Algorithms for three-dimensional rigidity analysis and a first-order percolation transition

An Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game

Phase stability and short range order in magnetic transition metal alloys: I. Electronic structure and interplay between chemical and magnetic cluster interactions

The frictional pebble game: An algorithm for rigidity percolation in saturated frictional assemblies

The conductivity of random resistor networks composed from percolating clusters of two-dimensional (2D) stick systems with anisotropic alignments is analyzed by using a finite-size scaling analysis for comparison to the conductivity of single-walled carbon-nanotube bundle film networks. For the conductivity analysis, we first calculate the critical properties of the percolation transition of 2D stick systems with anisotropic alignments. Even though the percolation transition stick density increases rapidly as the anisotropy is enhanced, the conductivity and the critical properties hardly vary. The resultant conductivity exponent of the stick networks at the percolation threshold is nearly the same as that of the lattice critical percolation clusters regardless of the anisotropy and the resistance ratio r = Rjct/RNT, where Rjct is the stick-to-stick junction resistance and RNT is the resistance of a stick.

Gaussian stochastic process (GaSP) has been widely used as a prior over functions due to its flexibility and tractability in modeling. However, the computational cost in evaluating the likelihood is , where n is the number of observed points in the process, as it requires to invert the covariance matrix. This bottleneck prevents GaSP being widely used in large-scale data. We propose a general class of nonseparable GaSP models for multiple functional observations with a fast and exact algorithm, in which the computation is linear (O(n)) and exact, requiring no approximation to compute the likelihood. We show that the commonly used linear regression and separable models are special cases of the proposed nonseparable GaSP model. Through the study of an epigenetic application, the proposed nonseparable GaSP model can accurately predict the genome-wide DNA methylation levels and compares favorably to alternative methods, such as linear regression, random forest, and localized Kriging method. The supplementary materials of this article are online and the algorithm for fast computation is implemented in the FastGaSP R package on CRAN. Supplemental materials for this article are available online.

Origami structures with a large number of excess folds are capable of storing distinguishable geometric states that are energetically equivalent. As the number of excess folds is reduced, the system has fewer equivalent states and can eventually become rigid. We quantify this transition from a floppy to a rigid state as a function of the presence of folding constraints in a classic origami tessellation, Miura-ori. We show that in a fully triangulated Miura-ori that is maximally floppy, adding constraints via the elimination of diagonal folds in the quads decreases the number of degrees of freedom in the system, first linearly and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via a redundancy in the assignment of constraints, and the degrees of freedom depend on constraint density in a scale-invariant manner. A percolation transition in the redundancy in the constraints as a function of constraint density suggests how excess folds in an origami structure can be used to store geometric information in a scale-invariant way.

Effect of selenium on the crystallization behaviour of AsTe glasses

In photosynthetic organisms the energy of the illuminating light is absorbed by the antenna complexes and transmitted by the excitons to the reaction centers (RCs). The energy of light is either absorbed by the RCs, leading to their "closing" or is emitted through fluorescence. The dynamics of the light absorption is described by a simple model developed for exciton migration that involves the exciton hopping probability and the exciton lifetime. During continuous illumination the fraction of closed RCs x continuously increases, and at a critical threshold x_{c}, a percolation transition takes place. Performing extensive Monte Carlo simulations, we study the properties of the transition in this correlated percolation model. We measure the spanning probability in the vicinity of x_{c}, as well as the fractal properties of the critical percolating cluster, both in the bulk and at the surface.

We investigate quantum phase transitions in the spin-1/2 Heisenberg antiferromagnet on square lattices with inhomogeneous bond dilution. It is shown that quantum fluctuations can be continuously tuned by inhomogeneous bond dilution, eventually leading to the destruction of long-range magnetic order on the percolating cluster. Two multicritical points are identified at which the magnetic transition separates from the percolation transition, introducing a novel quantum phase transition. Beyond these multicritical points a quantum-disordered phase appears, characterized by an infinite percolating cluster with short ranged antiferromagnetic order. In this phase, the low-temperature uniform susceptibility diverges algebraically with non-universal exponents. This is a signature that the novel quantum-disordered phase is a quantum Griffiths phase, as also directly confirmed by the statistical distribution of local gaps. This study thus presents evidence of a genuine quantum Griffiths phenomenon in a two-dimensional Heisenberg antiferromagnet.

