Exploring the diluted ferromagnetic p-spin model with a cavity master equation

From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics

We start from the theory of random point processes to derive n-point coupled master equations describing the continuous dynamics of discrete variables in random graphs. These equations constitute a hierarchical set of approximations that generalize and improve the cavity master equation (CME), a recently obtained closure for the usual master equation representing the dynamics. Our derivation clarifies some of the hypotheses and approximations that originally led to the CME, considered now as the first order of a more general technique. We tested the scheme in the dynamics of three models defined over diluted graphs: the Ising ferromagnet, the Viana-Bray spin-glass, and the susceptible-infectious-susceptible model for epidemics. In the first two, the equations perform similarly to the best-known approaches in literature. In the latter, they outperform the well-known pair quenched mean-field approximation.

Theory of Nonequilibrium Local Search on Random Satisfaction Problems

We study local search algorithms to solve instances of the random k-satisfiability problem, equivalent to finding (if they exist) zero-energy ground states of statistical models with disorder on random hypergraphs. It is well known that the best such algorithms are akin to nonequilibrium processes in a high-dimensional space. In particular, algorithms known as focused, and which do not obey detailed balance, outperform simulated annealing and related methods in the task of finding the solution to a complex satisfiability problem, that is to find (exactly or approximately) the minimum in a complex energy landscape. A physical question of interest is if the dynamics of these processes can be well predicted by the well-developed theory of equilibrium Gibbs states. While it has been known empirically for some time that this is not the case, an alternative systematic theory that does so has been lacking. In this Letter we introduce such a theory based on the recently developed technique of cavity master equations and test it on the paradigmatic random 3-satisfiability problem. Our theory predicts the qualitative form of the phase boundary between the satisfiable (SAT) and unsatisfiable (UNSAT) region of the phase diagram where the numerics of a focused Metropolis search and cavity master equation cannot be distinguished.