Abstract/Summary

Let σ σ be a non-atomic, infinite Radon measure on ℝd, for example, dσ(x)=zdx dσ(x)=zdx where z>0 z>0 . We consider a system of freely independent particles x 1 ,…,x N x1,…,xN in a bounded set Λ⊂R d Λ⊂ℝd , where each particle x i xi has distribution 1σ(Λ) σ 1σ(Λ)σ on Λ Λ and the number of particles, N N , is random and has Poisson distribution with parameter σ(Λ) σ(Λ) . If the particles were classically independent rather than freely independent, this particle system would be the restriction to Λ Λ of the Poisson point process on R d ℝd with intensity measure σ σ . In the case of free independence, this particle system is not the restriction of the free Poisson process on R d ℝd with intensity measure σ σ . Nevertheless, we prove that this is true in an approximative sense: if bounded sets Λ (n) Λ(n) (n∈N n∈ℕ ) are such that Λ (1) ⊂Λ (2) ⊂Λ (3) ⊂⋯ Λ(1)⊂Λ(2)⊂Λ(3)⊂⋯ and ⋃ ∞ n=1 Λ (n) =R d ⋃n=1∞Λ(n)=ℝd , then the corresponding particle system in Λ (n) Λ(n) converges (as n→∞ n→∞ ) to the free Poisson process on R d ℝd with intensity measure σ σ . We also prove the following (N/V) (N/V) -limit: Let N (n) N(n) be a deterministic sequence of natural numbers such that lim n→∞ N (n) /σ(Λ (n) )=1 limn→∞N(n)/σ(Λ(n))=1 . Then the system of N (n) N(n) freely independent particles in Λ (n) Λ(n) converges (as n→∞ 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.