Consider routing traffic on the N×N torus, simultaneously between all source-destination pairs, to minimize the cost ∑ec(e)f2(e), where f(e) is the volume of flow across edge e and the c(e) form an i.i.d. random environment. We prove existence of a rescaled N→∞ limit constant for minimum cost, by comparison with an appropriate analogous problem about minimum-cost flows across a M×M subsquare of the lattice.
"Optimal flow through the disordered lattice." Ann. Probab. 35 (2) 397 - 438, March 2007. https://doi.org/10.1214/009117906000000719