Ultrametric and tree potential

TL;DR

This paper investigates infinite tree and ultrametric matrices, establishing connections with symmetric random walks, harmonic functions, and potential theory, and introduces probabilistic methods for their simulation and analysis.

Contribution

It provides a new representation theorem for harmonic functions and a probabilistic framework for ultrametric matrices within their minimal tree extensions.

Findings

Existence of associated symmetric random walks for each tree matrix

Explicit representation of harmonic functions and Martin kernel

Probabilistic conditions for potential properties of ultrametric matrices

Abstract

We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. In the boundary, we construct the Markov kernel whose Green function is the extension of the matrix and we simulate it by using a cascade of killing independent exponential random variables and conditionally independent uniform variables. For ultrametric matrices we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension.