Gaussian random permutation and the boson point process

Abstract: We provide an explicit construction of the infinite volume spatial random permutation associated to the free Bose gas in Rd in both the subcritical and supercritical regimes. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman, in his study of the transition of Helium-4 from fluid to superfluid. For subcritical densities, a typical permutation can be decomposed into finite cycles. Above the density that determines the onset of Bose-Einstein condensation, a typical permutation has infinite cycles, which are described as a random interlacement of random walks with Gaussian increments.