Effective resistance of random trees
Louigi Addario-Berry, Nicolas Broutin and Gábor Lugosi
Annals of Applied Probability 2007.

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 = 2d X_e where the X_e are i.i.d. positive random variables bounded away from zero and infinity. It is shown that ER_n = nEX_e − (Var(X_e )/EX_e ) ln n + O(1) and 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.

[Edit]