Almost giant clusters for percolation on large trees with logarithmic heights

TL;DR

This paper surveys recent asymptotic results on Bernoulli bond percolation in large trees with logarithmic height and provides a criterion for the emergence of giant clusters, addressing a question by Croydon.

Contribution

It introduces a general criterion for giant cluster existence in large trees, advancing understanding of percolation on complex structures.

Findings

Asymptotic behavior of percolation clusters on large trees

A criterion for giant cluster emergence

Answers a question posed by Croydon

Abstract

This text is based on a lecture for the Sheffield Probability Day; its main purpose is to survey some recent asymptotic results about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.