# Random permutations without macroscopic cycles

@article{Betz2020RandomPW, title={Random permutations without macroscopic cycles}, author={Volker Betz and Helge Schafer and Dirk Zeindler}, journal={The Annals of Applied Probability}, year={2020} }

We consider uniform random permutations of length n conditioned to have no cycle longer than nβ 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…

## 7 Citations

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

- Mathematics
- 2020

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$ and study the asymptotic behaviour as $n\to\infty$. We obtain very precise…

The Cycle Structure of Permutations Without Long Cycles

- Mathematics
- 2019

We consider the cycle structure of a random permutation $\sigma$ chosen uniformly from the symmetric group, subject to the constraint that $\sigma$ does not contain cycles of length exceeding $r.$ We…

The Cycle Structure of Random Permutations without Macroscopic Cycles

- Mathematics
- 2018

We consider the Ewens measure on the symmetric group conditioned on the event that no cycles of macroscopic lengths occur and investigate the resulting cycle structure of random permutations without…

Random permutations with logarithmic cycle weights

- Mathematics
- 2018

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…

Long cycle of random permutations with polynomially growing cycle weights

- MathematicsRandom Struct. Algorithms
- 2021

The asymptotic behaviour of the long cycles under a multiplicative measure with polynomial growing cycle weights is determined and it is proved that the cumulative cycle numbers converge in the region of theLong cycles to a Poisson process.

Limit Shapes for Gibbs Partitions of Sets

- MathematicsJournal of Statistical Physics
- 2021

This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand…

A new approach to the characteristic polynomial of a random unitary matrix

- Mathematics
- 2020

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for…

## References

SHOWING 1-10 OF 29 REFERENCES

Local Probabilities for Random Permutations Without Long Cycles

- MathematicsElectron. J. Comb.
- 2016

The probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r is explored and saddle point method formulas valid in specified regions for the ratio n/r are obtained.

The number of cycles in random permutations without long cycles is asymptotically Gaussian

- Mathematics
- 2016

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior…

The Cycle Structure of Random Permutations without Macroscopic Cycles

- Mathematics
- 2018

We consider the Ewens measure on the symmetric group conditioned on the event that no cycles of macroscopic lengths occur and investigate the resulting cycle structure of random permutations without…

Cycle structure of random permutations with cycle weights

- MathematicsRandom Struct. Algorithms
- 2014

It is found that the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights are usually independent Poisson random variables.

Random permutations with cycle weights.

- Mathematics
- 2011

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows…

The limit shape of random permutations with polynomially growing cycle weights

- Mathematics
- 2015

In this work we are considering the behaviour of the limit shape of Young diagrams associated to random permutations on the set {1, . . . , n} under a particular class of multiplicative measures with…

Random Permutations of a Regular Lattice

- Mathematics
- 2013

Spatial random permutations were originally studied due to their connections to Bose–Einstein condensation, but they possess many interesting properties of their own. For random permutations of a…

Random A-permutations: Convergence to a Poisson process

- Mathematics
- 2007

Suppose that Sn is the permutation group of degree n, A is a subset of the set of natural numbers ℕ, and Tn(A) is the set of all permutations from Sn whose cycle lengths belong to the set A.…

Limit Theorems for Combinatorial Structures via Discrete Process Approximations

- MathematicsRandom Struct. Algorithms
- 1992

The power ofrete functional limitTheorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods are demonstrated.

Asymptotic Statistics of Cycles in Surrogate-Spatial Permutations

- Mathematics
- 2015

We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by…