arXiv subjects

Michele Castellana

Publications and source records attributed to Michele Castellana.

29 records · Page 2Linked to original sources

arXiv2014

Inverse spin glass and related maximum entropy problems

If we have a system of binary variables and we measure the pairwise correlations among these variables, then the least structured or maximum entropy model for their joint distribution is an Ising model with pairwise interactions among the spins. Here we consider inhomogeneous systems in which we constrain (for example) not the full matrix of correlations, but only the distribution from which these correlations are drawn. In this sense, what we have constructed is an inverse spin glass: rather than choosing coupling constants at random from a distribution and calculating correlations, we choose the correlations from a distribution and infer the coupling constants. We argue that such models generate a block structure in the space of couplings, which provides an explicit solution of the inverse problem. This allows us to generate a phase diagram in the space of (measurable) moments of the distribution of correlations. We expect that these ideas will be most useful in building models for systems that are nonequilibrium statistical mechanics problems, such as networks of real neurons.

cond-mat.dis-nn↗

arXiv2014

Rigorous Results for Hierarchical Models of Structural Glasses

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model (HREM), and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the $p$-spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.

math-ph↗

arXiv2014

Extreme-Value Distributions and the Freezing Transition of Structural Glasses

We consider two mean-field models of structural glasses, the random energy model (REM) and the $p$-spin model (PSM), and we show that the finite-size fluctuations of the freezing temperature are described by extreme-value statistics (EVS) distributions, establishing an unprecedented connection between EVS and the freezing transition of structural glasses. For the REM, the freezing-temperature fluctuations are described by the Gumbel EVS distribution, while for the PSM the freezing temperature fluctuates according to the Tracy-Widom (TW) EVS distribution, which has been recently discovered within the theory of random matrices. For the PSM, we provide an analytical argument showing that the emergence of the TW distribution can be understood in terms of the statistics of glassy metastable states.

cond-mat.dis-nn↗

arXiv2014

Free-energy bounds for hierarchical spin models

In this paper we study two non-mean-field spin models built on a hierarchical lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: Given that such fluctuations are not negligible in non-mean-field models, we develop a novel strategy to tackle part of OPF in hierarchical models. The method is based on absorbing part of OPF of a block of spins into an effective Hamiltonian of the underlying spin blocks. We illustrate this method for DHM and show that, compared to the mean-field bound for the free energy, it provides a tighter non-mean-field bound, with a critical temperature closer to the exact one. To extend this method to the HEA model, a suitable generalization of Griffith's correlation inequalities for Ising ferromagnets is needed: Since correlation inequalities for spin glasses are still an open topic, we leave the extension of this method to hierarchical spin glasses as a future perspective.

math-ph↗

arXiv2013

The Renormalization Group for Disordered Systems

In this thesis we investigate the Renormalization Group (RG) approach in finite-dimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems.

cond-mat.dis-nn↗

arXiv2011

Extreme Value Statistics Distributions in Spin Glasses

We study the probability distribution of the pseudo-critical temperature in a mean-field and in a short-range spin-glass model: the Sherrington-Kirkpatrick (SK) and the Edwards-Anderson (EA) model. In both cases, we put in evidence the underlying connection between the fluctuations of the pseudo-critical point and and the Extreme Value Statistics of random variables. For the SK model, both with Gaussian and binary couplings, the distribution of the pseudo-critical temperature is found to be the Tracy-Widom distribution. For the EA model, the distribution is found to be the Gumbel distribution. Being the EA model representative of uniaxial magnetic materials with quenched disorder like $Fe_{0.5} Mn_{0.5} Ti O_3$ or $Eu_{0.5} Ba_{0.5} Mn O_3$, its pseudo-critical point distribution should be a priori experimentally accessible.

cond-mat.dis-nn↗

arXiv2011

Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass

We investigate the finite-size fluctuations due to quenched disorder of the critical temperature of the Sherrington-Kirkpatrick spin glass. In order to accomplish this task, we perform a finite-size analysis of the spectrum of the susceptibility matrix obtained via the Plefka expansion. By exploiting results from random matrix theory, we obtain that the fluctuations of the critical temperature are described by the Tracy-Widom distribution with a non-trivial scaling exponent 2/3.

cond-mat.dis-nn↗

arXiv2011

Real space Renormalization Group analysis of a non-mean field spin-glass

A real space Renormalization Group approach is presented for a non-mean field spin-glass. This approach has been conceived in the effort to develop an alternative method to the Renormalization Group approaches based on the replica method. Indeed, non-perturbative effects in the latter are quite generally out of control, in such a way that these approaches are non-predictive. On the contrary, we show that the real space method developed in this work yields precise predictions for the critical behavior and exponents of the model.

cond-mat.dis-nn↗

arXiv2010

A renormalization group computation of the critical exponents of hierarchical spin glasses

The infrared behaviour of a non-mean field spin-glass system is analysed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the epsilon-expansion technique with two independent methods. Both methods yield the same result confirming that the infrared behaviour of the theory if well-defined and the underlying ideas of the Renormalization Group hold also in such non-mean field disordered model. By pushing such calculation to high orders in epsilon, a consistent and predictive non-mean field theory for such disordered system could be established.

cond-mat.dis-nn↗

arXiv2010

Adversarial Satisfiability Problem

We study the adversarial satisfiability problem, where the adversary can choose whether variables are negated in clauses or not in order to make the resulting formula unsatisfiable. This is one case of a general class of adversarial optimization problems that often arise in practice and are algorithmically much harder than the standard optimization problems. We use the cavity method to compute large deviations of the entropy in the random satisfiability problem with respect to the negation-configurations. We conclude that in the thermodynamic limit the best strategy the adversary can adopt is extremely close to simply balancing the number of times every variable is and is not negated. We also conduct a numerical study of the problem, and find that there are very strong pre-asymptotic effects that are due to the fact that for small sizes exponential and factorial growth is hardly distinguishable.

cs.CC↗

arXiv2010

The Hierarchical Random Energy Model

We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a non-mean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occuring in the usual mean field Random Energy Model. At variance with mean field, the high temperature branch of the free-energy is non-analytic at the transition point.

cond-mat.stat-mech↗