Numerical Results for Ground States of Mean-Field Spin Glasses at low Connectivities

An extensive list of results for the ground state properties of spin glasses on random graphs is presented. These results provide a timely benchmark for currently developing theoretical techniques based on replica symmetry breaking that are being tested on mean-field models at low connectivity. Comparison with existing replica results for such models verifies the strength of those techniques. Yet, we find that spin glasses on fixed-connectivity graphs (Bethe lattices) exhibit a richer phenomenology than has been anticipated by theory. Our data prove to be sufficiently accurate to speculate about some exact results.Comment: 4 pages, RevTex4, 5 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher

Extremal Optimization at the Phase Transition of the 3-Coloring Problem

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

Extremal Optimization for Graph Partitioning

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the NP-hard graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of runtime and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. Our numerical results demonstrate that on random graphs, extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over runtime roughly as a power law t^(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal, with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher

Jamming Model for the Extremal Optimization Heuristic

Extremal Optimization, a recently introduced meta-heuristic for hard optimization problems, is analyzed on a simple model of jamming. The model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed-valence graph. The numerical results for the spin system exhibit the same phenomena found in all earlier studies of extremal optimization, and our analytical results for the model reproduce many of these features.Comment: 9 pages, RevTex4, 7 ps-figures included, as to appear in J. Phys. A, related papers available at http://www.physics.emory.edu/faculty/boettcher

Photoelectrochemical water splitting: silicon photocathodes for hydrogen evolution

The development of low cost, scalable, renewable energy technologies is one of today's most pressing scientific challenges. We report on progress towards the development of a photoelectrochemical water-splitting system that will use sunlight and water as the inputs to produce renewable hydrogen with oxygen as a by-product. This system is based on the design principle of incorporating two separate, photosensitive inorganic semiconductor/liquid junctions to collectively generate the 1.7-1.9 V at open circuit needed to support both the oxidation of H_2O (or OH^-) and the reduction of H^+ (or H_2O). Si microwire arrays are a promising photocathode material because the high aspect-ratio electrode architecture allows for the use of low cost, earth-abundant materials without sacrificing energy-conversion efficiency, due to the orthogonalization of light absorption and charge-carrier collection. Additionally, the high surfacearea design of the rod-based semiconductor array inherently lowers the flux of charge carriers over the rod array surface relative to the projected geometric surface of the photoelectrode, thus lowering the photocurrent density at the solid/liquid junction and thereby relaxing the demands on the activity (and cost) of any electrocatalysts. Arrays of Si microwires grown using the Vapor Liquid Solid (VLS) mechanism have been shown to have desirable electronic light absorption properties. We have demonstrated that these arrays can be coated with earth-abundant metallic catalysts and used for photoelectrochemical production of hydrogen. This development is a step towards the demonstration of a complete artificial photosynthetic system, composed of only inexpensive, earth-abundant materials, that is simultaneously efficient, durable, and scalable

Broadband Records of Earthquakes in Deep Gold Mines and a Comparison with Results from SAFOD, California

For one week during September 2007, we deployed a temporary network of field recorders and accelerometers at four sites within two deep, seismically active mines. The ground-motion data, recorded at 200 samples/sec, are well suited to determining source and ground-motion parameters for the mining-induced earthquakes within and adjacent to our network. Four earthquakes with magnitudes close to 2 were recorded with high signal/noise at all four sites. Analysis of seismic moments and peak velocities, in conjunction with the results of laboratory stick-slip friction experiments, were used to estimate source processes that are key to understanding source physics and to assessing underground seismic hazard. The maximum displacements on the rupture surfaces can be estimated from the parameter Rv, where v is the peak ground velocity at a given recording site, and R is the hypocentral distance. For each earthquake, the maximum slip and seismic moment can be combined with results from laboratory friction experiments to estimate the maximum slip rate within the rupture zone. Analysis of the four M 2 earthquakes recorded during our deployment and one of special interest recorded by the in-mine seismic network in 2004 revealed maximum slips ranging from 4 to 27 mm and maximum slip rates from 1.1 to 6:3 m=sec. Applying the same analyses to an M 2.1 earthquake within a cluster of repeating earthquakes near the San Andreas Fault Observatory at Depth site, California, yielded similar results for maximum slip and slip rate, 14 mm and 4:0 m=sec

Quantum Search Algorithms on Hierarchical Networks

The "abstract search algorithm" is a well known quantum method to find a marked vertex in a graph. It has been applied with success to searching algorithms for the hypercube and the two-dimensional grid. In this work we provide an example for which that method fails to provide the best algorithm in terms of time complexity. We analyze search algorithms in degree-3 hierarchical networks using quantum walks driven by non-groverian coins. Our conclusions are based on numerical simulations, but the hierarchical structures of the graphs seems to allow analytical results.Comment: IEEE Information Theory Workshop 201

Continuous extremal optimization for Lennard-Jones Clusters

In this paper, we explore a general-purpose heuristic algorithm for finding high-quality solutions to continuous optimization problems. The method, called continuous extremal optimization(CEO), can be considered as an extension of extremal optimization(EO) and is consisted of two components, one is with responsibility for global searching and the other is with responsibility for local searching. With only one adjustable parameter, the CEO's performance proves competitive with more elaborate stochastic optimization procedures. We demonstrate it on a well known continuous optimization problem: the Lennerd-Jones clusters optimization problem.Comment: 5 pages and 3 figure