Schäfer, Helge (2018)
The Cycle Structure of Random Permutations without Macroscopic Cycles.
Technische Universität Darmstadt
Ph.D. Thesis, Primary publication
Abstract
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 macroscopic cycles when the system size tends to infinity. This probability measure can be represented by cycle weights which depend on the system size. We first establish that the joint distribution of the cycle counts of short cycles is not affected by the conditioning and converges to independent Poisson-distributed random variables in total variation distance. Cumulative cycle numbers of short cycles hence fulfil the same functional central limit theorem as under the classical Ewens measure. Then limit theorems are proved for (the joint distribution of) general individual cycle numbers where the limit strongly depends on the concrete choice of constraint in the conditioning and the cycle lengths in question. Having examined properties related to individual cycle numbers, we turn to the total number of cycles which satisfies a central limit theorem. For cumulative cycle and index numbers we prove the existence of limit shapes and functional limit theorems for the fluctuations about these limit shapes, the limit of the fluctuations being the Brownian bridge. The limit shapes also allow us to determine the asymptotic behaviour of a typical cycle. Lastly, we present findings concerning the distribution of the longest cycles in the model and in this context show convergence of cumulative cycle numbers in a certain regime to a Poisson process.
Item Type: | Ph.D. Thesis | ||||
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Erschienen: | 2018 | ||||
Creators: | Schäfer, Helge | ||||
Type of entry: | Primary publication | ||||
Title: | The Cycle Structure of Random Permutations without Macroscopic Cycles | ||||
Language: | English | ||||
Referees: | Betz, Prof. Dr. Volker ; Mörters, Prof. Dr. Peter ; Zeindler, Dr. Dirk | ||||
Date: | 2018 | ||||
Place of Publication: | Darmstadt | ||||
Refereed: | 30 October 2018 | ||||
URL / URN: | https://tuprints.ulb.tu-darmstadt.de/8148 | ||||
Abstract: | 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 macroscopic cycles when the system size tends to infinity. This probability measure can be represented by cycle weights which depend on the system size. We first establish that the joint distribution of the cycle counts of short cycles is not affected by the conditioning and converges to independent Poisson-distributed random variables in total variation distance. Cumulative cycle numbers of short cycles hence fulfil the same functional central limit theorem as under the classical Ewens measure. Then limit theorems are proved for (the joint distribution of) general individual cycle numbers where the limit strongly depends on the concrete choice of constraint in the conditioning and the cycle lengths in question. Having examined properties related to individual cycle numbers, we turn to the total number of cycles which satisfies a central limit theorem. For cumulative cycle and index numbers we prove the existence of limit shapes and functional limit theorems for the fluctuations about these limit shapes, the limit of the fluctuations being the Brownian bridge. The limit shapes also allow us to determine the asymptotic behaviour of a typical cycle. Lastly, we present findings concerning the distribution of the longest cycles in the model and in this context show convergence of cumulative cycle numbers in a certain regime to a Poisson process. |
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URN: | urn:nbn:de:tuda-tuprints-81487 | ||||
Classification DDC: | 500 Science and mathematics > 510 Mathematics | ||||
Divisions: | 04 Department of Mathematics 04 Department of Mathematics > Stochastik |
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Date Deposited: | 11 Nov 2018 20:55 | ||||
Last Modified: | 11 Nov 2018 20:55 | ||||
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Referees: | Betz, Prof. Dr. Volker ; Mörters, Prof. Dr. Peter ; Zeindler, Dr. Dirk | ||||
Refereed / Verteidigung / mdl. Prüfung: | 30 October 2018 | ||||
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