No Arabic abstract
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 $b_pto sqrt{2ln 2}$ as $puparrow +infty$. Additionally we study the formal $puparrow +infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at {it all} temperatures. For $b<sqrt{2ln2}$, 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 $sqrt{2ln 2}$, 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.
The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the free-energy barriers which are visible in the probability density of the Parisi overlap parameter. Assuming that the mean barrier height diverges with the number of spins N as N^alpha, our data show good agreement with the theoretical value alpha = 1/3.
In a region above the Almeida-Thouless line, where we are able to control the thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale N^{-1/2}, for N large. The method we employ is based on the idea, we recently developed, of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.
We investigate the fluctuations of the free energy of the $2$-spin spherical Sherrington-Kirkpatrick model at critical temperature $beta_c = 1$. When $beta = 1$ we find asymptotic Gaussian fluctuations with variance $frac{1}{6N^2} log(N)$, confirming in the spherical case a physics prediction for the SK model with Ising spins. We furthermore prove the existence of a critical window on the scale $beta = 1 +alpha sqrt{ log(N) } N^{-1/3}$. For any $alpha in mathbb{R}$ we show that the fluctuations are at most order $sqrt{ log(N) } / N$, in the sense of tightness. If $ alpha to infty$ at any rate as $N to infty$ then, properly normalized, the fluctuations converge to the Tracy-Widom$_1$ distribution. If $ alpha to 0$ at any rate as $N to infty$ or $ alpha <0$ is fixed, the fluctuations are asymptotically Gaussian as in the $alpha=0$ case. In determining the fluctuations, we apply a recent result of Lambert and Paquette on the behavior of the Gaussian-$beta$-ensemble at the spectral edge.
We present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature. It is proved that at the critical temperature the variance of the free energy is of $O((log N)^2).$ In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate $O(N^{-alpha})$ for some $alpha>0,$ then the variance is of $O((log N)^2+N^{1-alpha}).$
We study the problem of glassy relaxations in the presence of an external field in the highly controlled context of a spin-glass simulation. We consider a small spin glass in three dimensions (specifically, a lattice of size L=8, small enough to be equilibrated through a Parallel Tempering simulations at low temperatures, deep in the spin glass phase). After equilibrating the sample, an external field is switched on, and the subsequent dynamics is studied. The field turns out to reduce the relaxation time, but huge statistical fluctuations are found when different samples are compared. After taking care of these fluctuations we find that the expected linear regime is very narrow. Nevertheless, when regarded as a purely numerical method, we find that the external field is extremely effective in reducing the relaxation times.