbations of any given distribution, as well as almost every finitely supported distribution. Analogous results hold for $\mathbf{e}_1$-directed infinite geodesics. Our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach allows us to recover and extend a recent result of Krishnan, Rassoul-Agha, and Seppalainen regarding the convergence of geodesic lengths.