On a coloured tree with non i.i.d. random labels

TL;DR

This paper investigates the growth of vertices with large values in a coloured d-ary tree with non-i.i.d. random labels, extending previous models and analyzing how the number of such vertices behaves as the threshold approaches zero.

Contribution

It provides new probabilistic results for a coloured tree model with non-i.i.d. labels, expanding understanding of vertex value distributions in such structures.

Findings

Analyzes the asymptotic growth of vertices with large values as the threshold approaches zero.

Extends previous models to include non-i.i.d. random labels based on vertex colours.

Applies results to related probabilistic models.

Abstract

We obtain new results for the probabilistic model introduced in Menshikov et al (2007) and Volkov (2006) which involves a $d$-ary regular tree. All vertices are coloured in one of $d$ distinct colours so that $d$ children of each vertex all have different colours. Fix $d^2$ strictly positive random variables. For any two connected vertices of the tree assign to the edge between them {\it a label} which has the same distribution as one of these random variables, such that the distribution is determined solely by the colours of its endpoints. {\it A value} of a vertex is defined as a product of all labels on the path connecting the vertex to the root. We study how the total number of vertices with value of at least $x$ grows as $x\downarrow 0$, and apply the results to some other relevant models.