Eur. Phys. J. Special Topics 223, 2307–2321 (2014)
https://doi.org/10.1140/epjst/e2014-02266-y

Review

Recent advances and open challenges in percolation

¹ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal
Centro de Física Teórica e Computacional, Universidade de Lisboa, 1749-016 Lisboa, Portugal
² JSC, FZ Jülich, 52425 Jülich, Germany
³ Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
⁴ Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
⁵ Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA

^a e-mail: nmaraujo@fc.ul.pt
^b e-mail: p.grassberger@fz-juelich.de
^c e-mail: bkahng@snu.ac.kr
^d e-mail: kjs73@cam.ac.uk
^e e-mail: rziff@umich.edu

Received: 22 April 2014
Revised: 18 August 2014
Published online: 24 October 2014

Abstract

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation.