Spherical Sherrington-Kirkpatrick Model Research Articles

Spherical Sherrington–Kirkpatrick Model for Deformed Wigner Matrix with Fast Decaying Edges

We consider the 2-spin spherical Sherrington–Kirkpatrick model whose disorder is given by a deformed Wigner matrix of the form \(W+\lambda V\), where W is a Wigner matrix and V is a random diagonal matrix with i.i.d. entries. Assuming that the density function of the entries of V decays faster than a certain rate near the edges of its spectrum, we prove the sharp phase transition of the limiting free energy and its fluctuation. In the high temperature regime, the fluctuation of the free energy converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low temperature regime. We also prove several results for deformed Wigner matrices, including a local law for the resolvent entries, a central limit theorem of the linear spectral statistics, and a theorem on the rigidity of eigenvalues.

Free energy fluctuations of the two-spin spherical SK model at critical temperature

We investigate the fluctuations of the free energy of the two-spin spherical Sherrington–Kirkpatrick model at critical temperature βc = 1. When β = 1, we find asymptotic Gaussian fluctuations with variance 16N2log(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 β=1+αlog(N)N−1/3. For any α∈R, we show that the fluctuations are at most order log(N)/N in the sense of tightness. If α → ∞ at any rate as N → ∞, then, properly normalized, the fluctuations converge to the Tracy–Widom1 distribution. If α → 0 at any rate as N → ∞ or α < 0 is fixed, the fluctuations are asymptotically Gaussian as in the α = 0 case. In determining the fluctuations, we apply a recent result of G. Lambert and E. Paquette [“Strong approximation of Gaussian beta-ensemble characteristic polynomials: The edge regime and the stochastic airy function,” arXiv:2009.05003 (2020)] on the behavior of the Gaussian-β-ensemble at the spectral edge.

Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington–Kirkpatrick model

We revisit the gradient descent dynamics of the spherical Sherrington–Kirkpatrick (p = 2) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a first algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a faster algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations scaling as N −ν from perfect alignment the system escapes the initial configuration in a time-scale scaling as ln N after which the dynamics no longer ‘self-averages’ with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.

Overlaps of a spherical spin glass model with microscopic external field

We examine the behavior of the 2-spin spherical Sherrington-Kirkpatrick model with an external field by analyzing the overlap of a spin with the external field. Previous research has noted that, at low temperature, this overlap exhibits dramatically different behavior in the presence of an external field as compared to the model with no external field. The transition between those two settings was examined in a recent physics paper by Baik, Collins-Woodfin, Le Doussal, and Wu as well as a recent math paper by Landon and Sosoe. Both papers focus on the setting in which the external field strength, $h$, approaches zero as the dimension, $N$, approaches infinity. In particular, the paper of Baik et al studies the overlap with a microscopic external field ($h\sim N^{-1/2}$) but without a rigorous proof. This paper aims to give a proof of that result. The proof involves representing the generating function of the overlap as a ratio of contour integrals and then analyzing the asymptotics of those contour integrals using results from random matrix theory.

Free energy of bipartite spherical Sherrington–Kirkpatrick model

Nous considerons l’energie libre du modele spherique bipartite de Sherrington–Kirkpatrick et determinons l’energie libre limite a chaque temperature. Nous prouvons egalement la convergence de la loi des fluctuations de l’energie libre a temperature non critique. La limite est donnee par la distribution Gaussienne pour toutes les temperatures elevees et par la distribution de Tracy–Widom GOE pour toutes les temperatures basses. Le resultat est universel et l’analyse est applicable a un cadre plus general, y compris le cas ou le desordre est distribue de maniere non identique.

The TAP–Plefka Variational Principle for the Spherical SK Model

We reinterpret the Thouless-Anderson-Palmer approach to mean field spin glass models as a variational principle in the spirit of the Gibbs variational principle and the Bragg-Williams approximation. We prove this TAP-Plefka variational principle rigorously in the case of the spherical Sherrington-Kirkpatrick model.

