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A decade and a half ago Chatterjee established the first rigorous connectionbetween anomalous fluctuations and a chaotic behaviour of the ground state incertain Gaussian disordered systems. The purpose of this paper is to show thatChatterjee's work gives evidence of a more general principle, by establishingan analogous connection between fluctuations and chaos in the context offirst-passage percolation. The notion of `chaos' here refers to the sensitivityof the time-minimising path between two points when exposed to a slightperturbation. More precisely, we resample a small proportion of the edgeweights, and find that a vanishing fraction of the edges on thedistance-minimising path still belongs to the time-minimising path obtainedafter resampling. We also show that the chaotic behaviour implies the existenceof a large number of almost-optimal paths that are almost disjoint from thetime-minimising path, a phenomenon known as `multiple valleys'.

Chaos, concentration and multiple valleys in first-passage percolation