Gaussian random permutation and the boson point process

ARMENDARIZ INES; PABLO A. FERRARI; YUHJTMAN, SERGIO A.

COMMUNICATIONS IN MATHEMATICAL PHYSICS

SPRINGER

Lugar: Berlin; Año: 2021 vol. 387 p. 1515 - 1547

0010-3616

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. Each finite cycle of σ induces a loop of points of~χ. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman 1953. Bose-Einstein condensation occurs for dimension d≥3 and above a critical density ρc=ρc(α). For ρ≤ρc we define (χ,σ) 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≥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. In either case, we call the resulting (χ,σ) a Gaussian random permutation at density ρ and temperature α, and show that its χ-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.