Under external mechanical loading, glassy materials, ranging from soft matter systems to metallic alloys, often respond via formation of inhomogeneous flow patterns, during yielding. These inhomogeneities can be precursors to catastrophic failure, implying that a better understanding of their underlying mechanisms could lead to the design of smarter materials. Here, extensive molecular dynamics simulations are used to reveal the emergence of heterogeneous dynamics in a binary Lennard-Jones glass, subjected to a constant strain rate. At a critical strain, this system exhibits for all considered strain rates a transition towards the formation of a percolating cluster of mobile regions. We give evidence that this transition belongs to the universality class of directed percolation. Only at low shear rates, the percolating cluster subsequently evolves into a transient (but long-lived) shear band with a diffusive growth of its width. Finally, the steady state with a homogeneous flow pattern is reached. In the steady state, percolation transitions also do occur constantly, albeit over smaller strain intervals, to maintain the stationary plastic flow in the system.

The disorder-order transitions of layering and intralayer structural orders of three-dimensional Yukawa liquids, under the enhanced confinement effect with decreasing normal distance z to the confinement boundary, is investigated numerically. The liquid between the two flat boundaries is segmented into many slabs parallel to the boundary, with the same slab width as the layer width. In each slab, particle sites are binarized into sites with layering order (LOSs)/ layering disorder (LDSs) and with intralayer structural order (SOSs)/disorder (SDSs). It is found that with decreasing z, a small fraction of LOSs starts to heterogeneously emerge in the form of small clusters in the slab, followed by the emergence of the large percolating LOS clusters spanning over the system. The smooth rapid rise of the fraction of LOSs from small values followed by their gradual saturations, and the scaling behavior of multiscale LOS clustering, are similar to those of the nonequilibrium systems governed by the percolation theory. The disorder-order transition of intraslab structural ordering also exhibits a similar generic behavior as that of layering with the same transition slab number. The spatial fluctuations of local layering order and local intralayer structural order are uncorrelated in the bulk liquid and the outmost layer next to the boundary. Approaching the percolating transition slab, their correlation gradually increases to the maximum.

We disclose a precursor-driven order−order transition (OOT) between bcc and fcc lattices in a sphere-forming blend of a poly(ethylene oxide)-block-poly(1,4-butadiene) (PEO-b-PB) and a PB homopolymer (h-PB). The blend as cast from toluene exhibited a metastable trapped fcc (t-fcc) phase, which showed no sign of transformation into bcc phase on heating to 260 °C. Annealing the as-cast blend at −20 °C disturbed the long-range order of t-fcc due to crystallization of PEO within the spherical microdomains. A significant portion of the rather disorderly arranged crystalline domains relaxed to grains having the equilibrium bcc lattice, when the PEO crystals were melted at elevated temperatures. The bcc grains thus developed then acted as the precursor to induce a transformation of t-fcc phase to bcc phase on heating up to 205 °C. Since the t-fcc phase was not completely wiped out through this process, a bcc-to-fcc OOT assisted by the residual t-fcc precursor set in on heating from 205 to 260 °C where fcc became the stable packing symmetry. Subsequent cooling from 260 °C recovered the bcc phase and the blend cooled to 110 °C was essentially fcc free. Because of the absence of fcc grain, the thermodynamically prescribed bcc-to-fcc OOT could no longer take place on reheating. Our results revealed the necessity of bcc or fcc precursor for initiating the relevant OOT and hence implied a high activation barrier associated with the transition. The activation barrier was proposed to be entropic in origin stemming from the deformations of PB blocks and h-PB chains in the lattice transition proceeding through the Bain distortion mechanism.

2U: an exact interval propagation algorithm for polytrees with binary variables

We study the critical behavior of models for adsorbed layers in which particles reside on a square lattice and have infinite nearest-neighbor repulsions. Such particles are often described as ``hard squares.'' We consider both the equilibrium hard-square model and a nonequilibrium model. The latter involves dimer adsorption onto diagonally adjacent sites, and the desorption and possible hopping of adsorbed monomer particles (where neither adsorption nor hopping can create adjacent pairs of occupied sites). In the limit of high monomer mobility, one recovers the equilibrium model. Both models exhibit a continuous symmetry breaking transition in the Ising universality class, and also a percolation transition for $c(2\ifmmode\times\else\texttimes\fi{}2)$ clusters of particles connected with diagonal bonds. For the equilibrium model, extensive Monte Carlo simulations show that the two transitions coincide, supporting the claim of Hu and Mak. We also determine percolation exponents for $c(2\ifmmode\times\else\texttimes\fi{}2)$ clusters and vacancy clusters, and consider a correlated site-bond percolation problem which elucidates conditions for coincidence of symmetry-breaking and percolation. In contrast, for the nonequilibrium model with immobile adsorbed monomers, there is a gap between the symmetry-breaking and percolation transitions, and the random percolation universality class applies. Finally, we examine the crossover behavior with increasing mobility of adsorbed monomers.

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.