The first half of the talk will be devoted to probabilistic models of "spatial" random permutations, that involve points in $R^d$. Permutations are weighed according to the length of the jumps. The main question deals with the occurrence of infinite cycles. The second part of the talk will be devoted to the quantum Bose gas in the path-integral representation. Models of interacting spatial permutations are expected to shed light on the effects of the interactions on the Bose-Einstein condensation.