Generalized Percolation
نویسندگان
چکیده
A generalized model of percolation encompassing both the usual model, in which bonds are occupied with probability p and are vacant with probability (1−p), and the model appropriate to the statistics of lattice animals, in which the fugacity for occupied bonds is p and that for unoccupied bonds is unity, is formulated. Within this model we discuss the crossover between the two problems and we study the statistics of large clusters. We determine the scaling form which the distribution function for the number of clusters with a given number of sites n assumes as a function of both n and p. For p near pc we find that the distribution function depends on percolation exponents for u=n(pc−p)Δp small, where Δp is a crossover exponent, and on exponents appropriate to the lattice-animals problem for large values of u. We thus have displayed the relation between the two limits and show conclusively that the lattice-animals exponents cannot be obtained by any simple scaling arguments from the percolation exponents. We also demonstrate that essential singularities in the cluster distribution functions for p>pc arise from metastable states of the Potts model.
منابع مشابه
Equivalence of several generalized percolation models on networks.
In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, k-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts threshold model (WTM). We show that-except for bond percolation-each of these processes arises as a special case of...
متن کاملOn the largest-eigenvalue process for generalized Wishart random matrices
Using a change-of-measure argument, we prove an equality in law between the process of largest eigenvalues in a generalized Wishart random-matrix process and a last-passage percolation process. This equality in law was conjectured by Borodin and Péché (2008).
متن کامل0 Percolation and Magnetization for Generalized Continuous Spin Models
For the Ising model, the spin magnetization transition is equivalent to the percolation transition of Fortuin-Kasteleyn clusters. This result remains valid also for the continuous spin Ising model. We show on the basis of numerical simulations that such an equivalence can be generalized to a wider class of theories, including spin distribution functions, longer range interactions and self-inter...
متن کاملDensity Fluctuations for a Zero-range Process on the Percolation Cluster
We prove that the density fluctuations for a zero-range process evolving on the d-dimensional supercritical percolation cluster, with d ≥ 3, are given by a generalized Ornstein-Uhlenbeck process in the space of distributions S (R).
متن کاملCrossover from Isotropic to Directed Percolation
Percolation clusters are probably the simplest example for scale–invariant structures which either are governed by isotropic scaling–laws (“self–similarity”) or — as in the case of directed percolation — may display anisotropic scaling behavior (“self–affinity”). Taking advantage of the fact that both isotropic and directed bond percolation (with one preferred direction) may be mapped onto corr...
متن کاملImproved Bounds on Metastability Thresholds and Probabilities for Generalized Bootstrap Percolation
We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik’s k-percolation models, and in the process provide a unified and impr...
متن کامل