A novel form of percolation is considered which is motivated by models of the displacement of one fluid by another from a porous medium. The physical idea is that if the displaced phase is incompressible, then regions of it which are surrounded by the displacing fluid become ‘trapped’ and cannot subsequently be invaded. We thus consider a new percolation process, ‘percolation with trapping’, in...

We build a multifractal object and use it as a support to study perco-lation. We identify some differences between percolation in a multifractal and in a regular lattice. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can h...

We investigate properties of two-dimensional finite-scale percolation systems whose size along the current flow is smaller than the perpendicular size. Successive thresholds of appearing multiple percolation channels in such systems have been determined and dependencies of the conductance on their size and percolation parameter p have been calculated. Various experimental examples show that the...

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...