Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2007-11-21
J. Phys. A: Math. Theor. 41 (2008) 324008
Physics
Condensed Matter
Disordered Systems and Neural Networks
Contribution to the conference "Viewing the World through Spin Glasses" in honour of Professor David Sherrington on the occasi
Scientific paper
10.1088/1751-8113/41/32/324008
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.
Aspelmeier Timo
Billoire Alain
Marinari Enzo
Moore Anna M.
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