Random permutations with logarithmic cycle weights

TL;DR

This paper investigates the asymptotic properties of random permutations with logarithmic cycle weights, revealing convergence to Poisson distributions, central limit theorems, and shape theorems for associated Young diagrams using complex analysis and combinatorics.

Contribution

It introduces new asymptotic results for permutations with logarithmic cycle weights and applies novel singularity analysis to complex generating functions.

Findings

Cycle count process converges to independent Poisson variables

Central limit theorem for total number of cycles

Shape and functional CLT for Young diagrams

Abstract

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study 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.