Numerical Results for Spin Glass Ground States on Bethe Lattices: Gaussian Bonds

TL;DR

This study numerically investigates the ground state energies of spin glasses on Bethe lattices with Gaussian bonds, revealing smooth energy convergence to the SK model and finite-size correction behaviors.

Contribution

It provides high-quality numerical estimates of ground state energies for Gaussian bond spin glasses on Bethe lattices, extending understanding of finite-size effects and corrections.

Findings

Ground state energies approach the SK model as r increases.

Finite-size corrections decay approximately as N^{-4/5}.

Instance-to-instance fluctuations are asymmetric.

Abstract

The average ground state energies for spin glasses on Bethe lattices of connectivities r=3,...,15 are studied numerically for a Gaussian bond distribution. The Extremal Optimization heuristic is employed which provides high-quality approximations to ground states. The energies obtained from extrapolation to the thermodynamic limit smoothly approach the ground-state energy of the Sherrington-Kirkpatrick model for r->\infty. Consistently for all values of r in this study, finite-size corrections are found to decay approximately with ~N^{-4/5}. The possibility of ~N^{-2/3} corrections, found previously for Bethe lattices with a bimodal +-J bond distribution and also for the Sherrington-Kirkpatrick model, are constrained to the additional assumption of very specific higher-order terms. Instance-to-instance fluctuations in the ground state energy appear to be asymmetric up to the limit of the accuracy of our heuristic. The data analysis provides insights into the origin of trivial fluctuations when using continuous bonds and/or sparse networks.