Effective resistance of random trees

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 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.