arXiv subjects

Robert M. Ziff

Publications and source records attributed to Robert M. Ziff.

At least 19 recordsLinked to original sources

arXiv2026

Bond percolation in distorted simple cubic and body-centered cubic lattices

We investigate the effect of structural distortion on bond percolation in simple cubic and body-centered cubic lattices using extensive Monte Carlo simulations. Distortion is introduced through controlled random displacements of lattice sites, thereby modifying nearest-neighbor distances. Bond occupation is permitted only when the bond length is smaller than a prescribed connection threshold, directly coupling geometric disorder to connectivity. Finite-size scaling analysis is employed to determine percolation thresholds for finite systems and in the thermodynamic limit. We find that when the connection threshold exceeds the nearest-neighbor distance of the undistorted lattice, the percolation threshold increases monotonically with distortion strength, indicating a systematic suppression of spanning. In contrast, this monotonic behavior breaks down when the connection threshold is below the nearest-neighbor distance of the undistorted lattice, highlighting a nontrivial interplay between geometric distortion and connectivity. We further identify critical values of the connection threshold and the distortion amplitude required for global spanning when all the allowed bonds are occupied. All qualitative behaviors remain robust across both lattice geometries. These results clarify how geometric disorder reshapes percolation in three-dimensional crystalline networks.

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arXiv2026

Site and bond percolation in linearly distorted triangular and square lattices

We investigate site and bond percolation in triangular and square lattices subjected to linear distortion. In contrast to previously studied distortion schemes that preserve lattice geometry, linear distortion dislocates regular lattice sites along a fixed direction. Nearest-neighbors of a regular lattice need to satisfy a distance-based connection criterion to remain neighbors in the linearly distorted lattice. Using extensive Monte Carlo simulations and finite-size scaling analyses, we examine how site and bond percolation thresholds vary with the distortion parameter and the connection threshold. For triangular lattices, we observe pronounced directional dependence of both site and bond percolation thresholds, as well as of the critical connection threshold. This arises from the distortion-induced anisotropic modification of nearest-neighbor separations. In particular, bond percolation exhibits nontrivial behavior that cannot be explained solely in terms of changes in the average coordination number. In contrast, square lattices remain effectively isotropic under linear distortion, resulting in identical percolation thresholds for distortions applied along different directions. Percolation thresholds in the thermodynamic limit, evaluated for a selected set of values of distortion parameter and connection threshold, confirm that the results for large finite lattices provide reliable estimates of the infinite-system behavior.

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arXiv2026

Bond percolation in distorted square and triangular lattices

This article presents a Monte Carlo study on bond percolation in distorted square and triangular lattices. The distorted lattices are generated by dislocating the sites from their regular positions. The amount and direction of the dislocations are random, but can be tuned by the distortion parameter $α$. Once the sites are dislocated, the bond lengths $δ$ between the nearest neighbors change. A bond can only be occupied if its bond length is less than a threshold value called the connection threshold $d$. It is observed that when the connection threshold is greater than the lattice constant (assumed to be $1$), the bond percolation threshold $p_\mathrm{b}$ always increases with distortion. For $d\le 1$, no spanning configuration is found for the square lattice when the lattice is distorted, even very slightly. On the other hand, the triangular lattice not only spans for $d\le 1$, it also shows a decreasing trend for $p_\mathrm{b}$ in the low-$α$ range. These variation patterns have been linked with the average coordination numbers of the distorted lattices. A critical value $d_\mathrm{c}$ for the connection threshold has been defined as the value of $d$ below which no spanning configuration can be found even after occupying all the bonds satisfying the connection criterion $δ\le d$. The behavior of $d_\mathrm{c}(α)$ is markedly different for the two lattices.

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arXiv2025

Percolation of random compact diamond-shaped systems on the square lattice

We study site percolation on a square lattice with random compact diamond-shaped neighborhoods. Each site $s$ is connected to others within a neighborhood in the shape of a diamond of radius $r_s$, where $r_s$ is uniformly chosen from the set $\{i, i+1, \ldots, m\}$ with $i \leq m$. The model is analyzed for all values of $i = 0, \ldots, 7$ and $m = i, \ldots, 10$, where $\overline{z}(i,m)$ denotes the average number of neighbors per site and $p_c(i,m)$ is the critical percolation threshold. For each fixed $i$, the product $\overline{z}(i,m)\,p_c(i,m)$ is found to converge to a constant as $m \to \infty$. Such behavior is expected when $i=m$ (single diamond sizes), for which the product $\overline{z}(i)\,p_c(i)$ tends toward $2^dη_c$, where $η_c$ is the continuum percolation threshold for diamond-shaped regions or aligned squares in two dimensions ($d=2$). This case is further examined for $i = 1, \ldots, 10$, and the expected convergence is confirmed. The particular case $i = m$ was first studied numerically by Gouker and Family in 1983. We also study the relation to systems of deposited diamond-shaped objects on a square lattice. For monodisperse diamonds of radius $r$, there is a direct mapping to percolation with a diamond-shaped neighborhood of radius $2r+1$, but when there is a distribution of object sizes, there is no such mapping. We study the case of mixtures of diamonds of radius $r=0$ and $r=1$, and compare it to the continuum percolation of disks of two sizes.

