Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

TL;DR

This paper numerically investigates the ground states of diluted Sherrington-Kirkpatrick spin glasses, revealing how finite-size correction exponents vary with bond density, using heuristic optimization methods.

Contribution

It introduces a comprehensive numerical analysis of dilute SK spin glasses, showing the continuous variation of the correction exponent with bond density, which was previously unexpected.

Findings

Finite-size correction exponent varies continuously with bond density p.

For p approaching 1, the exponent matches the known SK value of approximately 2/3.

Ground-state energies are accurately computed across a range of bond densities.

Abstract

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond-diluted to an overall density $p$, such that the average degree diverges as $\sim pN$ with the system size $N$. Ground-state energies are obtained with high accuracy for random instances over a wide range of fixed $p$. Since this is a NP-hard combinatorial problem, we employ the Extremal Optimization heuristic to that end. We find that the exponent describing the finite-size corrections, $\omega$, varies continuously with $p$, a somewhat surprising result, as one would not expect that gradual bond-dilution would change the $T=0$ universality class of a statistical model. For $p\to1$, the familiar result of $\omega(p=1)\approx\frac{2}{3}$ for SK is obtained.