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… 
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References

SHOWING 1-10 OF 29 REFERENCES
Local Probabilities for Random Permutations Without Long Cycles
TLDR
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
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
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
TLDR
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.
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
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
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
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
TLDR
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
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
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