$p$-Modulus on radially symmetric trees

TL;DR

This paper develops the theory of p-modulus for infinite paths on infinite-rooted trees, linking it to classical concepts like cuts, resistance, and shortest paths, and introduces a critical p-value for binary trees.

Contribution

It extends p-modulus theory to infinite trees, providing formulas, properties, and a critical p-value concept specific to radially symmetric structures.

Findings

p-modulus on infinite trees as a limit of truncated trees

p-modulus relates to minimum cut, effective resistance, and shortest paths

existence of a critical p-value for binary trees

Abstract

In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a limit of $p$-modulus on truncated trees, with a formula given in terms of a series. Analogous to the existing theory for finite graphs, the $1$-modulus of a family of descending paths in an infinite tree is related to the minimum cut problem, the $2$-modulus is related to effective resistance, and the $\infty$-modulus is related to the length of shortest paths. Another key result is the existence of a critical $p$-value for radially symmetric infinite binary trees, which assigns a kind of dimension to the boundaries