Test of scaling exponents for percolation-cluster perimeters

Critical Percolation in the Ordering Kinetics of Twisted Nematic Phases

I report on the experimental confirmation that critical percolation statistics underlie the ordering kinetics of twisted nematic phases in the Allen-Cahn universality class. Soon after the ordering starts from a homogeneous disordered phase and proceeds toward a broken Z_{2}-symmetry phase, the system seems to be attracted to the random percolation fixed point at a special timescale t_{p}. At this time, exact formulas for crossing probabilities in percolation theory agree with the corresponding probabilities in the experimental data. The ensuing evolution for the number density of hull-enclosed areas is described by an exact expression derived from a percolation model endowed with curvature-driven interface motion. Scaling relation for hull-enclosed areas versus perimeters reveals that the fractal percolation geometry is progressively morphed into a regular geometry up to the order of the classical coarsening length. In view of its universality and experimental possibilities, the study opens a path for exploring percolation keystones in the realm of nonequilibrium, phase-ordering systems.

Global patterns of tropical forest fragmentation

Remote sensing enables the quantification of tropical deforestation with high spatial resolution. This in-depth mapping has led to substantial advances in the analysis of continent-wide fragmentation of tropical forests. Here we identified approximately 130 million forest fragments in three continents that show surprisingly similar power-law size and perimeter distributions as well as fractal dimensions. Power-law distributions have been observed in many natural phenomena such as wildfires, landslides and earthquakes. The principles of percolation theory provide one explanation for the observed patterns, and suggest that forest fragmentation is close to the critical point of percolation; simulation modelling also supports this hypothesis. The observed patterns emerge not only from random deforestation, which can be described by percolation theory, but also from a wide range of deforestation and forest-recovery regimes. Our models predict that additional forest loss will result in a large increase in the total number of forest fragments-at maximum by a factor of 33 over 50 years-as well as a decrease in their size, and that these consequences could be partly mitigated by reforestation and forest protection.

Stochastic Loewner evolution relates anomalous diffusion and anisotropic percolation

We disclose the origin of anisotropic percolation perimeters in terms of the stochastic Loewner evolution (SLE) process. Precisely, our results from extensive numerical simulations indicate that the perimeters of multilayered and directed percolation clusters at criticality are the scaling limits of the Loewner evolution of an anomalous Brownian motion, being superdiffusive and subdiffusive, respectively. The connection between anomalous diffusion and fractal anisotropy is further tested by using long-range power-law correlated time series (fractional Brownian motion) as the driving functions in the evolution process. The fact that the resulting traces are distinctively anisotropic corroborates our hypothesis. Under the conceptual framework of SLE, our study therefore reveals different perspectives for mathematical and physical interpretations of non-Markovian processes in terms of anisotropic paths at criticality and vice versa.

Lattice model of correlated forces in granular solids near jamming

We have devised a lattice model to study force correlations in jamming granular solids in d=2 dimensions. We perform biased Monte Carlo simulations, favoring configurations with more bonds that bear no force, to "starve" the network of bonds and thereby control the distance from the isostatic point J. Increasingly long-ranged correlations are visible as point J is approached, not in the structure of the network of force-bearing bonds but in the spatial extent of perturbations of the force magnitudes consistent with a given starved network. The correlation length so defined diverges as the isostatic point is approached as a power law with an exponent of about ξ~δZ(-5). This divergence is much stronger than for the length scale of "soft modes" observed in jammed systems approaching point J from above.

Domain growth morphology in curvature-driven two-dimensional coarsening

We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initial condition, by the scaling function n_{h}(A,t)=2c_{h}(A+lambda_{h}t);{2} , where c_{h}=18pi sqrt[3] approximately 0.023 is a universal constant and lambda_{h} is a material parameter. For a critical initial condition, the same form is obtained, with the same lambda_{h} but with c_{h} replaced by c_{h}2 . For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_{d}(A,t)=2c_{d}(lambda_{d}t);{tau'-2}(A+lambda_{d}t);{tau'} , where c_{d} and lambda_{d} are numerically very close to c_{h} and lambda_{h} , respectively, and tau'=18791 approximately 2.055 . For critical initial conditions, one replaces c_{d} by c_{d}2 and the exponent is tau=379187 approximately 2.027 . These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.

