Finite Connectivity Spin Glasses Research Articles

Ground-state configuration space heterogeneity of random finite-connectivity spin glassesand random constraint satisfaction problems

We demonstrate through two case studies, one on thep-spin interaction model and the other on the randomK-satisfiability problem, that a heterogeneity transition occurs to the ground-stateconfiguration space of a random finite-connectivity spin glass system at a certain criticalvalue of the constraint density. At the transition point, exponentially many configurationcommunities emerge from the ground-state configuration space, making the entropy densitys(q) of configuration-pairs a non-concave function of the configuration-pair overlapq. Each configuration community is a collection of relatively similar configurations and it formsa stable thermodynamic phase in the presence of a suitable external field. We calculates(q) by the replica-symmetric and the first-step replica-symmetry-broken cavity methods, andshow by simulations that the configuration space heterogeneity leads to dynamicalheterogeneity of particle diffusion processes because of the entropic trapping effect ofconfiguration communities. This work clarifies the fine structure of the ground-stateconfiguration space of random spin glass models, it also sheds light on the glassy behaviorof hard-sphere colloidal systems at relatively high particle volume fractions.

Solution space heterogeneity of the random K-satisfiability problem: Theory and simulations

The random K-satisfiability (K-SAT) problem is an important problem for studying typical-case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity spin-glasses. In this paper we review our recent efforts on the solution space fine structures of the random K-SAT problem. A heterogeneity transition is predicted to occur in the solution space as the constraint density α reaches a critical value αcm. This transition marks the emergency of exponentially many solution communities in the solution space. After the heterogeneity transition the solution space is still ergodic until α reaches a larger threshold value αd, at which the solution communities disconnect from each other to become different solution clusters (ergodicity-breaking). The existence of solution communities in the solution space is confirmed by numerical simulations of solution space random walking, and the effect of solution space heterogeneity on a stochastic local search algorithm SEQSAT, which performs a random walk of single-spin flips, is investigated. The relevance of this work to glassy dynamics studies is briefly mentioned.

Thermodynamic construction of a one-step replica-symmetry-breaking solution in finite-connectivity spin glasses

A one-step replica-symmetry-breaking solution for finite-connectivity spin-glass models with K body interaction is constructed at finite temperature using the replica method and thermodynamic constraints. In the absence of external fields, this construction provides a general extension of replica symmetric solution at finite replica number to one-step replica-symmetry-breaking solution. It is found that this result is formally equivalent to that of the one-step replica-symmetry-breaking cavity method. To confirm the validity of the obtained solution, Monte Carlo simulations are performed for K=2 and 3. The thermodynamic quantities of the Monte Carlo results extrapolated to a large-size limit are consistent with those estimated by our solution for K=2 at all simulated temperatures and for K=3 except near the transition temperature.

Boltzmann distribution of free energies in a finite-connectivity spin-glass system and the cavity approach

At sufficiently low temperatures, the configurational phase space of a large spin-glass system breaks into many separated domains, each of which is referred to as a macroscopic state. The system is able to visit all spin configurations of the same macroscopic state, while it can not spontaneously jump between two different macroscopic states. Ergodicity of the whole configurational phase space of the system, however, can be recovered if a temperature-annealing process is repeated an infinite number of times. In a heating-annealing cycle, the environmental temperature is first elevated to a high level and then decreased extremely slowly until a final low temperature T is reached. Different macroscopic states may be reached in different rounds of the annealing experiment; while the probability of finding the system in macroscopic state α decreases exponentially with the free energy F α (T) of this state. For finite-connectivity spin glass systems, we use this free energy Boltzmann distribution to formulate the cavity approach of Mézard and Parisi [Eur. Phys. J. B, 2001, 20: 217] in a slightly different form. For the ±J spin-glass model on a random regular graph of degree K = 6, the predictions of the present work agree with earlier simulational and theoretical results.

Finite-connectivity spin-glass phase diagrams and low-density parity check codes

We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check (LDPC) codes within the replica symmetric (RS) ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate , an RS critical transition point at while the critical RSB transition point is located at , to be compared with the corresponding Shannon bound . For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed.

Long-range frustration in finite connectivity spin glasses: a mean-field theory and its application to the random K-satisfiability problem

A mean-field theory of long-range frustration is constructed for spin glass systems with quenched randomness of vertex–vertex connections and of spin–spin coupling strengths. This theory is applied to a spin glass model of the random K-satisfiability (K-SAT) problem (K=2 or K=3). The satisfiability transition in a random 2-SAT formula occurs when the clauses-to-variables ratio α approaches αc(2)=1. However, long-range frustration among unfrozen variable nodes builds up only when α>αR(2)=4.4588. For the random 3-SAT problem, we find a long-range frustrated mean-field solution when α>αR(3)=4.1897. The long-range frustration order parameter R of this solution jumps from zero to a finite positive value at αR(3), while the energy density increases only gradually from zero as a function of α. The SAT–UNSAT transition point of this solution is lower than the value of αc(3)=4.267 obtained by the survey propagation algorithm. Two possible reasons for this discrepancy are suggested. The zero-temperature phase diagram of the ±J Viana–Bray model is also determined, which is identical to that of the random 2-SAT problem. The predicted phase transition between a non-frustrated and a long-range frustrated spin glass phase might also be observable in real materials at a finite temperature.

Finite connectivity attractor neural networks

We study a family of diluted attractor neural networks with a finite average number of (symmetric) connections per neuron. As in finite connectivity spin glasses, their equilibrium properties are described by order parameter functions, for which we derive an integral equation in replica symmetric (RS) approximation. A bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin-glass transition lines in the phase diagram. The line separating the retrieval phase from the spin-glass phase is calculated at zero temperature. All phase transitions are found to be continuous.

Optimization problems and replica symmetry breaking in finite connectivity spin glasses

A formalism capable of handling the first step of hierarchical replica symmetry breaking in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetisation distributions) inside the cluster of states. The approach is shown to coincide with the previous works in the replica symmetric case and in the two limit cases m=0,1 where m is Parisi's break-point. As an application to the study of optimization problems, the ground-state properties of the random 3-Satisfiability problem are investigated and we present a first RSB solution improving replica symmetric results.

The finite connectivity spin glass: investigation of replica symmetry breaking of the ground state

The free energy of the fixed finite connectivity spin glass is investigated in detail and the authors prove the equivalence of different forms which were obtained in the literature using completely different methods. Furthermore, the question of replica symmetry breaking is examined for low connectivity. They find by a variational calculation that even for a first-step breaking one can obtain a more optimal value for the ground state energy by going beyond the simple ansatz of Wong-Sherrington (1988) of totally uncorrelated groups of replicas.