arXiv subjects

Robert M. Ziff

Publications and source records attributed to Robert M. Ziff.

At least 91 records · Page 5Linked to original sources

arXiv2007

The density of critical percolation clusters touching the boundaries of strips and squares

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips, and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory, and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to $(\sin πy)^{-5/48}\{[\cos(πy/2)]^{1/3} +[\sin (πy/2)]^{1/3}-1\}$. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides, and compare with theory for the density along an edge.

cond-mat.dis-nn↗

arXiv2006

Exact bond percolation thresholds in two dimensions

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.

cond-mat.dis-nn↗

arXiv2006

The Universal Amplitude Ratio $Γ^{-}/Γ^{+}$ for Two-Dimensional Percolation

The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value $Γ^{-}/Γ^{+} = 162.5 \pm 2$, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.

cond-mat.stat-mech↗

arXiv2006

General flux to a trap in one and three dimensions

The problem of the flux to a spherical trap in one and three dimensions, for diffusing particles undergoing discrete-time jumps with a given radial probability distribution, is solved in general, verifying the Smoluchowski-like solution in which the effective trap radius is reduced by an amount proportional to the jump length. This reduction in the effective trap radius corresponds to the Milne extrapolation length.

cond-mat.stat-mech↗

arXiv2006

Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagomé and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagomé (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.

cond-mat.dis-nn↗

arXiv2005

Generalized cell/dual-cell transformation for percolation, and new exact thresholds

Suggested by Scullard's recent star-triangle relation for bond correlated systems, we propose a general "cell/dual-cell" transformation, which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated. We directly verify Scullard's new site percolation thresholds, and derive the bond thresholds for his "martini" lattice (pc = 1/sqrt(2)) and the "A" lattice (pc = 0.625457..., solution to p^5 - 4 p^4 + 3 p^3 + 2 p^2 - 1 = 0). We also present a precise Monte-Carlo test the site threshold for the "A" lattice.

cond-mat.dis-nn↗

arXiv2005

Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap

Two random-walk related problems which have been studied independently in the past, the expected maximum of a random walker in one dimension and the flux to a spherical trap of particles undergoing discrete jumps in three dimensions, are shown to be closely related to each other and are studied using a unified approach as a solution to a Wiener-Hopf problem. For the flux problem, this work shows that a constant c = 0.29795219 which appeared in the context of the boundary extrapolation length, and was previously found only numerically, can be derived explicitly. The same constant enters in higher-order corrections to the expected-maximum asymptotics. As a byproduct, we also prove a new universal result in the context of the flux problem which is an analogue of the Sparre Andersen theorem proved in the context of the random walker's maximum.

cond-mat.stat-mech↗

arXiv2005

Enclosed area distribution in percolation

The number of two-dimensional percolation clusters whose external hulls enclose an area greater than A, in a system of area Omega, behaves at the critical point as C Ω/A for large A, where C = 1/(8 pi sqrt(3)). Here we show that away from the critical point this factor is multiplied by a scaling function that is asymptotically proportional to a simple exponential exp(-A/A*) where A* scales as |p - pc|^(-2 nu). The fit is better than for Kunz and Souillard sub-critical scaling, which would predict the asymptotic behavior exp(-(A/A*)^(2/D) where D = 91/48 is the fractal dimension.

cond-mat.dis-nn↗

arXiv2005

A stochastic model for wound healing

We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p near 1. In both one and two dimensions for small p, front propagation for this model should approach that of the Fisher-Kolmogorov equation. However, as in other cases, this discrete model approaches Fisher-Kolmogorov behavior slowly.

q-bio.CB↗

arXiv2005

A simple algorithm to test for linking to Wilson loops in percolation

A simple burning or epidemic type of algorithm is developed in order to test whether any loops in percolation clusters link a fixed reference loop, a problem considered recently by Gliozzi, Lottini, Panero, and Rago in the context of gauge theory. We test our algorithm at criticality in both 2d, where the behavior agrees with a theoretical prediction, and in 3d.

cond-mat.dis-nn↗

arXiv2004

Response of a catalytic reaction to periodic variation of the CO pressure: Increased CO_2 production and dynamic phase transition

We present a kinetic Monte Carlo study of the dynamical response of a Ziff-Gulari-Barshad model for CO oxidation with CO desorption to periodic variation of the CO presure. We use a square-wave periodic pressure variation with parameters that can be tuned to enhance the catalytic activity. We produce evidence that, below a critical value of the desorption rate, the driven system undergoes a dynamic phase transition between a CO_2 productive phase and a nonproductive one at a critical value of the period of the pressure oscillation. At the dynamic phase transition the period-averged CO_2 production rate is significantly increased and can be used as a dynamic order parameter. We perform a finite-size scaling analysis that indicates the existence of power-law singularities for the order parameter and its fluctuations, yielding estimated critical exponent ratios $β/ν\approx 0.12$ and $γ/ν\approx 1.77$. These exponent ratios, together with theoretical symmetry arguments and numerical data for the fourth-order cumulant associated with the transition, give reasonable support for the hypothesis that the observed nonequilibrium dynamic phase transition is in the same universality class as the two-dimensional equilibrium Ising model.

