Abstract

In this chapter we present some results on numerical simulation in three-dimensional spin glasses. In fact, there are problems for which the analytical approach is out of reach. In such cases numerical simulations are a source of hints and suggestions. Their robustness is based on the fact that the asymptotic properties of the model for large volumes are identified. After describing the standard algorithm used (parallel tempering) we address the following problems: overlap equivalence among site and link overlaps, ultrametricity or hierarchical organization of the equilibrium states, decay of correlations, pure state identification, energy interfaces, and stiffness exponents related to the lower critical dimension.

Introduction

Numerical simulations have played a crucial role in the development of spin glass theory, especially in those cases in which exact results are unavailable due to formidable mathematical difficulties. Some of the first systematic works were by Bray and Moore (1984), Ogielski and Morgenstern (1985), and Bhatt and Young (1988). For a recent review, see Marinari et al. (1997) and the references therein. It is also interesting to check Newman and Stein (1996) and Marinari et al. (2000) for examples of the difficulties of reconciling theory with numerical simulations in finite-dimensional spin glasses.

The general outcome of these (and other) numerical studies is that many features of themean-field theory are seen in the ever larger finite-volume systems accessible to numerical simulations. However, no definite conclusions can be reached by means of the sole use of computers.