Improved belief propagation algorithm finds many Bethe states in the random-field Ising model on random graphs

Abstract

We first present an empirical study of the Belief Propagation (BP) algorithm, when run on the random field Ising model defined on random regular graphs in the zero temperature limit. We introduce the notion of extremal solutions for the BP equations, and we use them to fix a fraction of spins in their ground state configuration. At the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size. This in turn makes the associated optimization problem highly non trivial in the critical region. Using the bounds on the BP messages provided by the extremal solutions we design a new and very easy to implement BP scheme which is able to output a large number of stable fixed points. On one hand this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time. On the other hand we found that the number of fixed points of the BP algorithm grows with the system size in the critical region. This unexpected feature poses new relevant questions about the physics of this class of models.

Two critical localization lengths in the Anderson transition on random graphs

“…C typ gives the typical exponential decay with ξ ⊥ along an arbitrarily chosen branch, namely the 1D lattice, whereas C av , which agrees with [39], is dominated by the configurations where the rare populated branches coincide with the 1D lattice, and is thus controlled by ξ (see [75,76] for similar large deviations in random magnets). At criticality ξ ⊥ remains finite while ξ diverges, hence C typ C av for W ≥ W c .…”

“…We thank C. Castellani, N. Laflorencie, and T. Thiery for fruitful discussions. We are grateful to G. Parisi and F. Ricci Tersenghi for mentioning the references [75,76]. We acknowledge interesting discussions with K.S.…”

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

“…In what we did before, the RS instability has been detected by looking at growth rate λ BP (in the time domain) of the global norm of the perturbations to the BP fixed point P * [η i→j ] -both at finite and zero temperature -, which of course represents a reliable tool, providing results that match with the analytic predictions where the latter ones are available. However, once having in our hands connected correlation functions, we can study their exponential decay with the distance C(r) ∼ e −r/ξ (19) and hopefully match the divergence of their correlation length ξ with the critical point already detected before.…”

“…Indeed, given the positive couplings of the model, connected correlation functions are always nonnegative and hence spin glass susceptibility is always upper bounded by the ferromagnetic one. This crucial observation rules out the possibility of a proper * Corresponding author: cosimo.lupo89@gmail.com spin glass phase, while nothing can be deducted from it about the behaviour of the RFIM exactly at the critical point, where at variance some evidences of the presence of many states in the Gibbs measure [19] have been recently found.…”

Self-Guided Belief Propagation – a Homotopy Continuation Method

“…Moreover, many methods that minimize F B are particularly efficient for attractive models and thus for balanced models as well. One of the properties that we will use throughout is that for attractive models all fixed points corresponding to local minima of F B are stable [57,Theorem 6], [43], [53]. Although BP may still fail to converge on balanced models, it is thus at least guaranteed that BP will converge if the messages are initialized close enough to a fixed point.…”