[DMO] Random colourings of trees with constant down degree: David de Boer

Abstract: Denote \((T_n^d, r)\) for the rooted tree with root \(r\), where each vertex has \(d\) children and the leaves are at distance exactly \(n\) from \(r\). Fix a colouring with \( r\) colours of the leaves, and look at a random proper colouring of \(T_n^d\)  that agrees with the  colours we fixed on the leaves. What is the probability that the root gets colour 1 under such a proper colouring? Is it \(1/q\), or does it depend on the fixed colouring of the leaves? If we let the distance between the root and the leaves grow, will this tend to \(1/q\)? After looking at this as a warm up, we will allow non proper colourings and discuss how we reformulate the problem into a dynamical system, which we analysed to obtain our result.

This talk is based on joint work with Ferenc Bencs, Pjotr Buys and Guus Regts.