Replica Symmetry Research Articles

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Building upon the one-step replica symmetry breaking formalism, we show that the extreme values of a general class of Euclidean-space logarithmic correlated random energy models behave as a randomly shifted decorated exponential Poisson point process in the thermodynamic limit. The distribution of the random shift is determined solely by the large-distance ( “infra-red”, IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance (“ultraviolet”, UV) limit, in terms of the biased minimal process. We discuss the relations of our approach with that based on the freezing/duality conjecture, and connections to results in the probability literature. Our approach allowed us to derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the distribution of the second minimum is independent of UV data, and depends on IR behaviour via a single parameter, the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of 1/f-noise.

Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Replica symmetry breaking for anisotropic magnets with quenched disorder

We study critical behaviour of a magnet with cubic anisotropy and quenched scalar disorder which is taken into account by replica method. We derive to first order in ε approximation the renormalization group equations taking into account possible replica symmetry breaking. We study the stability of the replica symmetric fixed points with respect to perturbations without (in general case) replica symmetry. However, we find that if a fixed point is stable with respect to replica symmetric deviations, it is also stable with respect to deviations without replica symmetry.

Replica approach to mean-variance portfolio optimization

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r = N/T < 1, where N is the dimension of the portfolio and T the length of the time series used to estimate the covariance matrix. At the critical point r = 1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1 − r), independently of the covariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample variance is found to vanish at the critical point inversely proportional to the divergent estimation error.

Zero temperature limit for (1 + 1) directed polymers with correlated random potential**In the memory of Sergei Korshunov.

The zero temperature limit in (1 + 1) directed polymers with finite range correlated random potential is studied. In terms of the standard replica technique it is demonstrated that in this limit the considered system reveals the one-step replica symmetry breaking structure similar to the one which takes place in the random energy model. In particular, it is shown that at the temperature (where u and R are the strength and the correlation length of the random potential) there is a crossover from the high- to the low-temperature regime. Namely, in the high-temperature regime at the model is equivalent to the one with the δ-correlated potential where the non-universal prefactor of the free energy is proportional to T−2/3, while at this non-universal prefactor saturates at a finite (temperature independent) value.

Metastable minima of the Heisenberg spin glass in a random magnetic field

We have studied zero-temperature metastable minima in classical m-vector component spin glasses in the presence of m-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for m>2. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the m=2 and m=3 cases in the SK model the final energy reached in the quench is very close to the energy E_{c} at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states ρ(λ) of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual sqrt[λ] form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.

Robustness of replica symmetry breaking phenomenology in random laser.

Random lasers are optical sources where light is amplified by stimulated emission along random paths through an amplifying scattering medium. Connections between their physics and the one of quenched disordered nonlinear systems, notably spin glasses, have been recently suggested. Here we report a first experimental study of correlations of spectral fluctuations intensity in a random laser medium where the scatterers displacement significantly changes among consecutive shots. Remarkably, our results reveal that the replica symmetry breaking (RSB) phenomenology is robust with respect to an averaging over different realizations of the disorder. Moreover, besides opening new intriguing questions about the understanding of such a phenomenon, this work aims to clarify the connection between the RSB with the onset of the Lévy regime, i.e. the fluctuations regime that is a peculiar feature of the random lasing under critical conditions. Our results suggest that the former occurs independently of the latter and then the RSB phenomenology is a generic feature linked to the random laser threshold.

Effects of smooth random surface on fluid monolayer thermodynamics

We consider the lattice gas approach to statistical mechanics of fluid adsorbed on random surfaces with fluid-fluid and fluid-surface potentials. It was shown that effective Hamiltonian contains quenched random interactions and random site fields. Their statistical features combine the properties of random geometry and fluid-fluid pair interaction potential. The high-temperature expansion leads to infinite-ranged random field model and Sherrington–Kirkpatrick spin-glass model. Thermodynamic properties are evaluated using replica theory procedure widely used to analyze quenched disorder systems. On the other hand we consider the random field model in random graph with finite connectivity instead of previous “infinite-ranged” approximations. This model has been investigated using finite connectivity technique. The replica symmetry ansatz for the order function is expressed in terms of an effective-field distribution. Analysis of random geometry effects on thermodynamic properties in such approach was done for the first time.

Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions.

The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.

On the genealogy of branching random walks and of directed polymers

It is well known that the mean-field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite-size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.

Large fluctuations at the lasing threshold of solid- and liquid-state dye lasers.

Intensity fluctuations in lasers are commonly studied above threshold in some special configurations (especially when emission is fed back into the cavity or when two lasers are coupled) and related with their chaotic behaviour. Similar fluctuating instabilities are usually observed in random lasers, which are open systems with plenty of quasi-modes whose non orthogonality enables them to exchange energy and provides the sort of loss mechanism whose interplay with pumping leads to replica symmetry breaking. The latter however, had never been observed in plain cavity lasers where disorder is absent or not intentionally added. Here we show a fluctuating lasing behaviour at the lasing threshold both in solid and liquid dye lasers. Above and below a narrow range around the threshold the spectral line-shape is well correlated with the pump energy. At the threshold such correlation disappears, and the system enters a regime where emitted laser fluctuates between narrow, intense and broad, weak peaks. The immense number of modes and the reduced resonator quality favour the coupling of modes and prepares the system so that replica symmetry breaking occurs without added disorder.

Circular coloring of random graphs: statistical physics investigation

Circular coloring is a constraint satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.

