Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

TL;DR

This paper explores the properties of generalized negative binomial distributions linked to hyperbolic Landau levels, analyzing their moment generating functions, nonclassical states, and divisibility properties, and introduces a new infinitely-divisible distribution.

Contribution

It provides explicit computations of the moment generating function, a perturbation decomposition, and a new infinitely-divisible distribution related to hyperbolic Landau levels.

Findings

The distribution's moment generating function is explicitly computed.

The distribution can be decomposed as a perturbation of the negative binomial.

A new infinitely-divisible distribution is introduced.

Abstract

To each hyperbolic Landau level of the Poincar\'e disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its decomposition as a perturbation of the negative binomial distribution by a finitely-supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we determine the L\'evy-Kintchine decomposition its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-L\'evy measure, we introduce a new infinitely-divisible distribution for which we derive the characteristic function.