# Conductance of a subdiffusive random weighted tree

## TL;DR

This paper investigates how conductance decays in a subdiffusive weighted Galton--Watson tree, revealing that its expectation can decay faster than 1/n depending on model parameters, with convergence results linked to the additive martingale.

## Contribution

It provides a detailed analysis of conductance decay in subdiffusive regimes, highlighting the influence of the second zero of the characteristic function, and establishes convergence to the additive martingale limit.

## Key findings

- Expectation of conductance can decay faster than 1/n
- Decay rate depends on the second zero of the characteristic function
- Conductance divided by expectation converges to the martingale limit

## Abstract

We work on a Galton--Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$-th level of the tree, as $n$ goes to infinity, by a mostly analytic method. It turns out the order of magnitude of the expectation of this conductance can be less than $1/n$ (in contrast with the results of Addario-Berry-Broutin-Lugosi and Chen-Hu-Lin), depending on the value of the second zero of the characteristic function associated to the model. We also prove the almost sure (and in $L^p$ for some $p>1$) convergence of this conductance divided by its expectation towards the limit of the additive martingale.