Numerical Results for Ground States of Spin Glasses on Bethe Lattices

TL;DR

This paper numerically investigates the ground state energies and entropies of +/- J spin glasses on Bethe lattices with various connectivities, revealing differences between even and odd connectivities and comparing results with theoretical models.

Contribution

It provides high-accuracy numerical approximations of ground state energies and entropies for spin glasses on Bethe lattices, highlighting differences between even and odd connectivities and validating some theoretical predictions.

Findings

Ground state energies scale with connectivity in a simple functional form.

Significant differences in entropy behavior between even and odd connectivities.

Results align with one-step replica symmetry breaking calculations for energies.

Abstract

The average ground state energy and entropy for +/- J spin glasses on Bethe lattices of connectivities k+1=3...,26 at T=0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n=2048), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_{k+1} but also for their entropies s_{k+1}. The results show considerable quantitative differences between lattices of even and odd connectivities. The results for the ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k+1 to within a fraction of a percent onto a simple functional form, e_{k+1} = E_{SK} sqrt(k+1) - {2E_{SK}+sqrt(2)} / sqrt(k+1), where E_{SK} = -0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k+1, which do not distinguish between even and odd connectivities. We find non-zero entropies s_{k+1} at small connectivities. While s_{k+1} seems to vanish asymptotically with 1/(k+1) for even connectivities, it is indistinguishable from zero already for odd k+1 >= 9.