Effective resistance of random trees
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.