Gaussian Random Permutation and the Boson Point Process

We construct an infinite volume spatial random permutation $(\chi,\sigma)$, where $\chi\subset\mathbb R^d$ is a point process and $\sigma:\chi\to \chi$ is a permutation (bijection), associated to the formal Hamiltonian $ H(\chi,\sigma) = \sum_{x\in \chi} \|x-\sigma(x)\|^2$. The measures are parametrized by the density $\rho$ of points and the temperature $\alpha$. Each finite cycle of $\sigma$ induces a loop of points of~$\chi$. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman 1953. Bose-Einstein condensation occurs for dimension $d\ge 3$ and above a critical density $\rho_c=\rho_c(\alpha)$. For $\rho\le \rho_c$ we define $(\chi,\sigma)$ as a Poisson process of finite unrooted loops that we call Gaussian loop soup after the Brownian loop soup of Lawler and Werner 2004. We also construct the Gaussian random interlacements, a Poisson process of trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements introduced by Sznitman 2010. For $d\ge 3$ and $\rho>\rho_c$ we define $(\chi,\sigma)$ 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 the resulting $(\chi,\sigma)$ a Gaussian random permutation at density $\rho$ and temperature $\alpha$, and show that its $\chi$-marginal has the same distribution as the boson point process introduced by Macchi 1975 at the same density and temperature. This implies in particular that when Bose-Einstein condensation occurs the associated Gaussian random permutation exhibits infinite trajectories.