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arXiv2024

Entropy and chirality in sphinx tilings

As a toy model of chiral interactions in crowded spaces, we consider sphinx tilings in finite regions of the triangular lattice. The sphinx tiles, hexiamonds composed of six equilateral triangles in the shape of a stylized sphinx, come in left and right enantiomorphs. Regions scaled up from the unit sphinx by an integer factor ("Sphinx frames") require tiles of both chiral forms to produce tilings, including crystalline, quasicrystalline, and fully disordered tilings. For frames up to order 13, we describe methods that permit exact enumeration and computation of partition functions using "accelerated backtracking," "seam," and "dangler" algorithms. For larger frames, we introduce a Monte Carlo (MC) method to sample typical tilings. Key to the latter is the identification of fundamental shapes (polyads) that admit multiple tilings and which allow a rejection-free MC simulation.

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arXiv2023

Extended-range percolation in five dimensions

Percolation on a five-dimensional simple hypercubic (sc(5)) lattice with extended neighborhoods is investigated by means of extensive Monte Carlo simulations, using an effective single-cluster growth algorithm. The critical exponents, including $τ$ and $Ω$, the asymptotic behavior of the threshold $p_c$ and its dependence on coordination number $z$ are investigated. Using the bond and site percolation thresholds $p_c = 0.11817145(3)$ and $0.14079633(4)$ respectively given by Mertens and Moore [Phys. Rev. E 98, 022120 (2018)], we find critical exponents of $τ= 2.4177(3)$, $Ω= 0.27(2)$ through a self-consistent process. The value of $τ$ compares favorably with a recent five-loop renormalization predictions $2.4175(2)$ by Borinsky et al. [Phys. Rev. D 103, 116024 (2021)], the value 2.4180(6) that follows from the work of Zhang et al. [Physica A 580, 126124 (2021)], and the measurement of $2.419(1)$ by Mertens and Moore. We also confirmed the bond threshold, finding $p_c = 0.11817150(5)$. sc(5) lattices with extended neighborhoods up to 7th nearest neighbors are studied for both bond and site percolation. Employing the values of $τ$ and $Ω$ mentioned above, thresholds are found to high precision. For bond percolation, the asymptotic value of $zp_c$ tends to Bethe-lattice behavior ($z p_c \sim 1$), and the finite-$z$ correction is found to be consistent with both and $zp_{c} - 1 \sim a_1 z^{-0.88}$ and $zp_{c} - 1 \sim a_0(3 + \ln z)/z$. For site percolation, the asymptotic analysis is close to the predicted behavior $zp_c \sim 32η_c = 1.742(2)$ for large $z$, where $η_c = 0.05443(7)$ is the continuum percolation threshold of five-dimensional hyperspheres given by Torquato and Jiao [J. Chem. Phys 137, 074106 (2015)]; finite-$z$ corrections are accounted for by taking $p_c \approx c/(z + b)$ with $c=1.722(7)$ and $b=1$.

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arXiv2022

Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems

We combine percolation theory and Monte Carlo simulation to study in two dimensions the connectivity of an equilibrium lattice model of interacting Janus disks which self-assemble into an orientationally ordered stripe phase at low temperature. As the patch size is increased or the temperature is lowered, clusters of patch-connected disks grow, and a percolating cluster emerges at a threshold. In the stripe phase, the critical clusters extend longer in the direction parallel to the stripes than in the perpendicular direction, and percolation is thus anisotropic. It is found that the critical behavior of percolation in the Janus system is consistent with that of standard isotropic percolation, when an appropriate spatial rescaling is made. The rescaling procedure can be applied to understand other anisotropic systems, such as the percolation of aligned rigid rods and of the $q$-state Potts model with anisotropic interactions.

