Abstract:Random permutations with logarithmic cycle weights

The topic of this talk are random permutations on symmetric group $S_n$
with logarithmic growing cycles weights and their asymptotic behaviour
as the length $n$ tends to infinity.
More precisely, we assign to each cycle in a given permutation the
weight $\log^{k}(m)$, where $m$ is the cycle length and $k$ is an
integer greater or equal to $1$.
We then take the product over all cycles of these weights and normalise
them to obtain a probability measure on the symmetric group $S_n$.
We begin by studying the cycle counts $C_m$, where $C_m$ denotes the
number of cycles of length $m$.
We show that $C_m$ converges in distribution, as $n \to \infty$, to a
Poisson random variable with parameter $\log^{k}(m)$ and
the process consisting of the cycle counts converges to a process
constituting of  independent Poisson random variables.
Furthermore we compute also the total variation distance between both
processes.
Finally, we establish a functional central limit theorem for the Young
diagrams associated to random permutations under this measure.