Finite size corrections in the Sherrington-Kirkpatrick model

TL;DR

This paper investigates finite-size effects in the Sherrington-Kirkpatrick model, proposing a finite replica-symmetry breaking scheme that accurately predicts corrections to thermodynamic quantities and fluctuations based on system size.

Contribution

It introduces a finite-size correction method for the SK model using a truncated Parisi scheme with K(N) ~ N^{1/6}, validated by Monte Carlo simulations.

Findings

Finite size corrections scale with N^{1/6}.

Predictions match Monte Carlo results for order parameter and energy.

Sample-to-sample fluctuations also follow the N^{1/6} scaling.

Abstract

We argue that when the number of spins $N$ in the SK model is finite, the Parisi scheme can be terminated after $K$ replica-symmetry breaking steps, where $K(N) \propto N^{1/6}$. We have checked this idea by Monte Carlo simulations: we expect the typical number of peaks and features $R$ in the (non-bond averaged) Parisi overlap function $P_J(q)$ to be of order $2K(N)$, and our counting (for samples of size $N$ up to 4096 spins) gives results which are consistent with our arguments. We can estimate the leading finite size correction for any thermodynamic quantity by finding its $K$ dependence in the Parisi scheme and then replacing $K$ by K(N). Our predictions of how the Edwards-Anderson order parameter and the internal energy of the system approach their thermodynamic limit compare well with the results of our Monte Carlo simulations. The $N$-dependence of the sample-to-sample fluctuations of thermodynamic quantities can also be obtained; the total internal energy should have sample-to-sample fluctuations of order $N^{1/6}$, which is again consistent with the results of our numerical simulations.