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

Abstract

Let [Formula: see text] be a non-atomic, infinite Radon measure on [Formula: see text], for example, [Formula: see text] where [Formula: see text]. We consider a system of freely independent particles [Formula: see text] in a bounded set [Formula: see text], where each particle [Formula: see text] has distribution [Formula: see text] on [Formula: see text] and the number of particles, [Formula: see text], is random and has Poisson distribution with parameter [Formula: see text]. If the particles were classically independent rather than freely independent, this particle system would be the restriction to [Formula: see text] of the Poisson point process on [Formula: see text] with intensity measure [Formula: see text]. In the case of free independence, this particle system is not the restriction of the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. Nevertheless, we prove that this is true in an approximative sense: if bounded sets [Formula: see text] ([Formula: see text]) are such that [Formula: see text] and [Formula: see text], then the corresponding particle system in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. We also prove the following [Formula: see text]-limit: Let [Formula: see text] be a deterministic sequence of natural numbers such that [Formula: see text]. Then the system of [Formula: see text] freely independent particles in [Formula: see text] converges (as [Formula: see text]) 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.