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Register a new userOrder of Fluctuations of the Free Energy in the SK Model at Critical Temperature
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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 $O((log N)^2+N^{1-alpha}).$
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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 describe the fluctuations of the overlap between two replicas in the 2-spin spherical SK model about its limiting value in the low temperature phase. We show that the fluctuations are of order $N^{-1/3}$ and are given by a simple, explicit function of the eigenvalues of a matrix from the Gaussian Orthogonal Ensemble. We show that this quantity converges and describe its limiting distribution in terms of the Airy1random point field (i.e., the joint limit of the extremal eigenvalues of the GOE) from random matrix theory.
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.
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 partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(sqrt{N}t)/sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $tau_N$ is the extinction time of $i_N(t)$ (on the $sqrt{N}$ time scale) then $tau_N$ has a limit.
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