On a Dynamical-Like Replica-Symmetry-Breaking Scheme for the Spin Glass

Considering the unphysical result obtained in the calculation of the free-energy cost for twisting the boundary conditions in a spin glass, we trace it to the negative multiplicities associated with the Parisi replica-symmetry breaking (RSB). We point out that a distinct RSB, that keeps positive multiplicities, was proposed long ago, in the spirit of an ultra-long time dynamical approach due to Sompolinsky. For an homogeneous bulk system, both RSB schemes are known to yield identical free energies and observables. However, using the dynamical RSB, we have recalculated the twist free energy at the mean-field level. The free-energy cost of this twist is, as expected, positive in that scheme, as it should be

The crossover region between long-range and short-range interactions for the critical exponents

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.Comment: 17 pages, 6 figure