  We determine the asymptotic behavior of the maximum subgraph density of large
random graphs with a prescribed degree sequence. The result applies in
particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of
Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in
extending the notion of balanced loads from finite graphs to their local weak
limits, using unimodularity. This is a new illustration of the objective method
described by Aldous and Steele [In Probability on Discrete Structures (2004)
1-72 Springer].
