16 - Three problems for the clairvoyant demon

Summary

Abstract

A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of ‘Winkler’ type, that is, they are challenges for a clairvoyant demon.

Keywords clairvoyant demon, dependent percolation, multiscale analysis, percolation, percolation of words, random walk

AMS subject classification (MSC2010) 60K35, 82B20, 60E15

Introduction

Probability theory has emerged in recent decades as a crossroads where many sub-disciplines of mathematical science meet and interact. Of the many examples within mathematics, we mention (not in order): analysis, partial differential equations, mathematical physics, measure theory, discrete mathematics, theoretical computer science, and number theory. The International Mathematical Union and the Abel Memorial Fund have recently accorded acclaim to probabilists. This process of recognition by others has been too slow, and would have been slower without the efforts of distinguished mathematicians including John Kingman.

JFCK's work looks towards both theory and applications. To single out just two of his theorems: the subadditive ergodic theorem [22, 23] is a piece of mathematical perfection which has also proved rather useful in practice; his ‘coalescent’ [24, 25] is a beautiful piece of probability, now a keystone of mathematical genetics. John is also an inspiring and devoted lecturer, who continued to lecture to undergraduates even as the Bristol Vice-Chancellor, and the Director of the Isaac Newton Institute in Cambridge.