Recent advances in percolation theory and its applications

Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. In this review we will first outline the basic features of the ordinary model and take a glimpse at a number of selective variations and modifications of the original model. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition. After a short review on SLE, we will provide an overview on existence of the scaling limit and conformal invariance of the critical percolation. We will also establish a connection with the magnetic models. Recent applications of the percolation theory in natural and artificial landscapes are also reviewed.