Recent advances in percolation theory and its applications

TL;DR

This review summarizes recent advances in percolation theory, covering its fundamental models, variations, critical phenomena, conformal invariance, and diverse applications in natural and artificial systems.

Contribution

It provides a comprehensive overview of recent developments in percolation theory, including new models, theoretical insights, and practical applications.

Findings

Percolation exhibits universal critical behavior with fractal structures.

Directed percolation models display nonequilibrium phase transitions.

Recent applications extend to natural and artificial landscapes.

Abstract

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.