### Gaussian random permutations and the boson point process

Probability Seminar

23rd October 2020, 3:00 pm – 4:00 pm

online, https://zoom.us/j/93299847543

We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.This is joint work with P.A. Ferrari and S. Yuhjtman.

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