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

Abstract

Let $\sigma$ be a non-atomic, infinite Radon measure on ${\mathbb{R}}^d$, for example, $d\sigma (x)=zdx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots ,x_N$ in a bounded set $\Lambda \subset {\mathbb{R}}^d$, where each particle $x_i$ has distribution $\frac{1}{\sigma (\Lambda )}\sigma$ on $\Lambda$ and the number of particles, $N$, is random and has Poisson distribution with parameter $\sigma (\Lambda )$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $\Lambda$ of the Poisson point process on ${\mathbb{R}}^d$ with intensity measure $\sigma$. In the case of free independence, this particle system is not the restriction of the free Poisson process on ${\mathbb{R}}^d$ with intensity measure $\sigma$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets ${\Lambda }^{(n)}$ ($n\in \mathbb{N}$) are such that ${\Lambda }^{(1)}\subset {\Lambda }^{(2)}\subset {\Lambda }^{(3)}\subset \cdots$ and ${\bigcup }_{n=1}^{\infty }{\Lambda }^{(n)}={\mathbb{R}}^d$, then the corresponding particle system in ${\Lambda }^{(n)}$ converges (as $n\to \infty$) to the free Poisson process on ${\mathbb{R}}^d$ with intensity measure $\sigma$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that ${\lim }_{n\to \infty }{N}^{(n)}/\sigma ({\Lambda }^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in ${\Lambda }^{(n)}$ converges (as $n\to \infty$) 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.