This is a molecular analog to the common board game Jenga, and has relevance to virus disassembly. In two dimensions, it formed part of Kesten's proof that pc = 1/2. [15] In 11 or more dimensions, these facts are largely proved using a technique known as the lace expansion. The 2D bidisperse models composed of overlapping superellipses of two different sizes are constructed then. In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Percolation of traffic in cities was introduced by Daqing Li et al. Note that the percolation threshold for the site-percolation on high-dimensional hypercubic lattices, where loops become irrelevant, approaches that of the Bethe lattice 1=(z 1), if we substitute the coordination number zwith 2d. However, recently percolation has been performed on a weighted planar stochastic lattice (WPSL) and found that although the dimension of the WPSL coincides with the dimension of the space where it is embedded, its universality class is different from that of all the known planar lattices. neighbouring occupied sites (bonds). It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Impact of polydispersity of particle shape and size on percolation threshold of 3D particulate media composed of penetrable superellipsoids. Critical percolation threshold is a crucial parameter that describes the connectivity of heterogeneous structures. In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are removed. For percolation of networks see Cohen and Havlin.[14]. By using the generally continuum percolation algorithm, the critical percolation thresholds ϕc for different binary-sized superellipse systems with the ranges of the equivalent radii ratio λ in [0.1, 1.0] and the number fraction of smaller superellipses f in [0.0, 1.0] are studied. © 2019 Elsevier B.V. All rights reserved. , near Introducing recovery of nodes and links in percolation. Porous networks are approximated by the packing of binary penetrable superellipsoids. A generalization was next introduced as the. We use cookies to help provide and enhance our service and tailor content and ads. p c or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1 – p, and they are ass… . Impact of particle size ratio on the threshold of binary-sized superellipses is given. The maximum threshold is generally achieved in the case of (1−f) ≈ [31], Mathematical theory on behavior of connected clusters in a random graph, In biology, biochemistry, and physical virology, harvtxt error: multiple targets (2×): CITEREFKesten1982 (, harvtxt error: multiple targets (2×): CITEREFGrimmettMarstrand1990 (, harvtxt error: multiple targets (2×): CITEREFGrimmett1999 (, harvtxt error: multiple targets (2×): CITEREFHaraSlade1990 (, harvtxt error: multiple targets (2×): CITEREFSmirnov2001 (, CS1 maint: multiple names: authors list (, weighted planar stochastic lattice (WPSL), gravitational forces acting on the liquid, "Complex Networks: Structure, Robustness and Function", "Critical effect of dependency groups on the function of networks", "Localized attacks on spatially embedded networks with dependencies", "Percolation transition in dynamical traffic network with evolving critical bottlenecks", "Spontaneous recovery in dynamical networks", "Critical stretching of mean-field regimes in spatial networks", "Eradicating catastrophic collapse in interdependent networks via reinforced nodes", "Molecular Jenga: the percolation phase transition (collapse) in virus capsids", "A molecular breadboard: Removal and replacement of subunits in a hepatitis B virus capsid", "Habitat fragmentation, percolation theory and the conservation of a keystone species", Introduction to Percolation Theory: short course by Shlomo Havlin, Independent and identically distributed random variables, Stochastic chains with memory of variable length, Autoregressive conditional heteroskedasticity (ARCH) model, Autoregressive integrated moving average (ARIMA) model, Autoregressive–moving-average (ARMA) model, Generalized autoregressive conditional heteroskedasticity (GARCH) model, https://en.wikipedia.org/w/index.php?title=Percolation_theory&oldid=987872668, Short description is different from Wikidata, Wikipedia articles needing clarification from April 2016, Creative Commons Attribution-ShareAlike License, A limit case for lattices in high dimensions is given by the, There are no infinite clusters (open or closed), The probability that there is an open path from some fixed point (say the origin) to a distance of, The shape of a large cluster in two dimensions is.