cond-mat.stat-mech↗

arXiv2001

Excess number of percolation clusters on the surface of a sphere

Monte Carlo simulations were performed in order to determine the excess number of clusters b and the average density of clusters n_c for the two-dimensional "Swiss cheese" continuum percolation model on a planar L x L system and on the surface of a sphere. The excess number of clusters for the L x L system was confirmed to be a universal quantity with a value b = 0.8841 as previously predicted and verified only for lattice percolation. The excess number of clusters on the surface of a sphere was found to have the value b = 1.215(1) for discs with the same coverage as the flat critical system. Finally, the average critical density of clusters was calculated for continuum systems n_c = 0.0408(1).

cond-mat.dis-nn↗

arXiv2001

Percolation Threshold, Fisher Exponent, and Shortest Path Exponent for 4 and 5 Dimensions

We develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created. The memory required to grow a cluster of mass s is of the order of $s^θ$ bytes where $θ$ ranges from 0.4 for 2-dimensional lattices to 0.5 for 6- (or higher)-dimensional lattices. We use this method to estimate $d_{\scriptsize min}$, the exponent relating the minimum path $\ell$ to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find $d_{\scriptsize min}=1.607\pm 0.005$ (4D) and $d_{\scriptsize min}=1.812\pm 0.006$ (5D). In order to determine $d_{\scriptsize min}$ to high precision, and without bias, it was necessary to first find precise values for the percolation threshold, $p_c$: $p_c=0.196889\pm 0.000003$ (4D) and $p_c=0.14081\pm 0.00001$ (5D) for site and $p_c=0.160130\pm 0.000003$ (4D) and $p_c=0.118174\pm 0.000004$ (5D) for bond percolation. We also calculate the Fisher exponent, $τ$, determined in the course of calculating the values of $p_c$: $τ=2.313\pm 0.003$ (4D) and $τ=2.412\pm 0.004$ (5D).

cond-mat.stat-mech↗

arXiv1999

Re-examination of seven-dimensional site percolation thresholds

Monte Carlo simulations alone could not clarify the corrections to scaling for the size-dependent p_c(L) above the upper critical dimension. Including the previous series estimate for the bulk threshold $p_c(\infty)$ gives preference for the complicated corrections predicted by renormalization group and against the simple 1/L extrapolation. Additional Monte-Carlo simulations using the Leath method corroborate the series result for p_c.

cond-mat.dis-nn↗

arXiv1999

Exact critical exponent for the shortest-path scaling function in percolation

It is shown that the critical exponent $g_1$ related to pair-connectiveness and shortest-path (or chemical distance) scaling, recently studied by Porto et al., Dokholyan et al., and Grassberger, can be found exactly in 2d by using a crossing-probability result of Cardy, with the outcome $g_1 = 25/24$. This prediction is consistent with existing simulation results.

cond-mat.stat-mech↗

arXiv1999

Similarity of percolation thresholds on the hcp and fcc lattices

Extensive Monte-Carlo simulations were performed in order to determine the precise values of the critical thresholds for site ($p^{hcp}_{c,S} = 0.199 255 5 \pm 0.000 001 0$) and bond ($p^{hcp}_{c,B} = 0.120 164 0 \pm 0.000 001 0$) percolation on the hcp lattice to compare with previous precise measuremens on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of $p_c$ are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.

cond-mat.dis-nn↗

arXiv1998

Site percolation thresholds and universal formulas for the Archimedean lattices

The site percolation thresholds p_c are determined to high precision for eight Archimedean lattices, by the hull-walk gradient-percolation simulation technique, with the results p_c = 0.697043, honeycomb or (6^3), 0.807904 (3,12^{2}), 0.747806 (4,6,12), 0.729724 (4,8^{2}), 0.579498 (3^{4},6), 0.621819 (3,4,6,4), 0.550213 (3^{3},4^{2}), and 0.550806 (3^{2},4,3,4), and errors of about +/- 2\times10^{-6}. (The remaining Archimedean lattices are the square (4^4), triangular (3^6) and Kagomé (3,6,3,6), for which p_c is already known exactly or to a high degree of accuracy.) The numerical result for the (3,12^{2}) lattice is consistent with the exact value [1-2 sin(π/18)]^{1/2}, which we also derive. The values of p_c for all Archimedean lattices are found to be linearly related to the density of sites within an error of about 1%, which predicts p_c more accurately than correlations based just upon the coordination number can give. Comparison with anisotropic lattices is also made.

cond-mat.dis-nn↗

arXiv1998

Shape-dependent universality in percolation

The shape-dependent universality of the excess percolation cluster number and cross-configuration probability on a torus is discussed. Besides the aspect ratio of the torus, the universality class depends upon the twist in the periodic boundary conditions, which for example are generally introduced when triangular lattices are used in simulations.

cond-mat.dis-nn↗