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arXiv2022

Percolation in two-species antagonistic random sequential adsorption in two dimensions

We consider two-species random sequential adsorption (RSA) in which species A and B adsorb randomly on a lattice with the restriction that opposite species cannot occupy nearest-neighbor sites. When the probability $x_A$ of choosing an A particle for an adsorption trial reaches a critical value $0.626441(1)$, the A species percolates and/or the blocked sites X (those with at least one A and one B nearest neighbor) percolate. Analysis of the size-distribution exponent $τ$, the wrapping probabilities, and the excess cluster number shows that the percolation transition is consistent with that of ordinary percolation. We obtain an exact result for the low $x_B = 1 - x_A$ jamming behavior: $θ_A = 1 - x_B +b_2 x_B^2+\mathcal{O}(x_B^3)$, $θ_B = x_B/(z+1)+\mathcal{O}(x_B^2)$ for a $z$-coordinated lattice, where $θ_A$ and $θ_B$ are respectively the saturation coverages of species A and B. We also show how differences between wrapping probabilities of A and X clusters, as well as differences in the number of A and X clusters, can be used to find the transition point accurately. For the one-dimensional case a three-site approximation appears to provide exact results for the coverages.

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arXiv2022

Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods

The asymptotic behavior of the percolation threshold $p_c$ and its dependence upon coordination number $z$ is investigated for both site and bond percolation on four-dimensional lattices with compact extended neighborhoods. Simple hypercubic lattices with neighborhoods up to 9th nearest neighbors are studied to high precision by means of Monte-Carlo simulations based upon a single-cluster growth algorithm. For site percolation, an asymptotic analysis confirms the predicted behavior $zp_c \sim 16 η_c = 2.086$ for large $z$, and finite-size corrections are accounted for by forms $p_c \sim 16 η_c/(z+b)$ and $p_c \sim 1- \exp(-16 η_c/z)$ where $η_c \approx 0.1304$ is the continuum percolation threshold of four-dimensional hyperspheres. For bond percolation, the finite-$z$ correction is found to be consistent with the prediction of Frei and Perkins, $zp_{c} - 1 \sim a_{1} (\ln z)/z$, although the behavior $zp_{c} - 1 \sim a_1 z^{-3/4}$ cannot be ruled out.

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arXiv2021

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

Extended-range percolation on various regular lattices, including all eleven Archimedean lattices in two dimensions, and the simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number $z$ and site thresholds $p_c$ for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting $z$ versus $1/p_{c}$ and $z$ versus $-1/\ln(1-p_c)$, using the data of d'Iribarne et al. [J. Phys. A 32:2611, 1999] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of $zp_{c} \sim 4 η_c = 4.51235$, where $η_c$ is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for BCC and FCC lattices with 2nd and 3rd NN, and bond thresholds for the SC lattice with up to the 13th NN, are obtained by Monte-Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of $zp_{c} \sim 8 η_c = 2.7351$, where $η_c$ is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior $p_c = 1/(z-1)$ is expected to hold for large $z$, and the finite-$z$ correction is confirmed to satisfy $zp_{c} - 1 \sim a_{1}z^{-x}$, with $x=2/3$ for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21:56, 2016] and by Xu et al. [Phys. Rev. E, 103:022127, 2021]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of $zp_{c}$ is universal, depending only upon the dimension of the system and whether site or bond percolation, but not upon the type of lattice.

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arXiv2021

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Descriptors that characterize the geometry and topology of the pore space of porous media are intimately linked to their transport properties. We quantify such descriptors, including pore-size functions and the critical pore radius $δ_c$, for four different models: maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, and inherent structures of the quantizer energy. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. Often, $δ_c$ is used as the key characteristic length scale that determines the fluid permeability $k$. A recent study [Torquato. Adv. Wat. Resour. 140, 103565 (2020)] suggested for porous media with a well-connected pore space an alternative estimate of $k$ based on the second moment of the pore size $\langleδ^2\rangle$. Here, we confirm that, for all porosities and all models considered, $δ_c^2$ is to a good approximation proportional to $\langleδ^2\rangle$. However, unlike $\langleδ^2\rangle$, the permeability estimate based on $δ_c^2$ does not predict the correct ranking of $k$ for our models. Thus, we confirm $\langleδ^2\rangle$ to be a promising candidate for convenient and reliable estimates of $k$ for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the $τ$ order metric. We find that (effectively) hyperuniform models tend to have lower values of $k$ than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

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arXiv2021

Percolation and the pandemic

This paper is dedicated to the memory of Dietrich Stauffer, who was a pioneer in percolation theory and applications of it to problems of society, such as epidemiology. An epidemic is a percolation process gone out of control, that is, going beyond the critical transition threshold $p_c$. Here we discuss how the threshold is related to the basic infectivity of neighbors $R_0$, for trees (Bethe lattice), trees with triangular cliques, and in non-planar lattice percolation with extended-range connectivity. It is shown how having a smaller range of contacts increases the critical value of $R_0$ above the value $R_{0,c}=1$ appropriate for a tree, an infinite-range system or a large completely connected graph.

