# Random permutations with logarithmic cycle weight

Zeindler, Dirk and Robles, Nicolas (2019) Random permutations with logarithmic cycle weight. Annales de l'institut Henri Poincare (B) Probability and Statistics. ISSN 0246-0203 (In Press)

Text (RoZe_2019_03_05_post_round1No_color)
RoZe_2019_03_05_post_round1No_color.pdf - Accepted Version

## Abstract

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study the asymptotic behavior as the length $n$ tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form $\exp\left( (-\log(1-z))^{k+1} \right)$ with $k\geq 1$, which have not yet been studied in the literature.

Item Type:
Journal Article
Journal or Publication Title:
Annales de l'institut Henri Poincare (B) Probability and Statistics
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2613
Subjects:
Departments:
ID Code:
137230
Deposited By:
Deposited On:
01 Oct 2019 08:15
Refereed?:
Yes
Published?:
In Press