Fluctuations of the free energy in the REM and the p-spin SK models

Details Fluctuations of the free energy in the REM and the p-spin SK models Fluctuations of the free energy in the REM and the p-spin SK models

2000-07-11

arxiv.org/abs/cond-mat/0007175v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

40pp, AMSTeX

Scientific paper

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that (for $p$ even) the random corrections to the free energy are on a scale $N^{-(p-2)/4}$ only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, $\b$, smaller than a critical $\b_p$. We also show that $\b_p\to \sqrt{2\ln 2}$ as $p\uparrow +\infty$. Additionally we study the formal $p\uparrow +\infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at {\it all} temperatures. For $\b<\sqrt{2\ln2}$, fluctuations are found at an {\it exponentially small} scale, with two distinct limit laws above and below a second critical value $\sqrt{\ln 2/2}$: For $\b$ up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For $\b$ larger than the critical $\sqrt{2\ln 2}$, the fluctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1/2.

Affiliated with

Also associated with

No associations

Rating

Fluctuations of the free energy in the REM and the p-spin SK models does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Fluctuations of the free energy in the REM and the p-spin SK models, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fluctuations of the free energy in the REM and the p-spin SK models will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-28587