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

TL;DR

This paper establishes an upper bound on the length of geodesics in 2D critical first-passage percolation, showing they grow faster than linear but slower than quadratic in the distance, with implications for understanding their geometric complexity.

Contribution

It provides the first nontrivial upper bound on geodesic length in 2D critical FPP, confirming superlinear growth and refining previous lower bounds.

Findings

Geodesic length is at most R^{2+ε}π_3(R) with high probability.

For Bernoulli(1/2) weights, an expectation bound without the ε factor is obtained.

The bound implies geodesic length grows faster than linear but slower than quadratic in R.

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 $\tfrac{1}{2}$. 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($\tfrac{1}{2}$) edge-weights, we replace the additional factor of $R^\epsilon$ with a constant and give an expectation bound.