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arXiv2020

Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds $η_c$ for objects of those shapes. This mapping implies $zp_{c} \sim 4 η_c = 4.51235$ in 2D and $zp_{c} \sim 8 η_c = 2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c = c/(z+b)$ we find good agreement with $c = 2^d η_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.

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arXiv2020

Critical percolation on the kagome hypergraph

We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and $(3,12^2)$ lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.

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arXiv2020

Universal correlations in percolation

We discuss correlations in percolation and recent contributions of Xiaojun Tan, Youjin Deng and Jesper Lykke Jacobsen in their paper "N-cluster correlations in four- and five-dimensional percolation," Frontiers of Physics 15, 41501 (July, 2020). This is a View and Perspective in Frontiers of Physics.

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arXiv2020

Bond percolation on simple cubic lattices with extended neighborhoods

We study bond percolation on the simple cubic (SC) lattice with various combinations of first, second, third, and fourth nearest-neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number $z$ quite accurately according to a power law $p_{c} \sim z^{-a}$, with exponent $a = 1.111$. However, for large $z$, the threshold must approach the Bethe lattice result $p_c = 1/(z-1)$. Fitting our data and data for lattices with additional nearest neighbors, we find $p_c(z-1)=1+1.224 z^{-1/2}$.

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arXiv2020

Bond percolation between $k$ separated points on a square lattice

We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through adjacent connected points of a single cluster. These processes yield new thresholds $\overline p_{ck}$ defined as the average value of $p$ at which the desired connections first occur. These thresholds are not sharp as the distribution of values of $p_{ck}$ for individual samples remains broad in the limit of $L \to \infty$. We study $\overline p_{ck}$ for bond percolation on the square lattice, and find that $\overline p_{ck}$ are above the normal percolation threshold $p_c = 1/2$ and represent specific supercritical states. The $\overline p_{ck}$ can be related to integrals over powers of the function $P_\infty(p)$ equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of $P_\infty(p)$ on $L\times L$ systems that, for $L \to \infty$, $\overline p_{c1} = 0.51755(5)$, $\overline p_{c2} = 0.53219(5)$, $\overline p_{c3} = 0.54456(5)$, and $\overline p_{c4} = 0.55527(5).$ The percolation thresholds $\overline p_{ck}$ remain the same, even when the $k$ points are randomly selected within the lattice. We show that the finite-size corrections scale as $L^{-1/ν_k}$ where $ν_k = ν/(k β+1)$, with $β=5/36$ and $ν=4/3$ being the ordinary percolation critical exponents, so that $ν_1= 48/41$, $ν_2 = 24/23$, $ν_3 = 16/17$, $ν_4 = 6/7$, etc. We also study three-point correlations in the system, and show how for $p>p_c$, the correlation ratio goes to 1 (no net correlation) as $L \to \infty$, while at $p_c$ it reaches the known value of 1.022.

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arXiv2020

Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes

Simplicial complexes are gaining increasing scientific attention as they are generalized network structures that can represent the many-body interactions existing in complex systems raging from the brain to high-order social networks. Simplicial complexes are formed by simplicies, such as nodes, links, triangles and so on. Cell complexes further extend these generalized network structures as they are formed by regular polytopes such as squares, pentagons etc. Pseudo-fractal simplicial and cell complexes are a major example of generalized network structures and they can be obtained by gluing $2$-dimensional $m$-polygons ($m=2$ triangles, $m=4$ squares, $m=5$ pentagons, etc.) along their links according to a simple iterative rule. Here we investigate the interplay between the topology of pseudo-fractal simplicial and cell complexes and their dynamics by characterizing the critical properties of link percolation defined on these structures. By using the renormalization group we show that the pseudo-fractal simplicial and cell complexes have a continuous percolation threshold at $p_c=0$. When the pseudo-fractal structure is formed by polygons of the same size $m$, the transition is characterized by an exponential suppression of the order parameter $P_{\infty}$ that depends on the number of sides $m$ of the polygons forming the pseudo-fractal cell complex, i.e., $P_{\infty}\propto p\exp(-α/p^{m-2})$. Here these results are also generalized to random pseudo-fractal cell-complexes formed by polygons of different number of sides $m$.

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