Spin Glass

A spin glass is a spin model with disordered interactions, often with both positive and negative couplings. For binary spins on a graph, a typical energy is , with random . Conflicting interaction signs can prevent simultaneous minimization of every edge energy. Conditional on the disorder, the Gibbs measure is a probability law on spin configurations. Two independently sampled configurations from that conditional law are called replicas, and their overlap is

Overlap hierarchies enter variational descriptions of the limiting thermodynamic pressure. The Sherrington-Kirkpatrick model has interactions between every pair, whereas a diluted spin glass has a random number of interactions proportional to the number of spins. Orthogonally invariant spin glasses instead prescribe a limiting spectral law and random orthogonal eigenvectors.