Table of contents

When we called for contributions to this special issue we stressed the fact that we were interested in having the topic interpreted broadly. We asked for contributions ranging from equilibrium and dynamical studies of spin glasses, glassy behaviour in amorphous materials and low temperature physics, to applications in non-conventional areas such as error-correcting codes, image analysis and reconstruction, optimization and algorithms based on statistical mechanical ideas. This was because we believe that we have arrived at a very exciting moment for the development of this multidisciplinary approach, and that this issue should bear witness to, and summarize, such an exciting situation.

Even a cursory look at the index of this issue shows, we believe, that our hopes have been completely fulfilled; we have a large variety of papers giving new insights into the whole range of fields. Our hope is that it will play a double role. On the one hand it will carry, as good journals always should, a number of good physics papers containing important results. On the other it should be a good summary of the state of the art for some time.

The larger section of the issue is about slow dynamics. This is understandable, since slow dynamics is such a ubiquitous phenomenon. Here recent progress deals with glasses, spin glasses and far more general situations. Modifications of the celebrated fluctuation--dissipation theorem also play a crucial role.

Finite dimensional systems (mainly spin glasses) are attracting a lot of attention since their behaviour is not yet well defined from a theoretical point of view. Here we have papers about Ising, Heisenberg and Potts spin glasses, together with the discussion of different types of disorder.

Even if the mean field theory of spin glasses is well understood, important questions (about, for example, the value of the complexity and the detailed nature of the solution of the model) are still open, and they are discussed in this issue.

This period has also seen very rich developments in rigorous results concerning complex disordered systems. We present some new rigorous results in this issue.

As we have said before, we have focused on the strong paradigmatic and interdisciplinary nature of the recent developments in the subject. Maybe the spin glass theory is more important to some people since it allows us to study error-correcting codes and similar problems than because of the study of the spin glass materials themselves. Here we present a number of new results in many of these emerging directions.

We then have new developments in image processing. One uses the mean field theory, the Bethe approximation and ideas from the dynamical approach.

Also very important is the relation among statistical mechanics of disordered systems and optimization. We have here new results about colouring and the analysis of disordered systems ground states, together with a short review on vertex covering. The same ideas applied to codes are also finding many applications; here we have new work about low-density parity check codes and CDMA multi-user detection codes.

Finally we have new results about the application of statistical mechanical ideas to game theory and to the so-called econophysics. We believe that this topic is also experiencing a fast and solid progress, and we are happy to be able to witness it here.

We thank the authors who have been collaborative, open-minded toward improvements and punctual (as much as one could hope). We also appreciate the efforts of the referees who have worked hard towards ensuring high quality; we believe they have succeeded. We thank all the staff of Journal of Physics A: Mathematical and General for their exceptional work. Without all that this issue would not have been possible.

The existence of a generalized fluctuation–dissipation theorem observed in simulations and experiments performed in various glassy materials is related to the concepts of local equilibration and heterogeneity in space. Assuming the existence of a dynamic coherence length scale up to which the system is locally equilibrated, we extend previous generalizations of the FDT relating static to dynamic quantities to the physically relevant domain where asymptotic limits of large times and sizes are not reached. The formulation relies on a simple scaling argument and thus does not have the character of a theorem. Extensive numerical simulations support this proposition. Our results quite generally apply to systems with slow dynamics, independently of the space dimensionality, the chosen dynamics or the presence of disorder.

We study the emergence of a cross-over from entropically driven to thermally activated dynamics in different versions of the 'entropic' phase space model introduced by Barrat and Mézard, and previously considered in the zero temperature limit. We first focus on the low temperature (T ⩽ T_g) ageing phase of the original model and show that a short time singularity appears in the correlation function for T_g/2 < T < T_g, leading to dynamical ultrametricity at T = T_g. We then consider the finite size version of this model, showing that the long time dynamics is always thermally activated beyond a size dependent cross-over time scale. We also generalize the model, introducing a threshold energy so as to mimic a phase space composed of saddles above the threshold, and minima below. In this case, the cross-over time scale becomes much smaller than the equilibration time, and both kinds of ageing dynamics are successively found, inducing a non-trivial ageing scaling which does not reduce to the usual t/t_w (or even t/t^ν_w) one.

