Short cycles of random permutations with cycle weights: point processes approach
Oleksii Galganov, Andrii Ilienko
TL;DR
The paper studies the asymptotics of short cycles in random permutations with cycle weights by embedding cycles into a multilevel metric space and analyzing the induced point process $\Psi_n$. Under a stability condition on the normalizing constants, $\Psi_n$ converges in the vague sense to a Poisson random measure $\Psi$ with intensity $\lambda=\sum_k \theta_k \lambda_k$, yielding a tractable framework to obtain limit laws for diverse cycle statistics. Through Laplace-functionals, it derives limit theorems for additive cycle statistics and provides explicit formulas, including the distribution for sums over cycles and for fixed-point statistics, as well as results on ranges of $k$-cycles. This point-process approach unifies and extends previous results (e.g., Ewens-type models) and facilitates derivation of a broad class of asymptotic cycle laws in non-uniform random permutations.
Abstract
We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on $\{1,\ldots,n\}$. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as $n\to\infty$ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.