arXiv subjects

Chuang Wang

Publications and source records attributed to Chuang Wang.

60 records · Page 4Linked to original sources

arXiv2014

Topological invariant tensor renormalization group method for spin glasses

Tensor renormalization group method (TRG) is a real space renormalization group approach. It has been successfully applied to both classical and quantum systems. In this paper, we study a disordered and frustrated system, the two-dimensional Edward-Anderson model, by a new topological invariant TRG scheme. We propose an approach to calculate the local magnetizations and nearest pair correlations simultaneously. The Nishimori multi-critical point predicted by the topological invariant TRG agrees well with the recent Monte-Carlo results. The TRG schemes outperform the mean field methods on the calculation of the partition function. We notice that it maybe obtain a negative partition function at sufficiently low temperatures. However, the negative contribution can be neglected if the systems is large enough. This topological invariant TRG can also be used to study three-dimensional spin glass systems.

cond-mat.dis-nn↗

arXiv2014

Cavity Method: Message Passing from a Physics Perspective

In this three-sections lecture cavity method is introduced as heuristic framework from a Physics perspective to solve probabilistic graphical models and it is presented both at the replica symmetric (RS) and 1-step replica symmetry breaking (1RSB) level. This technique has been applied with success on a wide range of models and problems such as spin glasses, random constrain satisfaction problems (rCSP), error correcting codes etc. Firstly, the RS cavity solution for Sherrington-Kirkpatrick model---a fully connected spin glass model---is derived and its equivalence to the RS solution obtained using replicas is discussed. Then, the general cavity method for diluted graphs is illustrated both at RS and 1RSB level. The latter was a significant breakthrough in the last decade and has direct applications to rCSP. Finally, as example of an actual problem, K-SAT is investigated using belief and survey propagation.

cond-mat.dis-nn↗

arXiv2013

Simplifying Generalized Belief Propagation on Redundant Region Graphs

The cluster variation method has been developed into a general theoretical framework for treating short-range correlations in many-body systems after it was first proposed by Kikuchi in 1951. On the numerical side, a message-passing approach called generalized belief propagation (GBP) was proposed by Yedidia, Freeman and Weiss about a decade ago as a way of computing the minimal value of the cluster variational free energy and the marginal distributions of clusters of variables. However the GBP equations are often redundant, and it is quite a non-trivial task to make the GBP iteration converges to a fixed point. These drawbacks hinder the application of the GBP approach to finite-dimensional frustrated and disordered systems. In this work we report an alternative and simple derivation of the GBP equations starting from the partition function expression. Based on this derivation we propose a natural and systematic way of removing the redundance of the GBP equations. We apply the simplified generalized belief propagation (SGBP) equations to the two-dimensional and the three-dimensional ferromagnetic Ising model and Edwards-Anderson spin glass model. The numerical results confirm that the SGBP message-passing approach is able to achieve satisfactory performance on these model systems. We also suggest that a subset of the SGBP equations can be neglected in the numerical iteration process without affecting the final results.

cs.IT↗

arXiv2012

Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.

cond-mat.dis-nn↗

arXiv2011

Partition Function Expansion on Region-Graphs and Message-Passing Equations

Disordered and frustrated graphical systems are ubiquitous in physics, biology, and information science. For models on complete graphs or random graphs, deep understanding has been achieved through the mean-field replica and cavity methods. But finite-dimensional `real' systems persist to be very challenging because of the abundance of short loops and strong local correlations. A statistical mechanics theory is constructed in this paper for finite-dimensional models based on the mathematical framework of partition function expansion and the concept of region-graphs. Rigorous expressions for the free energy and grand free energy are derived. Message-passing equations on the region-graph, such as belief-propagation and survey-propagation, are also derived rigorously.

cond-mat.stat-mech↗

arXiv2010

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

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

cond-mat.dis-nn↗