We discuss the 'memory effect' discovered in the 1960s by Kovacs in temperature shift experiments on glassy polymers, where the volume (or energy) displays a non-monotonic time behaviour. This effect is generic and is observed in a variety of different glassy systems (including granular materials). The aim of this paper is to discuss whether some microscopic information can be extracted from a quantitative analysis of the 'Kovacs hump'. We study analytically two families of theoretical models: domain growth and traps, for which detailed predictions of the shape of the hump can be obtained. Qualitatively, the Kovacs effect reflects the heterogeneity of the system: its description requires dealing not only with averages but with a full probability distribution (of domain sizes or of relaxation times). We conclude by some suggestions for a quantitative analysis of experimental results.

We analyse the relaxational dynamics of a system close to a saddle of the potential energy function, within an harmonic approximation. Our main aim is to relate the topological properties of the saddle, as encoded in its spectrum, to the dynamical behaviour of the system. In the context of the potential energy landscape approach, this represents a first formal step to investigate the belief that the dynamical slowing down at T_c is related to the vanishing of the number of negative modes found at the typical saddle point. In our analysis we keep the description as general as possible, using the spectrum of the saddle as an input. We prove the existence of a timescale t, which is uniquely determined by the spectrum, but is not simply related to the fraction of negative eigenvalues. The mean square displacement develops a plateau of length t, such that a two-step relaxation is obtained if t diverges at T_c. We analyse different spectral shapes and outline the conditions under which the mean square displacement exhibits a dynamical scaling identical to the β-relaxation regime of mode coupling theory, with a power-law approach to the plateau and power-law divergence of t at T_c.

We derive an exact expression of the response function to an infinitesimal magnetic field for an Ising–Glauber-like model with arbitrary exchange couplings. The result is expressed in terms of thermodynamic averages and does not depend on the initial conditions or on the dimension of the space. The response function is related to time-derivatives of a complicated correlation function and so the expression is a generalization of the equilibrium fluctuation–dissipation theorem in the special case of this model. Correspondence with the Ising–Glauber model is discussed. A discrete-time version of the relation is implemented in Monte Carlo simulations and then used to study the ageing regime of the ferromagnetic two-dimensional Ising–Glauber model quenched from the paramagnetic phase to the ferromagnetic one. Our approach has the originality to give direct access to the response function and the fluctuation–dissipation ratio.

The spin update engine (SUE) machine is used to extend, by a factor of 1000, the timescale of previous studies of the ageing, out-of-equilibrium dynamics of the Edwards–Anderson model with binary couplings, on large lattices (L = 60). The correlation function, C(t + t_w, t_w), t_w being the time elapsed under a quench from high-temperature, follows nicely a slightly-modified power law for t > t_w. Very small (logarithmic), yet clearly detectable deviations from the full-ageing t/t_w scaling can be observed. Furthermore, the t < t_w data show clear indications of the presence of more than one time sector in the ageing dynamics. Similar results are found in four dimensions, but a rather different behaviour is obtained in the infinite-dimensional z = 6 Viana–Bray model. Most surprisingly, our results in infinite dimensions seem incompatible with dynamical ultrametricity. A detailed study of the link correlation function is presented, suggesting that its ageing properties are the same as for the spin correlation function.

In this paper I introduce the probability distribution of the local overlaps in spin glasses. The properties of the local overlaps are studied in detail. These quantities are related to the recently proposed local version of the fluctuation dissipation relations: using the general principle of stochastic stability these local fluctuation dissipation relations can be proved in a way that is very similar to the usual proof of the fluctuation dissipation relations for intensive quantities. The local overlap and its probability distribution play a crucial role in this proof. Similar arguments can be used to prove that all sites in an ageing experiment remain at the same effective temperature at the same time.

We investigate violations of the fluctuation–dissipation theorem in two classes of trap models by studying the influence of the perturbing field on the transition rates. We show that for perturbed rates depending upon the value of the observable at the arrival trap, a limiting value of the fluctuation–dissipation ratio does exist. However, the mechanism behind the emergence of this value is different in both classes of models. In particular, for an entropically governed dynamics (where the perturbing field shifts the relative population of traps according to the value of the observable) perturbed rates are argued to take a form that guarantees the existence of a limiting value for the effective temperature, utterly related to the exponential character of the distribution of trap energies. Fluctuation–dissipation (FD) plots reproduce some of the patterns found in a broad class of glassy systems, reinforcing the idea that structural glasses self-generate a dynamical measure that is captured by phenomenological trap models.

