Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes

Abstract

Complex free-energy landscapes with many local minima separated by large barriers are believed to underlie glassy behavior across diverse physical systems. This is the heuristic picture associated with replica symmetry breaking (RSB) in spin glasses, but RSB has only been rigorously verified for certain mean-field models with all-to-all connectivity. In this work, we give a rigorous proof of finite temperature spin glass order for a family of models with local interactions on finite-connectivity, non-Euclidean expander graphs. To this end, we bypass the RSB formalism entirely, and instead exploit the mathematical equivalence of such models to certain low-density parity check (LDPC) codes. We use code expansion, a property of LDPC codes which guarantees extensive energy barriers around ground states. Together with mild additional assumptions, this allows us to construct an explicit decomposition of the low-temperature Gibbs state into disjoint components, each hosting an asymptotically long-lived state associated with a local minimum of the landscape. Each component carries at most an exponentially small fraction of the total weight, and almost all components do not contain ground states -- which we take together to define spin-glass order. The proof is elementary, and treats various expanding graph topologies on the same footing, including those with short loops where existing approaches such as the cavity method fail. Our results apply rigorously to diluted p-spin glasses for sufficiently large p, and while unproven, we also expect our assumptions to hold in a broader family of codes. Motivated by this, we numerically study two simple models, on random regular graphs and a regular tesselation of hyperbolic space. We show that both models undergo two transitions as a function of temperature, corresponding to the onset of weak ergodicity breaking and spin glass order, respectively.