Fluctuations of the free energy in the REM and the $p$-spin SK models
Bovier, Anton ; Kurkova, Irina ; Löwe, Matthias
Ann. Probab., Tome 30 (2002) no. 1, p. 605-651 / Harvested from Project Euclid
We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington–Kirkpatrick (SK) 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 the random corrections to the free energy are on a scale $N^{-(p-2)/2}$ only and, after proper rescaling, converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, $\beta$, smaller than a critical $\beta_p$. We also show that $\beta_p \to \sqrt{2 \ln 2}$ 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 (properly rescaled) partition function at all temperatures. For $\beta < \sqrt{2 \ln 2}$, fluctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value $\sqrt{\ln 2/2:}$ for $\beta$ 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 $\beta$ larger than the critical $\sqrt{2 \ln 2}$, the fluctuations of the logarithm of the partition function are on a scale of 1 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.
Publié le : 2002-04-14
Classification:  spin glasses,  Sherrington-Kirkpatrick model,  $p$-spin model,  random energy model,  central limit theorem,  extreme values,  martingales,  82C44,  60K35
@article{1023481004,
     author = {Bovier, Anton and Kurkova, Irina and L\"owe, Matthias},
     title = {Fluctuations of the free energy in the REM and the $p$-spin SK
			 models},
     journal = {Ann. Probab.},
     volume = {30},
     number = {1},
     year = {2002},
     pages = { 605-651},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1023481004}
}
Bovier, Anton; Kurkova, Irina; Löwe, Matthias. Fluctuations of the free energy in the REM and the $p$-spin SK
			 models. Ann. Probab., Tome 30 (2002) no. 1, pp.  605-651. http://gdmltest.u-ga.fr/item/1023481004/