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Register a new userFluctuations of the 2-spin SSK model with magnetic field
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We analyze the fluctuations of the free energy, replica overlaps, and overlap with the magnetic fields in the quadratic spherial SK model with a vanishing magnetic field. We identify several different behaviors for these quantities depending on the size of the magnetic field, confirming predictions by Fyodorov-Le Doussal and recent work of Baik, Collins-Wildman, Le Doussal and Wu.
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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.
We characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition.
We derive the Thouless-Anderson-Palmer (TAP) equations for the Ghatak and Sherrington model. Our derivation, based on the cavity method, holds at high temperature and at all values of the crystal field. It confirms the prediction of Yokota.
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
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