Critical Percolation Clusters Research Articles

The chemical distance distribution in percolation clusters

The authors study the conditional probability density p(r mod l), for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions. They argue that (i) p(r mod l)=Alxg exp(-axdelta ) where x=r/lnu , g=2.5+or-0.3, delta =9.8+or-0.5, and nu =0.88+or-0.02, and (ii) there is a relation delta =(1- nu )-1 which is in good agreement with numerical data. These results are derived by considering a special class of self-avoiding walks consisting of chains which are chemical paths (shortest paths) on critical percolation clusters.

A new kinetic walk and percolation perimeters.

We introduce the smart kinetic walk (SKW), a new kinetic-walk model which is in a different universality class from other such models. The SKW is strictly self-avoiding, yet is never forced to terminate, because it never starts down a path which would lead to its being trapped. We show that a ring-forming version of the model in two dimensions traces out the external perimeter of critical percolation clusters. Using previous results on these perimeters, we find that the SKW fractal dimension is ${D}_{\mathrm{SKW}\mathrm{\ensuremath{\approxeq}}}$1.75. The equivalence between the walk and percolation perimeters leads to a scaling form for the number of N-step rings. Finally, we see that the walk with a bias to turn to the left more often than to the right (or vice versa) traces out the perimeter of clusters that are not at the percolation threshold. This implies a maximum ring size depending upon the strength of the bias.

True self-avoiding walk on critical percolation clusters and lattice animal

The statistics of true-self-avoiding walk model on two dimensional critical percolation clusters and lattice animals are studied using real-space renormalization group method. The correlation length exponents ν's are found to beν TSAW pc ≈0.576 andν TSAW LA ≈0.623 respectively for the critical percolation clusters and lattice animals.

Kinetic growth walk on critical percolation clusters and lattice animals

The statistics of recently proposed kinetic growth walk (KGW) model for linear polymers (or growing self avoiding walk (GSAW)) on two dimensional critical percolation clusters and lattice animals are studied using real-space renormalization group method. The correlation length exponents ν's are found to be νKGWPc = 0.68 and νKGWLA respectively for the critical percolation clusters and lattice animals. Close agreements are found between these results and a generalized Flory formula for linear polymers at theta point νKGWF = 2/\(\bar d\)+1),, where\(\bar d\) is the fractal dimension of the fractal objectF.

Diffusion on percolation clusters at criticality

The concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters. The authors find from Monte Carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three-dimensional space is D=3.3+or-0.1 where the size of the cluster is restricted to be larger than the span of the walk, and is D'=3.9+or-0.1 for a walk on clusters not subject to this restriction. For two-dimensional space they find D approximately=D' approximately=2.7+or-P0.1. The exponent D (and D') is related to the scaling of the average length R of N steps via RD varies as N. The fracton dimensionality which is related to the density of states was found to be 1.26+or-0.1. These results are in good agreement with the predictions of Alexander and Orbach (1982).

Similarity laws for the density profile of percolation clusters and lattice animals

Analysis of Monte Carlo data shows the density profiles of critical percolation clusters (p=pc) to be similar to each other. The same is true for random animals (p=0). The exponents for these scaling laws are determined and agree with those expected from the cluster radii.

Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm

A new approach for the determination of the critical percolation concentration, percolation probabilities, and cluster size distributions is presented for the site percolation problem. The novel "cluster multiple labeling technique" is described for both two- and three-dimensional crystal structures. Its distinctive feature is the assignment of alternate labels to sites belonging to the same cluster. These sites are members of a simulated finite random lattice. An algorithm useful for the determination of the critical percolation concentration of a finite lattice is also presented. This algorithm is especially useful when applied in conjunction with the cluster multiple labeling technique. The basic features of this technique are illustrated by applying it to a small planar square lattice. Numerical results are given for a triangular subcrystal containing up to 9 000 000 sites. These results compare favorably with the exact value of the infinite lattice critical percolation concentration.

Phase Transitions in Random Graphs

Previous article Next article Phase Transitions in Random GraphsV. E. StepanovV. E. Stepanovhttps://doi.org/10.1137/1115027PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. E. Stepanov, Combinatorial algebra and random graphs, Theory Prob. Applications, 14 (1969), 373–399 10.1137/1114052 0239.05124 LinkGoogle Scholar[2] V. E. Stepanov, On the probability of connectedness of a random graph $\mathcal{G}_m(t)$, Theory Prob. Applications, 15 (1970), 55–67 10.1137/1115004 0233.60006 LinkGoogle Scholar[3] P. Erdo˝s and , A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl., 5 (1960), 17–61 MR0125031 Google Scholar[4] M. Hall, Combinatorial Analysis, IL, Moscow, 1963, (Russian translation.) Google Scholar[5] R. A. Minloss, Lectures in statistical physics, Uspekhi matem. nauk, XXIII (1968), 134–190, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Split Trees – A Unifying Model for Many Important Random Trees of Logarithmic Height: A Brief Survey16 May 2021 Cross Ref Critical percolation clusters in seven dimensions and on a complete graph7 February 2018 | Physical Review E, Vol. 97, No. 2 Cross Ref When language breaks into pieces A conflict between communication through isolated signals and languageBiosystems, Vol. 84, No. 3 Cross Ref Systems of Containers and Enumeration Problems Cross Ref On the Largest Component of the Random Graph at a Nearcritical StageJournal of Combinatorial Theory, Series B, Vol. 82, No. 2 Cross Ref A phase transition phenomenon in a random directed acyclic graph1 January 2001 | Random Structures and Algorithms, Vol. 18, No. 2 Cross Ref Phase transition phenomena in random discrete structuresDiscrete Mathematics, Vol. 136, No. 1-3 Cross Ref Components of Random Forests12 September 2008 | Combinatorics, Probability and Computing, Vol. 1, No. 1 Cross Ref On Some Features of the Structure of a Random Graph Near a Critical PointV. E. Stepanov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 4AbstractPDF (2474 KB)On the Behavior of a Random Graph Near a Critical PointV. F. Kolchin28 July 2006 | Theory of Probability & Its Applications, Vol. 31, No. 3AbstractPDF (1004 KB)A review of random graphsJournal of Graph Theory, Vol. 6, No. 4 Cross Ref Volume 15, Issue 2| 1970Theory of Probability & Its Applications History Submitted:17 March 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1115027Article page range:pp. 187-203ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics