Correlations among ‘special’ and ‘general’ grain boundaries are studied on two-dimensional networks, by examining the configurational entropy of boundary structures as well as percolation thresholds. Consideration of crystallographic constraints at various length scales reveals that higher-order constraints play a role in boundary connectivity and network structure. Implications for grain bound...

We consider critical site percolation on the triangular lattice, that is, we choose X(v) = 0 or 1 with probability 1/2 each, independently for all vertices v of the triangular lattice. We say that a word (ξ1, ξ2, . . . ) ∈ {0, 1}N is seen in the percolation configuration if there exists a selfavoiding path (v1, v2, . . . ) on the triangular lattice with X(vi) = ξi, i ≥ 1. We prove that with pro...

We prove the existence of a (random) Lipschitz function F : Z → Z such that, for every x ∈ Z, the site (x, F (x)) is open in a site percolation process on Z. The Lipschitz constant may be taken to be 1 when the parameter p of the percolation model is sufficiently close to 1.

The existence (or not) of infinite clusters is explored for two stochastic models intersecting line segments in $d \ge 2$ dimensions. Salient features the phase diagram are established each case. based on site percolation ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For occupied $v$, and $2d$ possible coordinate directions, declare entire segment from $v$ to next given direction be either blue ...

Dense suspensions display complex flow properties, intermediate between solid and liquid. When sheared, a suspension self-organizes and forms particle clusters that are likely to percolate, possibly leading to significant changes in the overall behavior. Some theoretical conjectures on percolation in suspensions were proposed by de Gennes some 35 years ago. Although still used, they have not re...

We consider the problem of distinguishing between different rates of percolation under noise. A statistical model of percolation is constructed allowing for the birth and death of edges as well as the presence of noise in the observations. This graph-valued stochastic process is composed of a latent and an observed nonstationary process, where the observed graph process is corrupted by type-I a...

Bootstrap percolation is a simple but nontrivial model. It has applications in many areas of science and has been explored on random networks for several decades. In single-layer (simplex) networks, it has been recently observed that bootstrap percolation, which is defined as an incremental process, can be seen as the opposite of pruning percolation, where nodes are removed according to a conne...

We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nano...