Gaussian Random Permutation and the Boson Point Process

Abstract

We construct an infinite volume spatial random permutation $({{\textsf {X}}},\sigma )$, where ${{\textsf {X}}}\subset {\mathbb {R}}^d$ is locally finite and $\sigma :{{\textsf {X}}}\rightarrow {{\textsf {X}}}$ is a permutation, associated to the formal Hamiltonian

$$\begin{aligned} H({{\textsf {X}}},\sigma ) = \sum _{x\in {{\textsf {X}}}} \Vert x-\sigma (x)\Vert ^2. \end{aligned}$$

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 (Phys Rev 91:1291–1301, 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 ${{\textsf {X}}}$. For $\rho \le \rho _c$ we define $({{\textsf {X}}}, \sigma )$ 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 (Probab Theory Related Fields 128(4):565–588, 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 (Ann Math 2 171(3):2039–2087, 2010). For $d\ge 3$ and $\rho >\rho _c$ we define $({{\textsf {X}}},\sigma )$ 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 $({{\textsf {X}}}, \sigma )$ 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 ${{\textsf {X}}}$ has the same distribution as the boson point process introduced by Shirai-Takahashi (J Funct Anal 205(2):414–463, 2003) in the subcritical case, and by Tamura-Ito (J Funct Anal 243(1): 207–231, 2007) in the supercritical case.