Free Poisson distribution

Definition of Free Poisson Law

Definition. The free Poisson law with jump size $\alpha$ and rate $\lambda$ arises in free probability theory as the limit of repeated free convolution $\left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}$ as $N\to \infty$.

In other words, let $X_{N}$ be random variables so that $X_{N}$ has value $\alpha$ with probability ${\frac {\lambda }{N}}$ and value $0$ with the remaining probability. Assume also that the family $X_{1},X_{2},\ldots$ are freely independent. Then the limit as $N\to \infty$ of the law of $X_{1}+\cdots +X_{N}$ is given by the Free Poisson law with parameters $\lambda ,\alpha$.

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

The measure associated to the free Poisson law is given by

$\mu ={\begin{cases}(1-\lambda )\delta _{0}+\lambda \nu ,&{\textrm {if}}\ 0\leq \lambda \leq 1\\\nu ,&{\textrm {if}}\ \lambda >1,\end{cases}}$

where

$\nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}dt$

and has support $\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}]$.

This law also arises in random matrix theory as the Marchenko-Pastur law.

Some transforms of this law

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher

The R-transform of the free Poisson law is given by

$R(z)={\frac {\lambda \alpha }{1-\alpha z}}$

The Stieltjes transformation (also known as the Cauchy transform) is given by

$G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda ))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}$

The S-transform is given by

$S(z)={\frac {1}{z+\lambda }}$

in the case that $\alpha =1$.