Chaos, concentration and multiple valleys in first-passage percolation

Daniel Ahlberg, Maria Deijfen, Matteo Sfragara

Published 2023-02-22Version 1

A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations and a chaotic behaviour of the ground state in certain Gaussian disordered systems. The purpose of this paper is to show that Chatterjee's work gives evidence of a more general principle, by establishing an analogous connection between fluctuations and chaos in the context of first-passage percolation. The notion of `chaos' here refers to the sensitivity of the time-minimising path between two points when exposed to a slight perturbation. More precisely, we resample a small proportion of the edge weights, and find that a vanishing fraction of the edges on the distance-minimising path still belongs to the time-minimising path obtained after resampling. We also show that the chaotic behaviour implies the existence of a large number of almost-optimal paths that are almost disjoint from the time-minimising path, a phenomenon known as `multiple valleys'.