Comparison of rigidity and connectivity percolation in two dimensions

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold $p_c\approx 0.655$ is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension $d_f\approx 1.733$. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that ...

We investigate site percolation on a weighted planar stochastic lattice (WPSL) which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by percolation threshold $p_c$ and by a set of critical exponents $\beta$, $\gamma$, $\nu$ which describe the critical behavior of percolation probability $P(p)\sim (p_c-p)^\beta$, mean cluster size $S\sim (p_c-p)^{-\gamma}$ and the correlation length $\xi\sim (p_c-p)^{-\nu}$. Besides, the exponen...

Using particle-scale models to accurately describe property enhancements and phase transitions in macroscopic behavior is a major engineering challenge in composite materials science. To address some of these challenges, we use the graph theoretic property of rigidity to model me- chanical reinforcement in composites with stiff rod-like particles. We develop an efficient algorithmic approach called rigid graph compression (RGC) to describe the transition from floppy to rigid ...

We have investigated both site and bond percolation on two dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of p...

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter and the connectedness criterion, and invoke bimodal, Gaussian and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation it depends only on cer...

Percolation processes on random networks have been the subject of intense research activity over the last decades: the overall phenomenology of standard percolation on uncorrelated and unclustered topologies is well known. Still some critical properties of the transition, in particular for heterogeneous substrates, have not been fully elucidated and contradictory results appear in the literature. In this paper we present, by means of a generating functions approach, a thoroug...

We show that the infinite percolating cluster (with density P_inf) of central-force networks is composed of: a fractal stress-bearing backbone (Pb) and; rigid but unstressed ``dangling ends'' which occupy a finite volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations of triangular lattices give B...

Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecula...

We study the statistics of the backbone cluster between two sites separated by distance $r$ in two-dimensional percolation networks subjected to spatial long-range correlations. We find that the distribution of backbone mass follows the scaling {\it ansatz}, $P(M_B)\sim M_B^{-(\alpha+1)}f(M_B/M_0)$, where $f(x)=(\alpha+ \eta x^{\eta}) \exp(-x^{\eta})$ is a cutoff function, and $M_0$ and $\eta$ are cutoff parameters. Our results from extensive computational simulations indicat...

We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the leaf-free and the bridge-free clusters using the breadth-first-search algorithm. From the geometric properties of these clusters, we determine a set of four critical exponents, including the thermal exponent $y_{\rm t} \equiv 1/\nu$, the frac...