Superconcentration and chaos in Bernoulli percolation

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.

💡 Research Summary

The paper investigates the fluctuations of the chemical (graph) distance in supercritical Bernoulli percolation on the lattice $\mathbb Z^{d}$, and establishes a precise equivalence between the phenomenon of superconcentration and the chaotic response of geodesics to small perturbations of the edge configuration.

Model and notation.
Each edge $e$ of the nearest‑neighbor graph on $\mathbb Z^{d}$ is open with probability $p>p_{c}(d)$ and closed otherwise, independently. The open subgraph $G_{p}$ contains a unique infinite cluster $C_{\infty}$. For any vertex $z$, $