Percolation is the study of connectedness in a randomly-chosen subset of an infinite graph. Following Grimmett[4], we set up percolation on a square lattice. We establish the existence of a critical edge-density, prove several results about the behavior of percolation systems above and below this critical density, and use these results to find the critical density of percolation on the two-dime...

We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these i...

Recently, new results on percolation of interdependent networks have shown that the percolation transition can be first order. In this paper we show that, when considering antagonistic interactions between interacting networks, the percolation process might present a bistability of the equilibrium solution. To this end, we introduce antagonistic interactions for which the functionality, or acti...

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli perc...

Many different concepts of percolation exist for networks. We show that bond percolation, site percolation, k-core percolation, and bootstrap percolation are all special cases of the Watts Threshold model. We show that the “heterogeneous k-core” and a corresponding heterogeneous bootstrap model are equivalent to one another and the Watts Threshold Model. A more recent model of a “Generalized Ep...

A method for analyzing a N-component percolation model in terms of one parameter p1 is presented. In Monte Carlo simulations on 16, 32, 64, and 128 simple cubic lattices the percolation threshold p1 c is determined for N52. Continuous transitions of p1 c are reported in two limits for the bond existence probabilities p5 and pÞ . In the same limits, empirical formulas for the percolation thresho...

We study families of dependent site percolation models on the triangular lattice T and hexagonal lattice H that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost...

Preferential Mode of gas invasion in sediments: Grain-scale mechanistic model of coupled multiphase fluid flow and sediment mechanics. " J. Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits ...

How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse the microscopic mechanisms underlying discontin...