We study percolation and conduction in a randomly diluted network in which the occupied bonds may be either resistor-like, transmitting connectivity in both directions along a bond, or diode-like, transmitting in only one direction. This system exhibits novel phase transitions signalling the onset of infinite clusters with either isotropic or unidirectional connectivity. Position-space renormal...

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, V ertex Separation, Edge Bisection and V ertex Bisection. For full square lattices, we give optimal layouts for the problems still...

We first introduce the percolation problems associated with the graph theoretical concepts of (k, l)-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for (k, l)-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of (k, l)-percolation probl...

We study percolation in a random resistor-diode network on the simple cubic lattice. Here the occupied bonds joining nearest-neighbour sites may be either resistor-like, transmitting information or connectivity in either direction along a bond, or diode-like, transmitting in only one direction. We consider a model in which there exist both positive diodes which transmit in the fx, + y or + z di...

A general method is given whereby m-connectedness correlation functions can be studied in the percolation problem. The method for m=1 involves calculating the kth power of the correlation function ⟨σ(x)σ(x′)⟩ for a randomly dilute Ising model at nonzero temperature and subsequently averaging over configurations. The final step is to take the limit k→0. This method is tested by reproducing the s...

Campaigners are increasingly using online social networking platforms for promoting products, ideas and information. A popular method of promoting a product or even an idea is incentivizing individuals to evangelize the idea vigorously by providing them with referral rewards in the form of discounts, cash backs, or social recognition. Due to budget constraints on scarce resources such as money ...

We propose a phenomenological field theoretical approach to the chemical etching of a disordered solid. The theory is based on a recently proposed dynamical etching model. Through the introduction of a set of Langevin equations for the model evolution, we are able to map the problem into a field theory related to isotropic percolation. To the best of the author's knowledge, this constitutes the...

The ‘analysis on fractals’ and ‘analysis on metric spaces’ communities have tended to work independently. Metric spaces such as the Sierpinski carpet fail to satisfy some of the properties which are generally assumed for metric spaces. This survey discusses analysis on the Sierpinski carpet, with particular emphasis on the properties of the heat kernel. 1. Background and history Percolation was...

We carry out Monte Carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation. These studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation. We use splitting, a technique that has not yet been fully exploited in the physics literature, to increase the speed of our simulations. This technique...

Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In r-neighbour bootstrap percolation on a graph G we assign a state, infected or healthy, to every vertex of G and then update these states in successive rounds, according to the following simple local update rule: infected vertices of G remain infected forever and a healthy vertex b...