Gaussian random permutation and the boson point process

We construct an infinite volume spatial random permutation $(\mathsf X,\sigma)$, where $\mathsf X\subset\mathbb R^d$ is locally finite and $\sigma:\mathsf X\to \mathsf X$ is a permutation, associated to the formal Hamiltonian $$H(\mathsf X,\sigma) = \sum_{x\in \mathsf X} \|x-\sigma(x)\|^2.$$ The measures are parametrized by the point density $\rho$ and the temperature $\alpha$. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Let $\rho_c=\rho_c(\alpha)$ be the critical density for Bose-Einstein condensation in Feynman's representation. Each finite cycle of $\sigma$ induces a loop of points of~$\mathsf X$. For $\rho\le \rho_c$ we define $(\mathsf X, \sigma)$ 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 $d\ge 3$ and $\rho>\rho_c$ we define $(\mathsf X,\sigma)$ as the superposition of independent realizations of the Gaussian loop soup at density $\rho_c$ and the Gaussian random interlacements at density $\rho-\rho_c$. In either case we call $(\mathsf X, \sigma)$ a Gaussian random permutation at density $\rho$ and temperature $\alpha$. The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian $H$. Its point marginal $\mathsf X$ has the same distribution as the boson point process introduced by Shirai-Takahashi (2003) in the subcritical case, and by Tamura-Ito (2007) in the supercritical case.