Spherical Spin-glass Model Research Articles

Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices

In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions goes to infinity. We consider a general framework containing the cases where A and/or B are taken from an invariant ensemble or are fixed diagonal matrices. We show that the tilting method introduced in Guionnet and Maïda (2020 Electron. J. Probab. 25 1–24) can be extended to our general setting and is equivalent to the study of a spherical spin glass model specific to the operation—sum of symmetric matrices/product of symmetric matrices/sum of rectangular matrices—we are considering.

Marginals of a spherical spin glass model with correlated disorder

In this paper we prove the weak convergence, in the high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions in a subset of the high-temperature phase. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.

Concentration of the complexity of spherical pure p-spin models at arbitrary energies

We consider critical points of the spherical pure p-spin spin glass model with Hamiltonian HNσ=1Np−1/2∑i1,…,ip=1NJi1,…,ipσi1…σip, where σ=σ1,…,σN∈SN−1≔σ∈RN:σ2=N and Ji1,…,ip are i.i.d. standard normal variables. Using a second moment analysis, we prove that for p ≥ 32 and any E > −E⋆, where E⋆ is the (normalized) ground state, the ratio of the number of critical points σ with HN(σ) ≤ NE and its expectation asymptotically concentrate at 1. This extends to arbitrary E, a similar conclusion of Subag [Ann. Probab. 45, 3385–3450 (2017)].

Spherical Spin Glass Model with External Field

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

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.

A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise

We define a (symmetric key) encryption of a signal mathbf{s}in {mathbb {R^N}} as a random mapping mathbf{s}mapsto mathbf y =(y_1,ldots ,y_M)^Tin mathbb {R}^M known both to the sender and a recipient. In general the recipients may have access only to images mathbf{y} corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) mu =M/Nge 1 and the signal strength parameter R=sqrt{sum _i {s_i^2/N}}, we consider the problem of reconstructing mathbf{s} from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap p_{infty }in [0,1] between the original signal and its recovered image (known as ’estimate’) as Nrightarrow infty , for a given (’bare’) noise-to-signal ratio (NSR) gamma ge 0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full p_{infty } (gamma ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with p_{infty }>0 for any mu >1 and any gamma <infty , with p_{infty }sim gamma ^{-1/2} as gamma rightarrow infty . In contrast, for the case of purely quadratic nonlinearity, for any ERP mu >1 there exists a threshold NSR value gamma _c(mu ) such that p_{infty }=0 for gamma >gamma _c(mu ) making the reconstruction impossible. The behaviour close to the threshold is given by p_{infty }sim (gamma _c-gamma )^{3/4} and is controlled by the replica symmetry breaking mechanism.

Ferromagnetic to Paramagnetic Transition in Spherical Spin Glass

We consider the spherical spin glass model defined by a combination of the pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian and the ferromagnetic Curie-Weiss Hamiltonian. In the large system limit, there is a two-dimensional phase diagram with respect to the temperature and the coupling strength. The phase diagram is divided into three regimes; ferromagnetic, paramagnetic, and spin glass regimes. The fluctuations of the free energy are known in each regime. In this paper, we study the transition between the ferromagnetic regime and the paramagnetic regime in a critical scale.

The geometry of the Gibbs measure of pure spherical spin glasses

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature.

The extremal process of critical points of the pure p-spin spherical spin glass model

Recently, sharp results concerning the critical points of the Hamiltonian of the p-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points—that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.

The decoupling of the glass transitions in the two-component p-spin spherical model

Binary mixtures of large and small particles with a disparate size ratio exhibit a rich phenomenology at their glass transition points. In order to gain insights on such systems, we introduce and study a two-component version of the p-spin spherical spin glass model. We employ the replica method to calculate the free energy and the phase diagram. We show that when the strengths of the interactions of each component are not widely separated, the model has only one glass phase characterized by the conventional one-step replica symmetry breaking. However when the strengths of the interactions are well separated, the model has three glass phases depending on the temperature and component ratio. One is the ‘single’ glass phase in which only the spins of one component are frozen while the spins of the other component remain mobile. This phase is characterized by the one-step replica symmetry breaking. The second is the ‘double’ glass phase obtained by cooling the single glass phase further, in which the spins of the remaining mobile component are also frozen. This phase is characterized by the two-step replica symmetry breaking. The third is also the ‘double’ glass phase, which, however, is formed by the simultaneous freezing of the spins of both components at the same temperatures and is characterized by the one-step replica symmetry breaking. We discuss the implications of these results for the glass transitions of binary mixtures.

Almost sure convergence in quantum spin glasses

Recently, Keating, Linden, and Wells [Markov Processes Relat. Fields 21(3), 537-555 (2015)] showed that the density of states measure of a nearest-neighbor quantum spin glass model is approximately Gaussian when the number of particles is large. The density of states measure is the ensemble average of the empirical spectral measure of a random matrix; in this paper, we use concentration of measure and entropy techniques together with the result of Keating, Linden, and Wells to show that in fact the empirical spectral measure of such a random matrix is almost surely approximately Gaussian itself with no ensemble averaging. We also extend this result to a spherical quantum spin glass model and to the more general coupling geometries investigated by Erdős and Schröder [Math. Phys., Anal. Geom. 17(3-4), 441–464 (2014)].

