No percolation at criticality

TL;DR

This paper discusses the longstanding open problem of whether the phase transition in bond percolation on integer lattices at criticality is continuous or discontinuous, a fundamental question in percolation theory.

Contribution

The paper provides insights into the nature of the phase transition at criticality in bond percolation on integer lattices, addressing a major open problem in the field.

Findings

Highlights the open problem of phase transition continuity at criticality

Reviews historical context and significance in percolation theory

Suggests directions for future research on the problem

Abstract

Ever since J.M. Hammersley showed the existence of phase-transitions regarding independent bond percolation on general $d \geq 2$ dimensional integer-lattices in the late 50's, the continuity (or discontinuity) of which is perhaps the most prominent and long-standing basic open problem in the subsequently extensively developed theory of percolation.