Sixty years of percolation

TL;DR

This paper reviews sixty years of percolation theory, highlighting its development from a niche probabilistic model to a multidisciplinary field with recent progress and ongoing challenges.

Contribution

It provides a concise overview of the historical evolution, key results, and open problems in percolation theory for a broad mathematical audience.

Findings

Major results achieved in Bernoulli percolation during the 1980s

Recent advances in understanding percolation in various domains

Remaining open challenges in the mathematical theory of percolation

Abstract

Percolation models describe the inside of a porous material. The theory emerged timidly in the middle of the twentieth century before becoming one of the major objects of interest in probability and mathematical physics. The golden age of percolation is probably the eighties, during which most of the major results were obtained for the most classical of these models, named Bernoulli percolation, but it is really the two following decades which put percolation theory at the crossroad of several domains of mathematics. In this short (and very partial) review, we propose to describe briefly some of the recent progress as well as some famous challenges remaining in the field. This review is not intended to probabilists (and a fortiori not to specialists in percolation theory): the target audience is mathematicians of all kinds.