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Register a new userMetastability in stochastic dynamics of disordered mean-field models
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We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of ``admissible transitions. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
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We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the Sherrington-Kirkpatrick and p-spin models, to a wider class of mean field spin glass systems, including models with multi-component and non-Ising type spins, mean field spin glasses with an additional Curie-Weiss interaction, and systems consisting of several replicas of the spin glass model, where replicas are coupled with terms depending on the mutual overlaps.
We apply the Kovacs experimental protocol to classical and quantum p-spin models. We show that these models have memory effects as those observed experimentally in super-cooled polymer melts. We discuss our results in connection to other classical models that capture memory effects. We propose that a similar protocol applied to quantum glassy systems might be useful to understand their dynamics.
We consider the complexity of random ferromagnetic landscapes on the hypercube ${pm 1}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $mathcal G(N,p)$. Previous results had shown that, with high probability as $Ntoinfty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $Ntoinfty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.
Approximating marginals of a graphical model is one of the fundamental problems in the theory of networks. In a recent paper a method was shown to construct a variational free energy such that the linear response estimates, and maximum entropy estimates (for beliefs) are in agreement, with implications for direct and inverse Ising problems[1]. In this paper we demonstrate an extension of that method, incorporating new information from the response matrix, and we recover the adaptive-TAP equations as the first order approximation[2]. The method is flexible with respect to applications of the cluster variational method, special cases of this method include Naive Mean Field (NMF) and Bethe. We demonstrate that the new framework improves estimation of marginals by orders of magnitude over standard implementations in the weak coupling limit. Beyond the weakly coupled regime we show there is an improvement in a model where the NMF and Bethe approximations are known to be poor for reasons of frustration and short loops.
We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $mathcal{D}(omega)$ of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness $kappa$ drawn from a distribution $p(kappa)$, subjected to a constant field $h$ and interacting bilinearly with a coupling of strength $J$. We investigate the vibrational properties of its ground state at zero temperature. When $p(kappa)$ is gapped, the emergent $mathcal{D}(omega)$ is also gapped, for small $J$. Upon increasing $J$, the gap vanishes on a critical line in the $(h,J)$ phase diagram, whereupon replica symmetry is broken. At small $h$, the form of this pseudogap is quadratic, $mathcal{D}(omega)simomega^2$, and its modes are delocalized, as expected from previously investigated mean-field spin glass models. However, we determine that for large enough $h$, a quartic pseudogap $mathcal{D}(omega)simomega^4$, populated by localized modes, emerges, the two regimes being separated by a special point on the critical line. We thus uncover that mean-field disordered systems can generically display both a quadratic-delocalized and a quartic-localized spectrum at the glass transition.
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