The glassy random laser: replica symmetry breaking in the intensity fluctuations of emission spectra.

The behavior of a newly introduced overlap parameter, measuring the correlation between intensity fluctuations of waves in random media, is analyzed in different physical regimes, with varying amount of disorder and non-linearity. This order parameter allows to identify the laser transition in random media and describes its possible glassy nature in terms of emission spectra data, the only data so far accessible in random laser measurements. The theoretical analysis is performed in terms of the complex spherical spin-glass model, a statistical mechanical model describing the onset and the behavior of random lasers in open cavities. Replica Symmetry Breaking theory allows to discern different kinds of randomness in the high pumping regime, including the most complex and intriguing glassy randomness. The outcome of the theoretical study is, eventually, compared to recent intensity fluctuation overlap measurements demonstrating the validity of the theory and providing a straightforward interpretation of qualitatively different spectral behaviors in different random lasers.

Replica Fourier Transform: Properties and applications

The Replica Fourier Transform is the generalization of the discrete Fourier Transform to quantities defined on an ultrametric tree. It finds use in conjunction of the replica method used to study thermodynamics properties of disordered systems such as spin glasses. Its definition is presented in a systematic and simple form and its use illustrated with some representative examples. In particular we give a detailed discussion of the diagonalization in the Replica Fourier Space of the Hessian matrix of the Gaussian fluctuations about the mean field saddle point of spin glass theory. The general results are finally discussed for a generic spherical spin glass model, where the Hessian can be computed analytically.

Free Energy and Complexity of Spherical Bipartite Models

We investigate both free energy and complexity of the spherical bipartite spin glass model. We first prove a variational formula in high temperature for the limiting free energy based on the well-known Crisanti-Sommers representation of the mixed p-spin spherical model. Next, we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy.

Ensemble Inequivalence in the Spherical Spin Glass Model with Nonlinear Interactions

We investigate the ensemble inequivalence of the spherical spin glass model with nonlinear interactions of polynomial order $p$. This model is solved exactly for arbitrary $p$ and is shown to have first-order phase transitions between the paramagnetic and spin glass or ferromagnetic phases for $p \geq 5$. In the parameter region around the first-order transitions, the solutions give different results depending on the ensemble used for the analysis. In particular, we observe that the microcanonical specific heat can be negative and the phase may not be uniquely determined by the temperature.

Simplicity of the spherical spin-glass model

We revisit an approach to the replica-based analysis of the spherical spin-glass model that makes use of a mapping of the problem onto a one-dimensional interacting charge system. A saddle point approximation leads to the conclusion that the interaction between charges is irrelevant in the thermodynamic limit, and as a consequence, that there is no nontrivial correlation between replicas for this model. This allows us to show that quenched and annealed disorder averages agree for the spherical spin glass. We demonstrate this result within two different mathematical frameworks, and we also relate our analysis to the conclusions that follow from the replica symmetry ansatz.

Quantum spherical spin glass with random short-range interactions

In the present paper we analyze the critical properties of a quantum spherical spin glass model with short range, random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with help of the replica method for the spin glass fluctuating fields Qαγ(~kω1ω2) separates into a mean field contribution for the Qαα(0;ω;−ω) and a strictly short range partition function for the fields Qα6=γ(~kω1ω2).Here α, γ = 1..n are replica indices. The mean field part WMF coincides with previous results. The short range part WSR describes a phase transition in a Q -field theory, where the fluctuating fields depend on a space variable ~r and two times τ1 and τ2. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc = 5 at very low temperatures due to the dimensionality shift Dc = dc + 1 = 6. We then perform a ǫ ′ expansion to order one loop to calculate the critical exponents by solving the renormalization group equations. PACS numbers: 64.60.Cn; 75.30.M; 75.10.N

Antiferromagnetic spherical spin-glass model

We study the thermodynamic properties and the phase diagrams of a multi-spin antiferromagnetic spherical spin-glass model using the replica method. It is a two-sublattice version of the ferromagnetic spherical p-spin glass model. We consider both the replica-symmetric and the one-step replica-symmetry-breaking solutions, the latter being the most general solution for this model. We find paramagnetic, spin-glass, antiferromagnetic and mixed or glassy antiferromagnetic phases. The phase transitions are always of second order in the thermodynamic sense, but the spin-glass order parameter may undergo a discontinuous change.

Nonlinear susceptibilities of spherical models

The static and dynamic susceptibilities for a general class of mean-field random orthogonal spherical spin-glass models are studied. We show how the static and dynamical properties of the linear and nonlinear susceptibilities depend on the behaviour of the density of states of the two-body interaction matrix in the neighbourhood of the largest eigenvalue. Our results are compared with experimental results and also with those of the droplet theory of spin glasses.

Slow interaction dynamics in the spherical spin-glass model

A surprising physical interpretation of the replica number n was unveiled in the study of the equilibrium properties of the coupled dynamics of fast Ising spins and slow interactions: n is the ratio of the characteristic temperatures of the spin and the interaction systems and can take on any positive or negative real value, depending on the choice of the coupling dynamics. Here we investigate analytically the thermodynamics of the spherical spin-glass model for general n and derive explicit equations for the transition lines separating the replica symmetric, replica symmetry broken and paramagnetic phases.