Correction to: Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Ferromagnetic Interaction

In Theorem 1.4 (iii) of the original article, we stated that 1 $$\begin{aligned} \sqrt{N} (F_N - F(\beta )) \Rightarrow \mathcal {N}(0, \alpha _2) \end{aligned}$$ in the ferromagnetic regime $$J > 1$$ and $$\beta > \frac{1}{2J}$$ . The proof was based on Theorem 1.5 (iii), which we proved in the paper, and a known random matrix theory result, given in the second part of (1.19) which reads 2 $$\begin{aligned} N^{1/2} \left( \mu _1 - \left( J + \frac{1}{J}\right) \right) \Rightarrow \mathcal {N}\left( 0, 2\left( 1- \frac{1}{J^2}\right) \right) \end{aligned}$$ for $$J>1$$ .

Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Ferromagnetic Interaction

We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices.

Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model

The spherical Sherrington-Kirkpatrick model is a spherical mean field model for spin glass. We consider the fluctuations of the free energy at arbitrary non-critical temperature for the 2-spin model with no magnetic field. We show that in the high temperature regime the law of the fluctuations converges to the Gaussian distribution just like in the Sherrington-Kirtkpatrick model. We show, on the other hand, that the law of the fluctuations is given by the GOE Tracy-Widom distribution in the low temperature regime. The orders of the fluctuations are markedly different in these two regimes. A universality of the limit law is also proved.

Instantons in the Langevin dynamics: An application to spin glasses

We develop a general technique to calculate the probability of transitions over the barriers in spin-glasses in the framework of the dynamical theory. We use Lagrangian formulation of the instanton dynamics in which the transitions are represented by instantons. We derive the full set of the equations that determine the instantons but instead of solving them directly we prove that an instanton process can be mapped into a usual process going back in time which simplifies the problem significantly. We apply this general considerations to a simple example of the spherical Sherrington-Kirkpatrick model and we find the probability of the transition between the metastable states which is in agreement with physical expectations.

Closure of the Monte Carlo dynamical equations in the spherical Sherrington-Kirkpatrick model.

We study the analytical solution of the Monte Carlo dynamics in the spherical Sherrington-Kirkpatrick model using the technique of the generating function. Explicit solutions for one-time observables (like the energy) and two-time observables (like the correlation and response function) are obtained. We show that the crucial quantity which governs the dynamics is the acceptance rate. At zero temperature, an adiabatic approximation reveals that the relaxational behavior of the model corresponds to that of a single harmonic oscillator with an effective renormalized mass. \textcopyright{} 1996 The American Physical Society.

Full dynamical solution for a spherical spin-glass model

We present a detailed analysis for the Langevin dynamics of a spherical spin-glass model (the spherical Sherrington-Kirkpatrick model). All the spins in the system are coupled by pairs via a random interaction matrix taken from the Gaussian ensemble. One finds that for a general initial configuration the system never reaches an equilibrium state and the theorems associated to `equilibrium dynamics' are violated. Only very particular initial conditions drive the system to equilibrium. The weak ergodicity breaking scenario is demonstrated for general `non-equilibrium' initial conditions. Two-time quantities such as the auto-correlation function explicitly depend on both times. When the time difference is short compared to the smaller time one finds stationary dynamics with time translation invariance and the fluctuation-dissipation theorem satisfied. Instead, when the time difference is of the order of the smaller time one finds non-stationary dynamics with aging phenomena, the system {\em remembers} the time spent after the initial time (the quench below the critical temperature). Interestingly enough the short time-difference dynamics (FDT regime) for non-equilibrium initial conditions is identical to the relaxation within the equilibrium states obtained for the particular `equilibrium' initial conditions. This points to a self-similaririty of the energy landascape. In addition we analyse the effect of temperature variations on the behaviour of the correlation function and energy density. In this way we are able to make some comparisons between experimentally observed results and exact calculations for the model. Somewhat surprisingly, this simple model captures some of the effects seen in laboratory spin-glasses.