  This paper initiates a limit theory of permutation valued processes, building
on the recent theory of permutons. We apply this to study the asymptotic
behaviour of random sorting networks. We prove that the Archimedean path, the
conjectured limit of random sorting networks, is the unique path from the
identity to the reverse permuton having minimal energy in an appropriate
metric. Together with a recent large deviations result (Kotowski, 2016), it
implies the Archimedean limit for the model of relaxed random sorting networks.
