Minimal spanning trees at the percolation threshold: a numerical calculation.

The fractal dimension of minimal spanning trees on percolation clusters is estimated for dimensions d up to d=5. A robust analysis technique is developed for correlated data, as seen in such trees. This should be a robust method suitable for analyzing a wide array of randomly generated fractal structures. The trees analyzed using these techniques are built using a combination of Prim's and Kruskal's algorithms for finding minimal spanning trees. This combination reduces memory usage and allows for simulation of larger systems than would otherwise be possible. The path length fractal dimension d_{s} of MSTs on critical percolation clusters is found to be compatible with the predictions of the perturbation expansion developed by T. S. Jackson and N. Read [Phys. Rev. E 81, 021131 (2010)].

Numerically exact correlations and sampling in the two-dimensional Ising spin glass.

A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.

Zero- and low-temperature behavior of the two-dimensional ±J Ising spin glass.

Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.

Exact algorithm for sampling the two-dimensional Ising spin glass.

A sampling algorithm is presented that generates spin-glass configurations of the two-dimensional Edwards-Anderson Ising spin glass at finite temperature with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proceedings of the Eighth Symposium on Discrete Algorithms (SIAM, Philadelphia, 1997), p 258] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin-glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination rather than Gaussian elimination for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n(3/2)); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128(2), with fixed and periodic boundary conditions.

Statistics of static avalanches in a random pinning landscape.

We study the minimum-energy configuration of a d -dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or "static avalanches." We obtain analytically the distribution of avalanche sizes and its cumulants within an =4-d expansion from a tree and one-loop resummation using functional renormalization. This is compared with exact numerical minimizations of interface energies for random-field disorder in d=2,3 . Connections to dynamic avalanches are mentioned.

Irrational mode locking in quasiperiodic systems.

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.

Measuring functional renormalization group fixed-point functions for pinned manifolds.

Exact numerical minimization of interface energies is used to test the functional renormalization group analysis for interfaces pinned by quenched disorder. The fixed-point function R(u) (the correlator of the coarse-grained disorder) is computed. In dimensions D=d+1, a linear cusp in R''(u) is confirmed for random bond (d=1, 2, 3), random field (d=0, 2, 3), and periodic (d=2, 3) disorders. The functional shocks that lead to this cusp are seen. Small, but significant, deviations from the 1-loop calculation are compared to 2-loop corrections and chaos is measured.

Improved extremal optimization for the Ising spin glass.

A version of the extremal optimization (EO) algorithm introduced by Boettcher and Percus is tested on two- and three-dimensional spin glasses with Gaussian disorder. EO preferentially flips spins that are locally "unfit"; the variant introduced here reduces the probability of flipping previously selected spins. Relative to EO, this adaptive algorithm finds exact ground states with a speedup of order 10(4) (10(2) ) for 16(2) - (8(3) -) spin samples. This speedup increases rapidly with system size, making this heuristic a useful tool in the study of materials with quenched disorder.

Driven depinning of strongly disordered media and anisotropic mean-field limits.

Extended systems driven through strong disorder are modeled generically using coarse-grained degrees of freedom that interact elastically in the directions parallel to the drive and slip along at least one of the directions transverse to the motion. In the limit of infinite-range elastic and viscous coupling this model has a tricritical point separating a region where the depinning is continuous, in the universality class of elastic depinning, from a region where depinning is hysteretic. Many of the collective transport models discussed in the literature are special cases of the generic model.

Critical slowing down in polynomial time algorithms.

Combinatorial optimization algorithms that compute exact ground states for disordered magnets are seen to exhibit critical slowing down at zero temperature phase transitions. Using the physical features of the models, such as vanishing stiffness on one side of the transition and the ground state degeneracy, the number of operations needed in the push-relabel algorithm for the random field Ising model and in the algorithm for the 2D spin glass is estimated. These results strengthen the connections between algorithms and the physical picture and may be used to improve the speed of computations.