Free Self-decomposability and Unimodality of the Fuss-Catalan Distributions
Abstract
We study properties of the Fuss-Catalan distributions μ (p ,r ) , p ≥1 , 0 <r ≤p : free infinite divisibility, free self-decomposability, free regularity and unimodality. We show that the Fuss-Catalan distribution μ (p ,r ) is freely self-decomposable if and only if 1 ≤p =r ≤2 . We verify numerically the following phase-transition conjecture: For every p >1 there exists r0(p ) , with p -1 <r0(p ) <p , such that the Fuss-Catalan distribution μ (p ,r ) is unimodal if and only if either r =p or 0 <r ≤r0(p ) . We prove rigorously some partial results.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- January 2020
- DOI:
- 10.1007/s10955-020-02488-1
- arXiv:
- arXiv:1908.07887
- Bibcode:
- 2020JSP...178.1055M
- Keywords:
-
- Fuss-Catalan distributions;
- Free cumulant transform;
- Voiculescu transform;
- Free Lévy measures;
- Free cumulants;
- Free infinite divisibility;
- Free self-decomposability;
- Free L<SUB>1</SUB>;
- Free regularity;
- Unimodality;
- Mathematics - Probability;
- Mathematics - Combinatorics
- E-Print:
- 16 pages, 2 figures