We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in p...

A hybrid model for soils, which combines percolation theory and finite element method is presented. The internal soil structure is modelled via the finite element method, and percolation networks are used for analyzing its mechanical behaviour. Through a microscopic characterization of elastic properties of soil grains, the model is generated. The effective percolation threshold obtained is low...

Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena. Dynamical percolation is a model where a simple time dynamics is added to the (ordinary) percolation model. This dynamical model exhibits very interesting behavior. Our goal in this survey is to give an overview of the work in dynamical percolation that has been do...

Let Ln denote the lowest crossing of the 2n × 2n square box B(n) centered at the origin for critical site percolation on Z 2 or critical site percolation on the triangular lattice imbedded in Z 2 , and denote by Qn the set of pivotal sites along this crossing. On the event that a pivotal site exists, denote the minimum height that a pivotal site attains above the bottom of B(n) by Mn := min{m ≥...

We consider the simple random walk on Z evolving in a potential of independent and identically distributed random variables taking values in [0,+∞]. We give optimal conditions for the existence of the quenched point-to-point Lyapunov exponent, and for different versions of a shape theorem. The method of proof applies as well to first-passage percolation, and builds up on an approach of Cox and ...

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d is fixed and n grows. We prove mean-field behavior around the critical probability pc = 1 d−1 . In particular, we show that there is a scaling window of width n around pc in which the sizes of the largest components are roughly n 2/3 and we describe their limiting joint ...

Knowledge can be seen as a complex system that does not reveal clearly how it works. However, some mathematical models have proved day after day their relevance to the description of complex phenomena in fields that are apparently unrelated. This is especially the case for Percolation Theory. After a survey of the percolation mechanism, it will be seen that Percolation Theory has proved to be r...