Gaussian random permutation and the boson point process

Abstract

We construct an infinite volume spatial random permutation (χ,σ), where χ⊂ℝd is a point process and σ:χ→χ is a permutation (bijection), associated to the formal Hamiltonian H(χ,σ)=∑_x∈χ‖x−σ(x)‖2. The measures are parametrized by the density ρ of points and the temperature α. Feynman (1953) related spatial random permutations with boson systems and proposed that Bose-Einstein condensation occurs precisely when infinite cycles appear in the corresponding random permutation. Each finite cycle of σ induces a loop of points of χ. For ρ ≤ ρc we define (χ, σ) as a Poisson process of finite unrooted loops 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 double-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2007). For d ≥ 3 and ρ > ρc we define (χ, σ) as the superposition of independent realizations of the Gaussian loop soup at density ρc and the Gaussian random interlacements at density ρ − ρc and call it a Gaussian random permutation at density ρ and temperature α. The resulting measure is Gibbs for the Hamiltonian H and the point marginal χ has the same distribution as the boson point process introduced by Macchi (1975) in the subcritical case and by Tamura-Ito (2007) in the supercritical case. Bose-Einstein condensation occurs when the Gaussian random permutation exhibits infinite trajectories.