Convergence of Monte Carlo Simulations to Equilibrium

We give two direct, elementary proofs that a Monte Carlo simulation converges to equilibrium provided that appropriate conditions are satisfied. The first proof requires detailed balance while the second is quite general.Comment: 4 pages. v2: published versio

Spin glasses in the non-extensive regime

Spin systems with long-range interactions are "non-extensive" if the strength of the interactions falls off sufficiently slowly with distance. It has been conjectured for ferromagnets, and more recently for spin glasses, that, everywhere in the non-extensive regime, the free energy is exactly equal to that for the infinite range model in which the characteristic strength of the interaction is independent of distance. In this paper we present the results of Monte Carlo simulations of the one-dimensional long-range spin glasses in the non-extensive regime. Using finite-size scaling, our results for the transition temperatures are consistent with this prediction. We also propose, and provide numerical evidence for, an analogous result for diluted long-range spin glasses in which the coordination number is finite, namely that the transition temperature throughout the non-extensive regime is equal to that of the infinite-range model known as the Viana-Bray model.Comment: 8 pages; corrected typos, additional background and references relating to FSS correction