Glassy behavior in a one-dimensional continuous-wave erbium-doped random fiber laser

The analogue of the paramagnetic to spin-glass phase transition in disordered magnetic systems, leading to the phenomenon of replica symmetry breaking, has been recently demonstrated in a two-dimensional random laser consisting of an organic-based amorphous solid-state thin film. We report here the first demonstration of replica symmetry breaking in a one-dimensional photonic system consisting of an erbium-doped random fiber laser operating in the continuous-wave regime based on a unique random fiber grating system, which plays the role of the random scatterers and operates in the Anderson localization regime. The clear transition from a photonic paramagnetic to a photonic spin glass phase, characterized by the probability distribution function of the Parisi overlap, was verified and characterized. In this unique system, the radiation field interacts only with the gain medium, and the fiber grating, which provides the disordered feedback mechanism, does not interfere with the pump.

Observation of photonic paramagnetic to spin-glass transition in a specially designed TiO2 particle-based dye-colloidal random laser.

Colloidal-based random lasers (RLs) are highly efficient and have been exploited in a wide range of geometries. However, in the particular case of ethanol solutions of rhodamines and TiO2 particles, the RL behavior is quite unstable due to the fast precipitation of the particles. In this Letter, specially designed amorphous TiO2 particles were synthesized by a sol-gel method, preventing the degradation of the RL for long operating lifetimes of over 105 shots. As a consequence, this modified colloidal RL allowed the observation of a clear replica-symmetry-breaking phase transition from the paramagnetic fluorescent to spin-glass RL behavior, which has not been observed in the system with nonfunctionalized TiO2 particles.

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.

On one-step replica symmetry breaking in the Edwards–Anderson spin glass model

We consider a one-step replica symmetry breaking description of the Edwards–Anderson spin glass model in 2D. The ingredients of this description are a Kikuchi approximation to the free energy and a second-level statistical model built on the extremal points of the Kikuchi approximation, which are also fixed points of a generalized belief propagation (GBP) scheme. We show that a generalized free energy can be constructed where these extremal points are exponentially weighted by their Kikuchi free energy and a Parisi parameter y, and that the Kikuchi approximation of this generalized free energy leads to second-level, one-step replica symmetry breaking (1RSB), GBP equations. We then proceed analogously to the Bethe approximation case for tree-like graphs, where it has been shown that 1RSB belief propagation equations admit a survey propagation solution. We discuss when and how the one-step-replica symmetry breaking GBP equations that we obtain also allow a simpler class of solutions which can be interpreted as a class of generalized survey propagation equations for the single instance graph case.

Observation of Lévy distribution and replica symmetry breaking in random lasers from a single set of measurements.

Random lasers have been recently exploited as a photonic platform for studies of complex systems. This cross-disciplinary approach opened up new important avenues for the understanding of random-laser behavior, including Lévy-type distributions of strong intensity fluctuations and phase transitions to a photonic spin-glass phase. In this work, we employ the Nd:YBO random laser system to unveil, from a single set of measurements, the physical origin of the complex correspondence between the Lévy fluctuation regime and the replica-symmetry-breaking transition to the spin-glass phase. A novel unexpected finding is also reported: the trend to suppress the spin-glass behavior for high excitation pulse energies. The present description from first principles of this correspondence unfolds new possibilities to characterize other random lasers, such as random fiber lasers, nanolasers and small lasers, which include plasmonic-based, photonic-crystal and bio-derived nanodevices. The statistical nature of the emission provided by random lasers can also impact on their prominent use as sources for speckle-free laser imaging, which nowadays represents one of the most promising applications of random lasers, with expected progress even in cancer research.

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover (min-VC), a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical mechanical model with a one-parameter family. Statistical mechanical analyses reveal for α=2 that the LP optimal solution is typically equal to that given by the IP below the critical average degree c=e in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result c=1 and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with α≥3, minimum vertex covers on α-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree c=e/(α-1) where the replica symmetry is broken.

Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution

.The state-following technique allows the study of metastable glassy states under external perturbations. Here we show how this construction can be used to study the behavior of glassy states of Hard Spheres in infinite dimensions under compression or shear strain. In Rainone et al (2015 Phys. Rev. Lett. 114 015701) it has been shown that in both cases, when the external perturbation is sufficiently strong, glassy states undergo a second-order transition, called the Gardner transition, whereupon a hierarchical structure of marginal micro-states manifests within the original glass state. The purpose of this work is to study the solution of the state-following construction in this marginal phase. We show that upon compression, close to the jamming transition, the metastable states are described by a scaling solution characterized by a set of non-trivial critical exponents that agree with the results of Charbonneau et al (2014 Nat. Commun. 5 3725), and we compute the value of the jamming density for various glassy states. Moreover we show that under the action of the shear strain, beyond the Gardner point, the metastable states can be followed in the marginal phase and we detect an overshoot in the stress-strain curve in agreement with numerical and experimental observations. Finally we further characterize the Gardner transition point by computing both the susceptibility and the exponent parameter λ that characterize the critical slowing down of the dynamics within a glassy state close to the transition.

Statistical mechanics models for multimode lasers and random lasers

Referring to recent approaches to multimode laser theory, including Monte Carlo simulations of effective models and statistical mechanical analytic computations, the status for a complete nonperturbative theory in open and disordered cavities is discussed and the derivation of the general statistical models in this framework is presented. When light is propagating in a disordered medium, the relevant models can be analysed via the replica method. For high degrees of disordered-induced frustration and nonlinearity, a glassy behaviour is expected beyond the lasing threshold, providing a suggestive link between glasses and photonics. We describe in detail the results for the general Hamiltonian model in the mean field approximation and we analytically justify an available test for replica symmetry breaking from intensity spectra measurements. Finally, we draw perspectives for such approaches.