The nature of attractors in an asymmetric spin glass with deterministic dynamics

The authors study the attractors in an infinite-range Ising spin-glass model with deterministic dynamics where the interactions have asymmetry, specified by a parameter k. They find a duality relation between the attractors for models with asymmetry parameters k and 1/k. The attractors are fixed points or limit cycles of short length, except for k=1, at which the average cycle length diverges, reminiscent of a phase transition, and the model has many similarities to the random map model as well as to the infinite-range symmetric spin glass in thermal equilibrium, including the fact that a few attractors dominate the weight. The extent of this dominance varies from sample to sample and so is given by a non-trivial probability distribution, Pi (Y), which they compute numerically.