Superconcentration and chaos in Bernoulli percolation

TL;DR

This paper links superconcentration of chemical distance in Bernoulli percolation to chaotic behavior of geodesics under small perturbations, confirming a general principle and introducing refined methods to measure edge influence.

Contribution

It establishes an equivalence between superconcentration and chaos in Bernoulli percolation, using a dynamical effective radius and lattice animal techniques.

Findings

Confirmed the link between superconcentration and chaos in Bernoulli percolation.

Developed a dynamical effective radius to measure edge influence.

Quantified edge influence via geodesic overlap analysis.

Abstract

We study the chemical distance of supercritical Bernoulli percolation on $\mathbb{Z}^d$. Recently, Dembin [Dem22] showed that the chemical distance exhibits sublinear variance, a phenomenon now referred to as superconcentration. In this article, we establish an equivalence between this phenomenon and chaotic behavior of geodesics under small perturbations of the configuration, thereby confirming Chatterjee's general principle relating anomalous fluctuations to chaos in the context of Bernoulli percolation. Our methods rely on a dynamical version of the effective radius, refining the notion first proposed in [CN25], in order to measure the co-influence of a given edge whose weight may be infinite. Together with techniques from the theory of lattice animals, this approach allows us to quantify the total co-influence of edges in terms of the overlap between original and perturbed geodesics.