Tree Capacity and Splitting Isometries for Subinvariant Kernels

TL;DR

This paper explores the properties of subinvariant kernels on trees, linking them to electrical network analogies to derive bounds and convergence criteria, with implications for RKHS structure and boundary analysis.

Contribution

It introduces a novel connection between subinvariant kernels and electrical networks on trees, providing explicit bounds, convergence results, and a boundary martingale framework.

Findings

Derived explicit capacity and resistance bounds for subinvariant kernels.

Proved convergence of kernels under finite limit conditions.

Described the RKHS splitting and boundary martingale construction.

Abstract

Starting from a subinvariant positive definite kernel under a branching pullback, we attach to the resulting kernel tower a canonical electrical network on the word tree whose edge weights are the diagonal increments. This converts diagonal growth into effective resistance and capacity, giving explicit criteria and quantitative bounds, together with a matching upper bound under a mild level regularity condition. When the diagonal tower has finite limit at a point, we prove convergence of the full kernels and obtain an invariant completion with a minimality property. We also describe the associated RKHS splitting and a boundary martingale construction leading to weighted invariant majorants.