Rotationally invariant first passage percolation: Breaking the $n/\log n$ variance barrier

TL;DR

This paper proves that for rotationally invariant Riemannian first passage percolation in the plane, the variance of passage times grows slower than previously known, specifically as a polynomial rate less than linear.

Contribution

It introduces a novel multi-scale approach to show polynomial improvement in the variance bound for a class of rotationally invariant FPP models.

Findings

Variance of passage times is bounded by $O(n^{1- ext{positive constant}})$.

Geodesics exhibit disorder chaos upon small resampling of randomness.

Multi-scale analysis leads to polynomial variance bounds in rotationally invariant FPP.

Abstract

For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly smaller than $1$. Following Kesten's $O(n)$ upper bound (Ann. Appl. Probab., 1993) on the variance, Benjamini, Kalai and Schramm (Ann. Probab., 2003) used hypercontractivity to obtain an improvement of a factor of $\log n$ when passage times take two values with equal probability. This was later extended to more general classes of passage time distributions. However, unlike in exactly solvable planar models in last passage percolation where the variance is known to be $\Theta(n^{2/3})$, the best known upper bound for the variance of passage times has remained $O(n/\log n)$ in all non-trivial variants of FPP. For a class of rotationally invariant Riemannian FPP on the plane, we show that the variance is $O(n^{1-\varepsilon})$ for some $\varepsilon>0$. Our argument uses fluctuation estimates for passage times and geodesics derived in Basu, Sidoravicius and Sly (2023) together with a multi-scale argument to establish that the geodesic exhibits disorder chaos, i.e., upon resampling a small fraction of the underlying randomness, the updated geodesic has on average a small overlap with the original one; this, established at a large number of scales, leads to a polynomial improvement of the variance bound.