Recent advances in percolation theory and its applications

摘要

Percolation is the simplest fundamental model in statistical mechanics thatexhibits phase transitions signaled by the emergence of a giant connectedcomponent. Despite its very simple rules, percolation theory has successfullybeen applied to describe a large variety of natural, technological and socialsystems. Percolation models serve as important universality classes in criticalphenomena characterized by a set of critical exponents which correspond to arich fractal and scaling structure of their geometric features. In this reviewwe will first outline the basic features of the ordinary model and take aglimpse at a number of selective variations and modifications of the originalmodel. Directed percolation process will be also discussed as a prototype ofsystems displaying a nonequilibrium phase transition. After a short review onSLE, we will provide an overview on existence of the scaling limit andconformal invariance of the critical percolation. We will also establish aconnection with the magnetic models. Recent applications of the percolationtheory in natural and artificial landscapes are also reviewed.