 
Summary: The Annals of Probability
2007, Vol. 35, No. 2, 397438
DOI: 10.1214/009117906000000719
© Institute of Mathematical Statistics, 2007
OPTIMAL FLOW THROUGH THE DISORDERED LATTICE1
BY DAVID ALDOUS
University of California at Berkeley
Consider routing traffic on the N × N torus, simultaneously between all
sourcedestination pairs, to minimize the cost e c(e)f 2(e), where f (e) is
the volume of flow across edge e and the c(e) form an i.i.d. random environ
ment. 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.
1. Introduction. In highly abstracted models of transportation or commu
nication (e.g., roads, Internet) one is required (simultaneously for all source
destination pairs) to route a certain "volume of flow" from source to destination,
and one seeks to minimize some notion of cost subject to some constraints (e.g.,
edgecapacities). In contrast to queueing theory, we shall regard flows as determin
istic, but networks as random. A survey of such problems under various different
models of random networks will be given elsewhere. In this paper we focus on
