# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Limit distributions of many-particle spectra and $q$-deformed Gaussian variables

### Tom 73 / 2006

Banach Center Publications 73 (2006), 409-414 MSC: 81V35, 81V70, 15A52. DOI: 10.4064/bc73-0-32

#### Streszczenie

We find the limit distributions for a spectrum of a system of $n$ particles governed by a $k$-body interaction. The hamiltonian of this system is modelled by a Gaussian random matrix. We show that the limit distribution is a $q$-deformed Gaussian distribution with the deformation parameter $q$ depending on the fraction ${k}/{\sqrt{n}}$. The family of $q$-deformed Gaussian distributions include the Gaussian distribution and the semicircular law; therefore our result is a generalization of the results of Wigner \cite{W1,W2}, Mon and French \cite{MF}.