2025/06/09 by Prem Raj Prasain, Prasain, Prem Raj
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2506.07377
openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∞-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