Narrow Results

In [11], Talagrand gave a rigorous proof of the Parisi formula in the classical Sherrington–Kirkpatrick (SK) model. In this paper, we build upon the methodology developed in [11] and extend Talagrand's result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.

By an exact calculation of the ground states for the ±J Edwards–Anderson spin glass, one can extrapolate the ground state energy to infinite system sizes. We calculate the exact ground states for the three-dimensional spin glass with free boundaries for system sizes up to 10 and fit different finite-size functions. We cannot decide, only from the quality of the fit, which fitting function to choose. Relying on the literature values for the extrapolated energy, we find the finite-size corrections to vary as 1/L.

In this work we study, by means of numerical simulations, the out-of-equilibrium dynamics of the one-dimensional Edwards–Anderson model with long-range interactions of the form ± Jr^-α. In the limit α → 0 we recover the well known Sherrington–Kirkpatrick mean-field version of the model, which presents a very complex dynamical behavior. At the other extreme, for α → ∞ the model converges to the nearest-neighbor one-dimensional system. We focus our study on the dependence of the dynamics on the history of the sample (aging phenomena) for different values of α. The model is known to have mean-field exponents already for values of α = 2/3. Our results indicate that the crossover to the dynamic mean-field occurs at a value of α < 2/3.

Monte Carlo simulation techniques, like simulated annealing and parallel tempering, are often used to evaluate low-temperature properties and find ground states of disordered systems. Here we compare these methods using direct calculations of ground states for three-dimensional Ising diluted antiferromagnets in a field (DAFF) and three-dimensional Ising spin glasses (ISG). For the DAFF, we find that, with respect to obtaining ground states, parallel tempering is superior to simple Monte Carlo and to simulated annealing. However, equilibration becomes more difficult with increasing magnitude of the externally applied field. For the ISG with bimodal couplings, which exhibits a high degeneracy, we conclude that finding true ground states is easy for small systems, as is already known. But finding each of the degenerate ground states with the same probability (or frequency), as required by Boltzmann statistics, is considerably harder and becomes almost impossible for larger systems.

The multicanonical ensemble is reviewed. Simulations of this ensemble promise improvements for a wide range of systems. Applications to first order phase transitions and spin glasses are summarized.

We report a Monte Carlo simulation of the 2D Edwards-Anderson spin glass model within the recently introduced multicanonical ensemble. Replica on lattices of size L² up to L=48 are investigated. Once a true groundstate is found, we are able to give a lower bound on the number of statistically independent groundstates sampled. Temperature dependence of the energy, entropy and other quantities of interest are easily calculable. In particular we report the groundstate results. Our data indicate that the large L increase of the ergodicity time is reduced to an approximately V³ power law. Altogether the results suggest that the multicanonical ensemble improves the situation of simulations for spin glasses and other systems which have to cope with similar problems of conflicting constraints.

The multicanonical ensemble is reviewed. Simulations of this ensemble promise improvements for a wide range of systems. Applications to first order phase transitions and spin glasses are summarized.

A variety of problems in statistical physics, such as Ising-like systems, can be modeled as integer programs. Physicists have relied mostly on Monte Carlo methods to find approximate solutions to these computationally difficult problems. In some cases, optimal solutions to relatively small problems have been found using standard optimization techniques, e.g., cutting plane and branch-and-bound algorithms. Motivated by the success of tabu search (TS) in finding optimal or near-optimal solutions to combinatorial optimization problems in a number of different settings, we study the application of this methodology to Ising-like systems. Particularly, we develop a TS method to find ground states of two-dimensional spin glasses. Our method performs a search at different levels of resolution in the spin lattice, and it is designed to obtain optimal or near-optimal solutions to problem instances with several different characteristics. Results are reported for computational experiments with up to 64×64 lattices.

We investigate the crossover properties of the frustrated percolation model on a two-dimensional square lattice, with asymmetric distribution of ferromagnetic and antiferromagnetic interactions. We determine the critical exponents ν, γ, and β of the percolation transition of the model, for various values of the density of antiferromagnetic interactions π in the range 0≤π≤0.5. Our data is consistent with the existence of a crossover from random percolation behavior for π=0, to frustrated percolation behavior, characterized by the critical exponents of the ferromagnetic 1/2-state Potts model, as soon as π>0.

We have developed a combinatoric matching method to find the exact groundstate energy for the 2-D Ising spin glass with the ± J distribution and equal numbers of positive and negative bonds. For the largest size (1800×1800 plaquettes of spins), we averaged results from 278 samples and for the smaller ones up to 374, 375 samples. We also studied the behavior of the distributions of computer time (CPU) and memory as functions of sample size. We present finite size scaling leading to a groundstate energy estimate of E_∞=-1.40193±2 for the infinite system. We found that the memory scales as the square of sample length and that for a given size, the CPU time appears to have a skewed and high-tailed distribution.

After a brief review of the Survey Propagation equations defined over single instances of complex glassy-like Hamiltonians, we discuss their application to the search for minimum energy configurations in difficult combinatorial optimization problems. Furthermore, we show that the application of arbitrary external field is helpful in the investigation of the topology of the configuration space.

