Finite-size corrections in the Sherrington–Kirkpatrick model

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) ∝ N1/6. 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 PJ(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 N1/6, which is again consistent with the results of our numerical simulations.