Trap models are intuitively appealing and often solvable models of glassy dynamics. In particular, they have been used to study ageing and the resulting out-of-equilibrium fluctuation–dissipation relations between correlation and response functions. In this paper I show briefly that one such relation, first given by Bouchaud and Dean, is valid for a general class of mean-field trap models: it relies only on the way a perturbation affects the transition rates, but is independent of the distribution of trap depths and the form of the unperturbed transition rates, and holds for all observables that are uncorrelated with the energy. The model with Glauber dynamics and an exponential distribution of trap depths, as considered by Barrat and Mézard, does not fall into this class if the perturbation is introduced in the natural way by shifting all trap energies. I show that a similar relation between response and correlation nevertheless holds for the out-of-equilibrium dynamics at low temperatures. The results point to intriguing parallels between trap models with energetic and entropic barriers.

We revisit the problem of how spin-glasses 'heal' after being exposed to tortuous perturbations by the temperature/bond chaos effects in temperature/bond cycling protocols. Revised scaling arguments suggest that the amplitude of the order parameter within ghost domains recovers very slowly, compared with the rate it is reduced by the strong perturbations. The parallel evolution of the order parameter and the size of the ghost domains can be examined in simulations and experiments by measurements of a memory auto-correlation function which exhibits a 'memory peak' at the timescale of the age imprinted in the ghost domains. These expectations are confirmed by Monte Carlo simulations of an Edwards–Anderson Ising spin-glass model.

We study numerically the properties of local low-energy excitations in the two-dimensional Ising spin glass with Gaussian couplings. Given the ground state, we determine the lowest-lying connected cluster of flipped spins containing one given spin, either with a fixed volume, or with a volume constrained to lie in a certain range. Our aim is to understand corrections to the scaling predicted by the droplet picture of spin glasses and to resolve contradictory results reported in the literature for the stiffness exponent. We find no clear trace of corrections to scaling, and the obtained stiffness exponent is in relatively good agreement with standard domain-wall calculations.

We present the results of Monte Carlo simulations of two different three-dimensional-Potts glass models with short-range random interactions. In the first model, a ±J-distribution of the bonds is chosen, in the second model a Gaussian distribution. In both cases, the first two moments of the distribution are chosen to be J₀ = −1, ΔJ = +1, so that no ferromagnetic ordering of the Potts spins can occur. We find that for all temperatures investigated the spin glass susceptibility remains finite, that the spin glass order parameter remains zero, and that the specific heat has only a smooth Schottky-like peak. These results can be understood quantitatively by considering small but independent clusters of spins. Hence, we have evidence that there is no static phase transition at any nonzero temperature. Consistent with these findings, only very minor size effects are observed, which implies that all correlation lengths of the models remain very short. We also compute for both models the time auto-correlation function C(t) of the Potts spins. While in the Gaussian model C(t) shows a smooth uniform decay, the correlator for the ±J model has several distinct steps. These steps correspond to the breaking of bonds in small clusters of ferromagnetically coupled spins (dimers, trimers, etc). The relaxation times follow simple Arrhenius laws, with activation energies that are readily interpreted within the cluster picture, giving evidence that the system does not have a dynamic transition at a finite temperature. Hence we find that for the present models, all the transitions known for the mean-field version of the model are completely wiped out. Finally, we also determine the time auto-correlation functions of individual spins, and show that the system is dynamically very heterogeneous.

The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 (L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T_×, the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T_× to be ν_SG = 0.9 ± 0.2 and ν_CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T_×.

We re-examine the spin-glass (SG) phase transition of the ±J Heisenberg models with and without random anisotropy D in three dimensions (d = 3) using two complementary methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T_SG ∼ 0.19J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ_SG for D = 0 is obtained using almost the same transition temperature of T_SG ∼ 0.18J. Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.

We find the possibility of weak universality of spin-glass phase transitions in three-dimensional ±J models. The Ising, the XY and the Heisenberg models seem to undergo finite-temperature phase transitions with a ratio of the critical exponents γ/ν ∼ 2.4. Evaluated critical exponents may explain the corresponding experimental results. The analyses are based upon nonequilibrium relaxation from a paramagnetic state and finite-time scaling.

