  We consider the problem of undirected polymers (tied at the endpoints) in
random environment, also known as the unoriented first passage percolation on
the hypercube, in the limit of large dimensions. By means of the multiscale
refinement of the second moment method we obtain a fairly precise geometrical
description of optimal paths, i.e. of polymers with minimal energy. The picture
which emerges can be loosely summarized as follows. The energy of the polymer
is, to first approximation, uniformly spread along the strand. The polymer's
bonds carry however a lower energy than in the directed setting, and are
reached through the following geometrical evolution. Close to the origin, the
polymer proceeds in oriented fashion -- it is thus as stretched as possible.
The tension of the strand decreases however gradually, with the polymer
allowing for more and more backsteps as it enters the core of the hypercube.
Backsteps, although increasing the length of the strand, allow the polymer to
connect reservoirs of energetically favorable edges which are otherwise
unattainable in a fully directed regime. These reservoirs lie at mesoscopic
distance apart, but in virtue of the high dimensional nature of the ambient
space, the polymer manages to connect them through approximate geodesics with
respect to the Hamming metric: this is the key strategy which leads to an
optimal energy/entropy balance. Around halfway, the mirror picture sets in: the
polymer tension gradually builds up again, until full orientedness close to the
endpoint. The approach yields, as a corollary, a constructive proof of the
result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018)]
concerning the leading order of the ground state.
