Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices

Abstract

We study XY spin systems on small world lattices for a variety of graph structures, e.g. Poisson and scale-free, superimposed upon a one dimensional chain. In order to solve this model we extend the cavity method in the one pure-state approximation to deal with real-valued dynamical variables. We find that small-world architectures significantly enlarge the region in parameter space where synchronization occurs. We contrast the results of population dynamics performed on a truncated set of cavity fields with Monte Carlo simulations and find excellent agreement. Further, we investigate the appearance of replica symmetry breaking in the spin-glass phase by numerically analyzing the proliferation of pure states in the message passing equations. FoG 2005/04

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

“…for a C-RRG, as derived in Refs. [21,[35][36][37]. The exact location of the second-order critical line between paramagnetic and ferromagnetic phases can be found via the Susceptibility Propagation approach [20,21], namely the stability analysis of the BP fixed point P * [η i→j ].…”

Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin-glass XY model in a field on sparse random graphs

“…We clearly see that when increasing ∆, the population dynamics algorithm leaves sooner the unstable fixed point (e. g. the paramagnetic fixed point in the low-temperature region). For H = 0, a second-order phase transition occurs between the high-temperature RS-stable phase and the lowtemperature RS-unstable phase, with a critical temperature T c = 1/β c given by [19,20,24]:…”

Criticality in diluted ferromagnets

“…In the past few years a match among the study of systems defined on lattices by means of statistical mechanics [27] and the study of networks by means of graph theory [13] gave origin to very interesting models as the small world magnets [38] [36] or the scale free networks [14].…”