Resistance growth of branching random networks

TL;DR

This paper investigates how the effective resistance and conductance behave in an infinite random branching network modeled by a Galton-Watson tree with random edge resistances, extending previous binary tree results.

Contribution

It generalizes prior work on binary trees to more complex random branching networks, analyzing asymptotic resistance and conductance behavior.

Findings

Asymptotic behavior of resistance and conductance established

Results extend known binary tree models to general Galton-Watson trees

Provides insights into electrical properties of random branching structures

Abstract

Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi_e$ indexed by all the edges in the tree. We assign the resistance $m^d \xi_e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.