An upper bound on geodesic length in 2D critical first-passage percolation

Abstract

We consider i.i.d. first-passage percolation (FPP) on the two-dimensional square lattice, in the critical case where edge-weights take the value zero with probability $\frac{1}{2}$. Critical FPP is unique in that the Euclidean lengths of geodesics are superlinear -- rather than linear -- in the distance between their endpoints. This fact was speculated by Kesten in 1986 but not confirmed until 2019 by Damron and Tang, who showed a lower bound on geodesic length that is polynomial with degree strictly greater than $1$. In this paper, we establish the first nontrivial upper bound. Namely, we prove that for a large class of critical edge-weight distributions, the shortest geodesic from the origin to a box of radius $R$ uses at most $R^{2+\epsilon }{\pi }_3(R)$ edges with high probability, for any $\epsilon >0$. Here ${\pi }_3(R)$ is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to $R^{4/3+\epsilon }$. In any case, it is known that ${\pi }_3(R)\lesssim R^{-\delta }$ for some $\delta >0$, so our bound gives an exponent strictly less than $2$. In the special case of Bernoulli($\frac{1}{2}$) edge-weights, we replace the additional factor of $R^{\epsilon }$ with a constant and give an expectation bound.

76 pages, 16 figures. Version 3: fixed typos

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