We review the structure of the spin glass phase in the infinite-range Sherrington–Kirkpatrick model and the short-range Edwards–Anderson (EA) model. While the former is now believed to be understood, the nature of the latter remains unresolved. However, considerable insight can be gained through the use of the metastate, a mathematical construct that provides a probability measure on the space of all thermodynamic states. Using tools provided by the metastate construct, possibilities for the nature of the organization of pure states in short-range spin glasses can be considerably narrowed. We review the concept of the "ordinary" metastate, and also newer ideas on the excitation metastate, which has been recently used to prove existence of only a single pair of ground states in the EA Ising model in the half-plane. We close by presenting a new result, using metastate methods, on the number of mixed states allowed in the EA model.

In the last few years, the statistical mechanics of spin glasses has become one of the major frameworks for analyzing the macroscopical equilibrium properties of complex systems starting from the microscopical dynamics of their components. Recently, many advances in its rigorous formulation without the replica trick have been achieved, highlighting the importance of this field of research in our understanding of complex systems. In this framework we analyze the critical behavior of a Poissonian diluted network with random competitive interactions among gauge-invariant dichotomic variables pasted on the nodes (i.e., a suitable version of the Viana–Bray diluted spin glass). The model is described by an infinite series of order parameters (the multioverlaps) and has two degrees of freedom: the temperature (which can be thought of as the noise level) and the connectivity (the averaged number of links per node in the underlying network).

In this paper, we show that there are not several transition lines, one for every order parameter, as a naive approach would suggest but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique: we show the existence of a driving mechanism such that, as the first order parameter (the two-replica overlap) becomes different from zero due to a real second order phase transition, it enforces all the other multioverlaps toward positive values thanks to the strong correlations which develop among themselves and the two-replica overlap at the critical line. These correlations are ultimately related — within our framework — to the breaking of the gauge invariance of the Boltzmann state at the boundary of the ergodic region. A discussion on the structure of the free energy, fundamental macroscopical observable by which the whole thermodynamic can be achieved, is also presented.

The standard model defined in the Capital Adequacy Directive issued by the EEC in 1993 imposes nonlinear constraints on certain parts of the trading portfolios of financial institutions. It is shown that an institution that complies with the rules of the standard model but wants to optimize its portfolio according to some internal criteria, such as minimizing the variance or the VaR, faces a computational problem equivalent to finding the ground states of a long range spin glass. This problem is known to be NP-complete, and to have exponentially many solutions which are extremely sensitive to any changes of the input parameters.

With the aim of describing a general benchmark for several complex systems, we analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes. The model is described by an infinite series of order parameters (the multi-overlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multi-overlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line.

We discuss the competition between the Kondo effect, the spin glass state and a magnetic order observed in disordered Cerium systems. We present firstly the experimental situation of disordered alloys such as CeNi_1-xCu_x and then the different theoretical approaches based on the Kondo lattice model, with different descriptions of the intersite exchange interaction for the spin glass. After the gaussian approach of the Sherrington-Kirkpatrick model, we discuss the Mattis and the van Hemmen models. Then, we present simple cluster calculations in order to describe the percolative evolution of the clusters from the cluster spin glass to the inhomogeneous ferromagnetic order recently observed in CeNi_1-xCu_x disordered alloys and finally we discuss the effect of random and transverse magnetic field.

It has been recently demonstrated that optimal neural networks operate near the asymptotic edge of chaos for state-of-the-art feed-forward neural networks, where its generalization power is maximal due to the highest number of asymptotic metastable states. However, how to leverage this principle to improve the model training process remains open. Here, by mapping the model evolution during training to the phase diagram in the classic analytic result of Sherrington–Kirkpatrick model in spin glasses, we illustrate on a simple neural network model that one can provide principled training of the network without manually tuning the training hyper-parameters. In particular, we provide a semi-analytical method to set the optimal weight decay strength, such that the model will converge toward the edge of chaos during training. Consequently, such hyper-parameter setting leads the model to achieve the highest test accuracy. Another benefit for restricting the model at the edge of chaos is its robustness against the common practical problem of label noise, as we find that it automatically avoids fitting the shuffled labels in the training samples while maintaining good fitting to the correct labels, providing simple means of achieving good performance on noisy labels without any additional treatment.

After a brief review of the Survey Propagation equations defined over single instances of complex glassy-like Hamiltonians, we discuss their application to the search for minimum energy configurations in difficult combinatorial optimization problems. Furthermore, we show that the application of arbitrary external field is helpful in the investigation of the topology of the configuration space.

We discuss the competition between the Kondo effect, the spin glass state and a magnetic order observed in disordered Cerium systems. We present firstly the experimental situation of disordered alloys such as CeNi_{1 − x}Cu_x and then the different theoretical approaches based on the Kondo lattice model, with different descriptions of the intersite exchange interaction for the spin glass. After the gaussian approach of the Sherrington-Kirkpatrick model, we discuss the Mattis and the van Hemmen models. Then, we present simple cluster calculations in order to describe the percolative evolution of the clusters from the cluster spin glass to the inhomogeneous ferromagnetic order recently observed in CeNi_{1 − x}Cu_x disordered alloys and finally we discuss the effect of random and transverse magnetic field.

Some invariances under perturbations of the spin glass phase are introduced, their proofs outlined and their consequences illustrated as factorization rules for the overlap distribution. A comparison between the state of the art for mean field and finite dimensional models is shortly discussed.