In the zero-temperature Edwards-Anderson model, the spectral measure of the two-point spin correlation assigns exponentially small mass to all edge sets whose size is at most (1+ε) times the distance, so the spectral sample is strictly larger than a shortest path with high probability.
On absence of disorder chaos for spin glasses on z\^ d
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Spectral properties of the zero temperature Edwards-Anderson model
In the zero-temperature Edwards-Anderson model, the spectral measure of the two-point spin correlation assigns exponentially small mass to all edge sets whose size is at most (1+ε) times the distance, so the spectral sample is strictly larger than a shortest path with high probability.