We study the strong role played by structural (quenched) heterogeneities on static and dynamic properties of the frustrated Ising lattice gas in two dimensions, already in the liquid phase. Different from the dynamical heterogeneities observed in other glass models, in this case they may have infinite lifetime and be spatially pinned by the quenched disorder. We consider a measure of local frustration to show how it induces the appearance of spatial heterogeneities and how this reflects in the observed behaviour of equilibrium density distributions and dynamic correlation functions.

We show that phase transitions in Ising systems with planar defects, i.e., disorder perfectly correlated in two dimensions are destroyed by smearing. This is caused by effects similar to but stronger than the Griffiths phenomena: exponentially rare spatial regions can develop true static long-range order even when the bulk system is still in its disordered phase. Close to the smeared transition, the order parameter is very inhomogeneous in space, with the thermodynamic (average) order parameter depending exponentially on temperature. We determine the behaviour using extremal statistics, and we illustrate the results by computer simulations.

The Becchi–Rouet–Stora–Tyutin (BRST) supersymmetry is a powerful tool for the calculation of the complexity of metastable states in glassy systems, and it is particularly useful to uncover the relationships between complexity and standard thermodynamics. In this work we compute the Thouless–Anderson–Palmer (TAP) complexity of the Sherrington–Kirkpatrick model at the quenched level, by using the BRST supersymmetry. We show that the complexity calculated at K steps of replica symmetry breaking is strictly related to the static free energy at K + 1 steps of replica symmetry breaking. The supersymmetry therefore provides a prescription to obtain the complexity of the TAP states from the standard static free energy, even in models which are solved by more than one step of replica symmetry breaking. This recipe states that the complexity is given by the Legendre transform of the static free energy, where the Legendre parameter is the largest replica symmetry breaking point of the overlap matrix.

We show that there is no need to modify the Parisi replica symmetry breaking ansatz, by working with R steps of breaking and solving exactly the discrete stationarity equations generated by the standard 'truncated Hamiltonian' of spin glass theory.

After introducing and discussing the link-overlap between spin configurations we show that the Edwards–Anderson model has a replica-equivalent quenched equilibrium state, a property introduced by Parisi in the description of the mean-field spin-glass phase which generalizes ultrametricity. Our method is based on the control of fluctuations through the property of stochastic stability and works for all the finite-dimensional spin-glass models.

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda–Guerra ones to the multi-overlap case.

In this paper, we consider a particular aspect of the relationship between mean field and finite-dimensional spin glasses. By means of a simple interpolation method, we prove that the free energy of a class of finite-dimensional spin glass models with Kac-type interactions is bounded below by that of their mean field analogue. As a result, Parisi theory of replica symmetry breaking can be exploited in order to give bounds on their free energy and ground state energy. Similar results hold for diluted versions of the systems.

The EM algorithm for the Bayesian grey scale image restoration is investigated in the framework of the mean field theory. Our model system is identical to the infinite range random field -Ising model. The maximum marginal likelihood method is applied to the determination of hyper-parameters. We calculate both the data-averaged mean square error between the original image and its maximizer of posterior marginal estimate, and the data-averaged marginal likelihood function exactly. After evaluating the hyper-parameter dependence of the data-averaged marginal likelihood function, we derive the EM algorithm which updates the hyper-parameters to obtain the maximum likelihood estimate analytically. The time evolutions of the hyper-parameters and so-called Q function are obtained. The relation between the speed of convergence of the hyper-parameters and the shape of the Q function is explained from the viewpoint of dynamics.

We have investigated the relaxation dynamics of image restoration through a Bayesian approach. The relaxation dynamics is much faster at zero temperature than at the Nishimori temperature where the pixel-wise error rate is minimized in equilibrium. At low temperature, we observed non-monotonic development of the overlap. We suggest that the optimal performance is realized through premature termination in the relaxation processes in the case of the infinite-range model. We also performed Markov chain Monte Carlo simulations to clarify the underlying mechanism of non-trivial behaviour at low temperature by checking the local field distributions of each pixel.

The framework of Bayesian image restoration for multi-valued images by means of the Q-Ising model with nearest-neighbour interactions is presented. Hyperparameters in the probabilistic model are determined so as to maximize the marginal likelihood. A practical algorithm is described for multi-valued image restoration based on the Bethe approximation. The algorithm corresponds to loopy belief propagation in artificial intelligence. We conclude that, in real world grey-level images, the Q-Ising model can give us good results.

