Effective resistance of random trees

We investigate the effective resistance Rn and conductance Cn 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 re=2dXe where the Xe are i.i.d. positive random variables bounded away from zero and infinity. It is shown that ERn=nEXe.(Var.(Xe)/EXe)ln.n+O(1) and Var.(Rn)=O(1). Moreover, we establish sub-Gaussian tail bounds for Rn. We also discuss some possible extensions to supercritical Galton.Watson trees.