The best bounds in a theorem of Russell Lyons (Q1272305)

Consider a Bernoulli percolation on a tree which assigns to every edge \(\sigma\) a survival probability \(p_\sigma\) and associate to this percolation an electric network: \(A_\sigma\) be the set of edges connecting \(\sigma\) to the root, including \(\sigma\), and put on \(\sigma\) a resistance \(R_\sigma\) such that \(1+\sum_{\rho\in A_\sigma}R_\rho=\prod_{\rho\in A_\sigma}p^{-1}_\rho\). If \(C\) is the total conductance of the network between the root and the boundary, then \[ {C\over 1+C}\leq P\text{(a ray starting at the root survives) }\leq 1-e^{-2C} \] and these inequalities are optimal. This sharpens a result of \textit{R. Lyons} [Ann. Probab. 20, No. 4, 2043-2088 (1992; Zbl 0766.60091)].