Effective resistance of random trees

TL;DR

This paper analyzes the effective electrical resistance in random binary trees with edge resistances scaled by distance, deriving asymptotic expectations, variance bounds, and tail probabilities, with extensions to Galton--Watson trees.

Contribution

It provides explicit asymptotic formulas and probabilistic bounds for the effective resistance in random trees with distance-dependent edge resistances, extending to Galton--Watson trees.

Findings

Expected resistance grows linearly with tree height.

Variance of resistance remains bounded.

Sub-Gaussian tail bounds for resistance.

Abstract

We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by $r_e=2^dX_e$ where the $X_e$ are i.i.d. positive random variables bounded away from zero and infinity. It is shown that $\mathbf{E}R_n=n\mathbf{E}X_e-(\operatorname {\mathbf{Var}}(X_e)/\mathbf{E}X_e)\ln n+O(1)$ and $\operatorname {\mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for $R_n$. We also discuss some possible extensions to supercritical Galton--Watson trees.