Disorder chaos in spin glasses

We investigate numerically disorder chaos in spin glasses, i.e. the sensitivity of the ground state to small changes of the random couplings. Our study focuses on the Edwards-Anderson model in d=1,2,3 and in mean-field. We find that in all cases, simple scaling laws, involving the size of the system and the strength of the perturbation, are obeyed. We characterize in detail the distribution of overlap between ground states and the geometrical properties of flipped spin clusters in both the weak and strong chaos regime. The possible relevance of these results to temperature chaos is discussed.Comment: 7 pages, 8 figures, replaced with accepted versio

Distribution of partition function zeros of the ±J\pm J model on the Bethe lattice

The distribution of partition function zeros is studied for the $\pm J$ model of spin glasses on the Bethe lattice. We find a relation between the distribution of complex cavity fields and the density of zeros, which enables us to obtain the density of zeros for the infinite system size by using the cavity method. The phase boundaries thus derived from the location of the zeros are consistent with the results of direct analytical calculations. This is the first example in which the spin glass transition is related to the distribution of zeros directly in the thermodynamical limit. We clarify how the spin glass transition is characterized by the zeros of the partition function. It is also shown that in the spin glass phase a continuous distribution of singularities touches the axes of real field and temperature.Comment: 23 pages, 12 figure

Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems

We review the understanding of the random constraint satisfaction problems, focusing on the q-coloring of large random graphs, that has been achieved using the cavity method of the physicists. We also discuss the properties of the phase diagram in temperature, the connections with the glass transition phenomenology in physics, and the related algorithmic issues.Comment: 10 pages, Proceedings of the International Workshop on Statistical-Mechanical Informatics 2007, Kyoto (Japan) September 16-19, 200

DICE: Exploiting All Bivariate Dependencies in Binary and Multary Search Spaces

Although some of the earliest Estimation of Distribution Algorithms (EDAs) utilized bivariate marginal distribution models, up to now, all discrete bivariate EDAs had one serious limitation: they were constrained to exploiting only a limited O(d) subset out of all possible O(d2) bivariate dependencies. As a first we present a family of discrete bivariate EDAs that can learn and exploit all O(d2) dependencies between variables, and yet have the same run-time complexity as their more limited counterparts. This family of algorithms, which we label DICE (DIscrete Correlated Estimation of distribution algorithms), is rigorously based on sound statistical principles, and particularly on a modelling technique from statistical physics: dichotomised multivariate Gaussian distributions. Initially (Lane et al. in European Conference on the Applications of Evolutionary Computation, Springer, 1999), DICE was trialled on a suite of combinatorial optimization problems over binary search spaces. Our proposed dichotomised Gaussian (DG) model in DICE significantly outperformed existing discrete bivariate EDAs; crucially, the performance gap increasingly widened as dimensionality of the problems increased. In this comprehensive treatment, we generalise DICE by successfully extending it to multary search spaces that also allow for categorical variables. Because correlation is not wholly meaningful for categorical variables, interactions between such variables cannot be fully modelled by correlation-based approaches such as in the original formulation of DICE. Therefore, here we extend our original DG model to deal with such situations. We test DICE on a challenging test suite of combinatorial optimization problems, which are defined mostly on multary search spaces. While the two versions of DICE outperform each other on different problem instances, they both outperform all the state-of-the-art bivariate EDAs on almost all of the problem instances. This further illustrates that these innovative DICE methods constitute a significant step change in the domain of discrete bivariate EDAs

Potts glass on random graphs

We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean-field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q≥ 4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive- and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular, we argue that in the colorable region the one-step replica symmetry-breaking solution is stable towards more steps of replica symmetry breaking