Ground-state energy fluctuations in the Sherrington-Kirkpatrick model

TL;DR

This study numerically analyzes the distribution of ground-state energies in the Sherrington-Kirkpatrick model, revealing a non-Gaussian distribution that scales with system size and fits a Gumbel distribution.

Contribution

It provides the first detailed numerical characterization of the ground-state energy distribution and its finite-size scaling, identifying a non-Gaussian form well approximated by a Gumbel distribution.

Findings

Standard deviation scales as N^{-0.765} or N^{-3/4}

Ground-state energy distribution is non-Gaussian

Distribution fits a Gumbel distribution with m ≈ 6

Abstract

The probability distribution function (PDF) of the ground-state energy in the Sherrington-Kirkpatrick spin-glass model is numerically determined by collecting a large statistical sample of ground states, computed using a genetic algorithm. It is shown that the standard deviation of the ground-state energy per spin scales with the number of spins, N, as N^{-\rho} with \rho \simeq 0.765, but the value \rho=3/4 is also compatible with the data, while the previously proposed value \rho=5/6 is ruled out. The PDF satisfies finite-size scaling with a non-Gaussian asymptotic PDF, which can be fitted remarkably well by the Gumbel distribution for the m-th smallest element in a set of random variables, with m \simeq 6.