We study the problem of bicolouring random hypergraphs, both numerically and analytically. We apply the zero-temperature cavity method to find analytical results for the phase transitions (dynamic and static) in the one-step replica symmetry breaking (1RSB) approximation. These points appear to be in agreement with the results of the numerical algorithm. In the second part, we implement and test the survey propagation algorithm for specific bicolouring instances in the so-called HARD-SAT phase.

We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time, ω(c) = (ln T)/N, is quantitatively predicted, in agreement with simulations.

We review recent progress in the study of the vertex-cover problem (VC). The VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits a coverable–uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy–hard transitions in the typical running time of the algorithms occur.

We explain a statistical mechanics approach, which works by mapping the VC to a hard-core lattice gas, and then applying techniques such as the replica trick or the cavity approach. Using these methods, the phase diagram of the VC could be obtained exactly for connectivities c < e, where the VC is replica symmetric. Recently, this result could be confirmed using traditional mathematical techniques. For c > e, the solution of the VC exhibits full replica symmetry breaking.

The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for the VC. Finally, we describe recent results for the VC when studied on other ensembles of finite- and infinite-dimensional graphs.

We discuss the application of polynomial combinatorial optimization algorithms to extract the universal zero-temperature properties of various disordered systems. Dijkstras algorithm is used for models of non-directed elastic lines on general regular graphs with isotropically correlated random potentials. The successive shortest path algorithm for minimum-cost-flow problems is applied for the study of ground state properties and the entanglement of many elastic lines in a disordered environment and the disorder-induced loop percolation transition in a vortex glass model. The pre-flow-push algorithm for minimum-cut–maximum-flow problems is used for the investigation of a roughening transition occurring in a model for elastic manifolds in a periodic potential in the presence of point disorder.

An iterative algorithm for the multiuser detection problem that arises in code division multiple access (CDMA) systems is developed on the basis of Pearl's belief propagation (BP). We show that the BP-based algorithm exhibits nearly optimal performance in a practical time scale by utilizing the central limit theorem and self-averaging property appropriately, whereas direct application of BP to the detection problem is computationally difficult and far from practical. We further present close relationships of the proposed algorithm to the Thouless–Anderson–Palmer approach and replica analysis known in spin-glass research.

We apply statistical mechanics to an inverse problem of linear mapping to investigate the physics of the irreversible compression. We use the replica symmetry breaking (RSB) technique with a toy model to demonstrate the Shannon result. The rate distortion function, which is widely known as the theoretical limit of the compression with a fidelity criterion, is derived using the Parisi one step RSB scheme. The bound cannot be achieved in the sparsely-connected systems, where suboptimal solutions dominate the capacity.

We present a theoretical method for a direct evaluation of the average and reliability error exponents in low-density parity-check error-correcting codes using methods of statistical physics. Results for the binary symmetric channel are presented for codes of both finite and infinite connectivity.

Typical performance of low-density parity-check (LDPC) codes over a general binary-input output-symmetric memoryless channel is investigated using methods of statistical mechanics. Relationship between the free energy in statistical-mechanics approach and the mutual information used in the information-theory literature is established within a general framework; Gallager and MacKay–Neal codes are studied as specific examples of LDPC codes. It is shown that basic properties of these codes known for particular channels, including their potential to saturate Shannon's bound, hold for general symmetric channels. The binary-input additive-white-Gaussian-noise channel and the binary-input Laplace channel are considered as specific channel models.

We present an exact dynamical solution of a spherical version of the batch minority game (MG) with random external information. The control parameters in this model are the ratio of the number of possible values for the public information over the number of agents, and the radius of the spherical constraint on the microscopic degrees of freedom. We find a phase diagram with three phases: two without anomalous response (an oscillating versus a frozen state) and a further frozen phase with divergent integrated response. In contrast to standard MG versions, we can also calculate the volatility exactly. Our study reveals similarities between the spherical and the conventional MG, but also intriguing differences. Numerical simulations confirm our analytical results.

When neural networks are trained on their own output signals they generate disordered time series. In particular, when two neural networks are trained on their mutual output they can synchronize; they relax to a time-dependent state with identical synaptic weights. Two applications of this phenomenon are discussed for (a) econophysics and (b) cryptography. (a) When agents competing in a closed market (minority game) are using neural networks to make their decisions, the total system relaxes to a state of good performance. (b) Two partners communicating over a public channel can find a common secret key.