arXiv subjects

R. M. Ziff

Publications and source records attributed to R. M. Ziff.

14 recordsLinked to original sources

arXiv2026

Strongly-connected percolation on directed lattices

We study percolation on lattices with directed bonds, focusing on the behavior of strongly-connected percolation clusters -- clusters in which every site is reachable from every other along a directed path. We consider the two-dimensional square lattice and various globally isotropic arrangements of the directions of the bonds. Performing simulations using a range of algorithmic approaches, we calculate high-precision values for critical exponents, fractal dimensions, crossing probabilities, and percolation thresholds for bond percolation with each bond arrangement. We find that the critical behavior is in a distinctly different universality class from that of traditional undirected percolation, but that all bond arrangements appear to fall in the same universality class.

cond-mat.stat-mech↗

arXiv2014

Retention capacity of correlated surfaces

We extend the water retention model [C. L. Knecht et al., Phys. Rev. Lett. 108, 045703 (2012)] to correlated random surfaces. We find that the retention capacity of discrete random landscapes is strongly affected by spatial correlations among the heights. This phenomenon is related to the emergence of power-law scaling in the lake volume distribution. We also solve the uncorrelated case exactly for a small lattice and present bounds on the retention of uncorrelated landscapes.

cond-mat.stat-mech↗

arXiv2014

Recent advances and open challenges in percolation

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation.

cond-mat.stat-mech↗

arXiv2012

Ordinary Percolation with Discontinuous Transitions

Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a nontrivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here, we provide a simple example of this transition in form of a small-world network consisting of a one-dimensional lattice combined with a hierarchy of long-range bonds that reveals many features of the transition in a mathematically rigorous manner.

cond-mat.dis-nn↗

arXiv2010

Self-dual planar hypergraphs and exact bond percolation thresholds

Recent research on percolation has led to the construction of an infinite class of lattices for which the percolation thresholds can be determined exactly. We discuss the mathematical basis for the solutions of bond percolation models, and, in particular, attempt to address some misconceptions that might arise from the seminal articles by Ziff and Scullard. As one consequence, we show that there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.

cond-mat.stat-mech↗

arXiv2009

Small-World Bonds and Patchy Percolation on the Hanoi Network

The bond-percolation properties of the Hanoi networks are analyzed with the renormalization group. Unlike scale-free networks, they are meant to provide an analytically tractable interpolation between finite dimensional, lattice-based models and their mean-field limits. In percolation, the hierarchical small-world bonds in the Hanoi networks impose a new form of order by uniting otherwise disconnected, local clusters. This "patchy" order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability, unlike the usual 0-1 transition found on ordinary lattices. The various networks studied here exhibit a range of phase behaviors, depending on the prevalence of those long-range bonds. Fixed points in general exhibit non-universal behavior.

cond-mat.dis-nn↗

arXiv2009

The Harmonic Measure of Diffusion-Limited Aggregates including Rare Events

We obtain the harmonic measure of diffusion-limited aggregate (DLA) clusters using a biased random-walk sampling technique which allows us to measure probabilities of random walkers hitting sections of clusters with unprecedented accuracy; our results include probabilities as small as 10^(-80). We find the multifractal D(q) spectrum including regions of small and negative q. Our algorithm allows us to obtain the harmonic measure for clusters more than an order of magnitude larger than those achieved using the method of iterative conformal maps, which is the previous best method. We find a phase transition in the singularity spectrum f(alpha) at alpha approximately equal to 14 and also find a minimum q of D(q), q_{min} = 0.9 plus or minus 0.05.

cond-mat.stat-mech↗

arXiv2006

Anchored Critical Percolation Clusters and 2-D Electrostatics

We consider the densities of clusters, at the percolation point of a two-dimensional system, which are anchored in various ways to an edge. These quantities are calculated by use of conformal field theory and computer simulations. We find that they are given by simple functions of the potentials of 2-D electrostatic dipoles, and that a kind of superposition {\it cum} factorization applies. Our results broaden this connection, already known from previous studies, and we present evidence that it is more generally valid. An exact result similar to the Kirkwood superposition approximation emerges.

cond-mat.dis-nn↗

arXiv2005

Decay of Metastable Nonequilibrium Phases, Enhanced Reaction Rate, and Dynamic Phase Transition in a Model of CO Oxidation with CO Desorption

We present a computational study of the dynamic behavior of a Ziff-Gulari-Barshad model of CO oxidation with CO desorption on a catalytic surface. Our results provide further evidence that below a critical desorption rate the model exhibits a non-equilibrium, first-order phase transition between low and high CO coverage phases. Our kinetic Monte Carlo simulations indicate that the transition process between these phases follows a decay mechanism very similar to the one described by the classic Kolmogorov-Johnson-Mehl-Avrami theory of phase transformation by nucleation and growth. We measure the lifetimes of the metastable phases on each side of the transition line and find that they are strongly dependent on the direction of the transformation, i.e., from low to high coverage or vice versa. Inspired by this asymmetry, we introduce a square-wave periodic external forcing, whose two parameters can be tuned to enhance the catalytic activity. At CO desorption rates below the critical value, we find that this far-from-equilibrium system undergoes a dynamic phase transition between a CO_2 productive phase and a nonproductive one. In the space of the parameters of the periodic external forcing, this nonequilibrium phase transition defines a line of critical points. The maximum enhancement rate for the CO_2 production rate occurs near this critical line.

cond-mat.mtrl-sci↗

arXiv2002

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 so-called "average-probability" estimate is described by a non-trivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90(2). 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.

cond-mat.stat-mech↗

arXiv2001

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.

cond-mat.stat-mech↗

arXiv2001

A fast Monte Carlo algorithm for site or bond percolation

We describe in detail a new and highly 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 which 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.

cond-mat.stat-mech↗

arXiv2000

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 occupation probability 0.59274621(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.

cond-mat.stat-mech↗

arXiv1997

Exact results at the 2-D percolation point

We derive exact expressions for the excess number of clusters b and the excess cumulants b_n of a related quantity at the 2-D percolation point. High-accuracy computer simulations are in accord with our predictions. b is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site n_c in the infinite system limit; the bn correct bulk cumulants. b and b_n are universal, and thus depend only on the system's shape. Higher-order corrections show no apparent dependence on fractional powers of the system size.

cond-mat.dis-nn↗