Undirected Polymers in Random Environment: path properties in the mean field limit.

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