Random environment on coloured trees

TL;DR

This paper investigates a probabilistic model of coloured trees with random edge labels influenced by endpoint colours, generalizing multiple existing models like random walks in random environments and branching processes.

Contribution

It introduces a new model of coloured trees with edge labels depending on endpoint colours, extending various classical probabilistic models.

Findings

Derived new results for the coloured tree model.

Showed the model's generalization of existing probabilistic frameworks.

Connected the model to random walks, recursive equations, and branching processes.

Abstract

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.