Approximation of a free poisson process by systems of freely independent particles

Let σ be a non-atomic, infinite Radon measure on Rd, for example, dσ(x) = zdx where z > 0. We consider a system of freely independent particles x1,...,xN in a bounded set Δ ⊂ Rd, where each particle xi has distribution 1 σ(Δ)σ on Δ and the number of particles, N, is random and has Poisson distribution with parameter σ(Δ). If the particles were classically independent rather than freely independent, this particle system would be the restriction to Δ of the Poisson point process on Rd with intensity measure σ. In the case of free independence, this particle system is not the restriction of the free Poisson process on Rd with intensity measure σ. Nevertheless, we prove that this is true in an approximative sense: If bounded sets Δ(n) (nN) are such that Δ(1)⊂Δ(2) ⊂Δ(3) ⋯ and Un=1∞Δ(n) = Rd, then the corresponding particle system in Δ(n) converges (as n → ∞) to the free Poisson process on Rd with intensity measure σ. We also prove the following (N/V)-limit: Let N(n) be a deterministic sequence of natural numbers such that limn→∞N(n)/σ(Δ(n)) = 1. Then the system of N(n) freely independent particles in Δ(n) converges (as n → ∞) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.