Spectral properties of the zero temperature Edwards-Anderson model

An Ising model with random couplings on a graph is a model of a spin glass. While the mean field case of the Sherrington-Kirkpatrick model is very well studied, the more realistic lattice setting, known as the Edwards-Anderson (EA) model, has witnessed rather limited progress. In (Chatterjee,'23) chaotic properties of the ground state in the EA model were established via the study of the Fourier spectrum of the two-point spin correlation. A natural direction of research concerns fractal properties of the Fourier spectrum in analogy with critical percolation. In particular, numerical findings (Bray, Moore,'87) seem to support the belief that the fractal dimension of the associated spectral sample drawn according to the Fourier spectrum is strictly bigger than one. Towards this, in this note we introduce a percolation-type argument, relying on the construction of ``barriers'', to obtain new probabilistic lower bounds on the size of the spectral sample.