Extremal Optimization for Sherrington-Kirkpatrick Spin Glasses

TL;DR

This paper applies Extremal Optimization to approximate ground states of the Sherrington-Kirkpatrick spin glass model, achieving high accuracy and revealing insights into finite-size corrections and energy distribution skewness.

Contribution

It extends EO to highly connected systems and provides highly accurate numerical estimates for ground state energies and correction exponents in the SK model.

Findings

EO achieves ~0.01% accuracy in ground state energy estimates.

Finite-size correction exponent approximates 2/3.

Ground state energy distribution is highly skewed and connectivity-dependent.

Abstract

Extremal Optimization (EO), a new local search heuristic, is used to approximate ground states of the mean-field spin glass model introduced by Sherrington and Kirkpatrick. The implementation extends the applicability of EO to systems with highly connected variables. Approximate ground states of sufficient accuracy and with statistical significance are obtained for systems with more than N=1000 variables using $\pm J$ bonds. The data reproduces the well-known Parisi solution for the average ground state energy of the model to about 0.01%, providing a high degree of confidence in the heuristic. The results support to less than 1% accuracy rational values of $\omega=2/3$ for the finite-size correction exponent, and of $\rho=3/4$ for the fluctuation exponent of the ground state energies, neither one of which has been obtained analytically yet. The probability density function for ground state energies is highly skewed and identical within numerical error to the one found for Gaussian bonds. But comparison with infinite-range models of finite connectivity shows that the skewness is connectivity-dependent.