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.