Random permutations without macroscopic cycles

Abstract

We consider uniform random permutations of length n conditioned to have no cycle longer than n β with 0 < β < 1, in the limit of large n. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

“…In the context of the methods we use, the dierence between the Ewens measure and uniform random permutations is not large, see [22] for details. What should be considered the main contribution of the present paper compared to [3,4] are the following three items: rstly, we obtain much more precise asymptotics for the distribution of the longest cycles in various regimes (Propositions 2.2 and 2.3, and Theorem 2.5). For this, we consider the number of cycles of length α(n) and distinguish the cases where this number is diverging, is converging to a positive number and is vanishing.…”

“…Here we recall some of the results from [4] and [22] that are crucial for the following. Let x n,α be the unique positive solution of the equation…”

“…[26,27]); our interest lies in the second case. Using precise asymptotic results by Manstavi£ius and Petuchovas [20], in [3,4] we investigated the case where a permutation of length n is prevented from having any cycles above a threshold α(n) that grows strictly slower than volume order. While the results in these papers were reasonably detailed, some interesting questions and ne details have been left out.…”

“…Secondly, we extend the validity of the joint Poisson approximation (in variation distance) to the whole regime of cycles of length (o(α(n)) (Theorem 2.8). We achieve this by replacing the Poisson random variables with xed expectation by Poisson random variables with an expectation depending on n. Finally, we remove a spurious additional assumption for the central limit theorem for cycle numbers that was present in [4], see Theorem 2.9. We achieve this by a more preciser saddle point solution.…”

Long cycle of random permutations with polynomially growing cycle weights

“…Another well studied case are the weights k ∼ k , see [6,10]. Further studied weights are for instance k = log m (k) and k = {k≤n } , see [19] and [3]. An overview can be found in [9].…”