Investigation of nodal domains in the chaotic microwave ray-splitting rough billiard

We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. For this aim the wave functions of the billiard were measured up to the level number . We show that in the regime of Shnirelman ergodicity wave functions of the chaotic half-circular microwave ray-splitting rough billiard are extended over the whole energy surface and the amplitude distributions are Gaussian. For such ergodic wave functions, the dependence of the number of nodal domains on the level number was found. We show that in the limit the least squares fit of the experimental data yields , which is close to the theoretical prediction . We demonstrate that for higher level numbers the variance of the mean number of nodal domains is scattered around the theoretical limit . We also found that the distribution of the areas of nodal domains has power behavior , where the scaling exponent is equal to . This result is in good agreement with the prediction of percolation theory.

Experimental investigation of nodal domains in the chaotic microwave rough billiard

We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave functions PsiN of the rough billiard were measured up to the level number N=435 . In this way the dependence of the number of nodal domains [symbol: see text]N on the level number N was found. We show that in the limit N-->infinity a least squares fit of the experimental data reveals the asymptotic number of nodal domains [symbol: see text]N/N approximately equal to 0.058+/-0.006 that is close to the theoretical prediction [symbol: see text]N/N approximately equal to 0.062 . We also found that the distributions of the areas s of nodal domains and their perimeters l have power behaviors ns is proportional to s(-tau) and nl is proportional to l(-tau'), where scaling exponents are equal to tau=1.99+/-0.14 and tau'=2.13+/-0.23 , respectively. These results are in a good agreement with the predictions of percolation theory. Finally, we demonstrate that for higher level numbers N approximately equal to 220-435 the signed area distribution oscillates around the theoretical limit SigmaA approximately 0.0386 N(-1) .

Surfaces of percolation systems in lattice problems

The internal and external surface area of a percolation cluster along with a full surface area of whole percolation system are investigated both analytically and numerically. Numerical simulation is performed by a Monte Carlo method for site and bond problems on square and simple cubic lattices. It is shown that both the external and full surface areas of a percolation cluster as well as the full surface area of the whole percolation system have maxima for a certain share of occupied sites (for the site problem) or permeable bonds (for a bond problem). On the basis of a probabilistic approach, analytical expressions are obtained which relate the surface area of percolation cluster to its density. The last value has been studied in more details at present that allows to analyze the behavior of the above-mentioned surface for various lattices. Two particular technological processes are discussed where the surface area of a percolation cluster plays an important part: generation of electric current in a fuel cell and self-propagating high-temperature synthesis in heterogeneous condensed systems.

Roughness exponent in two-dimensional percolation, Potts model, and clock model

We present a numerical study of the self-affine profiles obtained from configurations of the q-state Potts (with q=2,3, and 7) and p=10 clock models as well as from the occupation states for site percolation on the square lattice. The first and second order static phase transitions of the Potts model are located by a sharp change in the value of the roughness exponent alpha characterizing those profiles. The low temperature phase of the Potts model corresponds to flat (alpha approximately 1) profiles, whereas its high temperature phase is associated with rough (alpha approximately 0.5) ones. For the p=10 clock model, in addition to the flat (ferromagnetic) and rough (paramagnetic) profiles, an intermediate rough (0.5<alpha<1) phase-associated with a soft spin-wave one-is observed. Our results for the transition temperatures in the Potts and clock models are in agreement with the static values, showing that this approach is able to detect the phase transitions in these models directly from the spin configurations, without any reference to thermodynamical potentials, order parameters, or response functions. Finally, we show that the roughness exponent alpha is insensitive to geometric critical phenomena.

Percolation, Bose-Einstein condensation, and string proliferation

The close analogy between cluster percolation and string proliferation in the context of critical phenomena is studied. Like clusters in percolation theory, closed strings, which can be either finite-temperature worldlines or topological line defects, are described by a distribution parametrized by only two exponents. On approaching the critical point, the string tension vanishes and the loops proliferate, thereby signalling the onset of Bose-Einstein condensation (in the case of worldlines) or the disordering of the ordered state (in the case of vortices). The ideal Bose gas with modified energy spectrum is used as a stepping stone to derive general expressions for the critical exponents in terms of the two exponents parametrizing the loop distribution near criticality.