Dynamical Cavity Method for Hypergraphs and its Application to Quenches in the k-XOR-SAT Problem

Abstract: The dynamical cavity method and its backtracking version provide a powerful approach to studying the properties of dynamical processes on large random graphs. This paper extends these methods to hypergraphs, enabling the analysis of interactions involving more than two variables. We apply them to analyse the $k$-XOR-satisfiability ($k$-XOR-SAT) problem, an important model in theoretical computer science which is closely related to the diluted $p$-spin model from statistical physics. In particular, we examine whether the quench dynamics -- a deterministic, locally greedy process -- can find solutions with only a few violated constraints on $d$-regular $k$-uniform hypergraphs. Our results demonstrate that the methods accurately characterize the attractors of the dynamics. It enables us to compute the energy reached by typical trajectories of the dynamical process in different parameter regimes. We show that these predictions are accurate, including cases where a classical mean-field approach fails.