Biomedical subjects

R M Ziff

Publications and source records attributed to R M Ziff.

6 recordsLinked to original sources

PubMed2002

Convergence of threshold estimates for two-dimensional percolation.

Using a recently introduced algorithm for simulating percolation in microcanonical (fixed-occupancy) samples, we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary, specifically for site percolation on a square lattice. We show that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90+/-0.02. For the median and cell-to-cell estimates of the percolation threshold we verify that convergence does not depend on this exponent, having instead a slightly faster convergence with a trivial analytic leading exponent.

Journal Article↗

PubMed2002

Percolation and epidemics in a two-dimensional small world.

Percolation on two-dimensional small-world networks has been proposed as a model for the spread of plant diseases. In this paper we give an analytic solution of this model using a combination of generating function methods and high-order series expansion. Our solution gives accurate predictions for quantities such as the position of the percolation threshold and the typical size of disease outbreaks as a function of the density of "shortcuts" in the small-world network. Our results agree with scaling hypotheses and numerical simulations for the same model.

Biophysical Phenomena↗

PubMed2001

Percolation threshold, Fisher exponent, and shortest path exponent for four and five 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(straight theta) bytes where straight theta ranges from 0.4 for two-dimensional (2D) lattices to 0.5 for six (or higher)-dimensional lattices. We use this method to estimate d(min), the exponent relating the minimum path l to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find d(min)=1.607+/-0.005 (4D) and d(min)=1.812+/-0.006 (5D). In order to determine d(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+/-0.000003 (4D) and p(c)=0.14081+/-0.00001 (5D) for site and p(c)=0.160130+/-0.000003 (4D) and p(c)=0.118174+/-0.000004 (5D) for bond percolation. We also calculate the Fisher exponent tau determined in the course of calculating the values of p(c): tau=2.313+/-0.003 (4D) and tau=2.412+/-0.004 (5D).

Journal Article↗

PubMed2001

Fast Monte Carlo algorithm for site or bond percolation.

We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.

Journal Article↗

PubMed2000

Efficient Monte Carlo algorithm and high-precision results for percolation.

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.

Journal Article↗

PubMed1999

Site percolation thresholds for Archimedean lattices.

Precise thresholds for site percolation on eight Archimedean lattices are determined by the hull-walk gradient-percolation simulation method, with the results p(c)=0.697 043, honeycomb or (6(3)), 0.807 904 (3,12(2)), 0.747 806 (4,6,12), 0.729 724 (4,8(2)), 0.579 498 (3(4),6), 0.621 819 (3,4,6,4), 0.550 213 (3(3),4(2)), and 0.550 806 (3(2),4,3,4), with errors of about +/- 3 x 10(-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(pi/18)](1/2). The values of p(c) for all 11 Archimedean lattices, as well as a number of nonuniform lattices, are found to be well correlated by a nearly linear function of a generalized Scher-Zallen filling factor. This correlation is much more accurate than recently proposed correlations based solely upon coordination number.

Journal Article↗