Recent advances and open challenges in percolation

TL;DR

This paper reviews over 60 years of research on percolation, a fundamental model for random connectivity, highlighting recent advances, open challenges, and various models including classical, explosive, invasion, bootstrap, and correlated percolation.

Contribution

It provides a comprehensive overview of recent progress and open problems in the diverse field of percolation theory and models.

Findings

Percolation exhibits a robust continuous transition across all dimensions.

Multiple models of percolation, including classical and correlated, have unique open questions.

The review identifies key open challenges for future research in percolation theory.

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.