Continuum percolation of overlapping disks with a distribution of radii having a power-law tail

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

Percolation of binary disk systems: Modeling and theory

The dispersion and connectivity of particles with a high degree of polydispersity is relevant to problems involving composite material properties and reaction decomposition prediction and has been the subject of much study in the literature. This work utilizes Monte Carlo models to predict percolation thresholds for a two-dimensional systems containing disks of two different radii. Monte Carlo simulations and spanning probability are used to extend prior models into regions of higher polydispersity than those previously considered. A correlation to predict the percolation threshold for binary disk systems is proposed based on the extended dataset presented in this work and compared to previously published correlations. A set of boundary conditions necessary for a good fit is presented, and a condition for maximizing percolation threshold for binary disk systems is suggested.

Multifluorophore localization as a percolation problem: limits to density and precision

We show that the maximum desirable density of activated fluorophores in a superresolution experiment can be determined by treating the overlapping point spread functions as a problem in percolation theory. We derive a bound on the density of activated fluorophores, taking into account the desired localization accuracy and precision, as well as the number of photons emitted. Our bound on density is close to that reported in experimental work, suggesting that further increases in the density of imaged fluorophores will come at the expense of localization accuracy and precision.

Majority-vote model on spatially embedded networks: Crossover from mean-field to Ising universality classes

We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which directed long-range connections are randomly added according to the probability P_{ij}∼r^{-α}, where r_{ij} is the Manhattan distance between nodes i and j, and the exponent α is a controlling parameter [J. M. Kleinberg, Nature (London) 406, 845 (2000)NATUAS0028-083610.1038/35022643]. Our results show that the collective behavior of this system exhibits a continuous order-disorder phase transition at a critical parameter, which is a decreasing function of the exponent α. Precisely, considering the scaling functions and the critical exponents calculated, we conclude that the system undergoes a crossover among distinct universality classes. For α≤3 the critical behavior is described by mean-field exponents, while for α≥4 it belongs to the Ising universality class. Finally, in the region where the crossover occurs, 3<α<4, the critical exponents are dependent on α.

Continuum percolation of polydisperse hyperspheres in infinite dimensions

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are treelike for d→∞ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density η(c)→2(-d), independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality z(c) is smaller than unity, while z(c)→1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit, the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a log-normal distribution is no longer universal, and that it can be smaller than 2(-d) for d→∞.