On random graphs and the statistical mechanics of granular matter

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Abstract
The dynamics of spins on a random graph with ferromagnetic three-spin interactions is used to model the compaction of granular matter under a series of taps. Taps are modelled as the random flipping of a small fraction of the spins followed by a quench at zero temperature. We find that the density approached during a logarithmically slow compaction -the random-close-packing density-corresponds to a dynamical phase transition. We discuss the role of cascades of successive spin-flips in this model and link them with density-noise power fluctuations observed in recent experiments.