# Sublinear variance in Euclidean first-passage percolation

## TL;DR

This paper proves that the variance of passage times in Euclidean first-passage percolation grows sublinearly, specifically as Cn divided by log n, improving previous bounds and employing advanced probabilistic techniques.

## Contribution

It introduces a novel approach to establish sublinear variance bounds in Euclidean first-passage percolation using Bernoulli encoding and lattice animal arguments.

## Key findings

- Variance of passage time is bounded by Cn/ log n.
- Method adapts lattice-based variance proofs to Euclidean models.
- Technical innovations handle the model's geometric complexity.

## Abstract

The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several "greedy lattice animal" arguments.