Bogachev, LV orcid.org/0000-0002-2365-2621 and Zeindler, D (2015) Asymptotic Statistics of Cycles in Surrogate-Spatial Permutations. Communications in Mathematical Physics, 334 (1). pp. 39-116. ISSN 0010-3616
Abstract
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 a natural approximation to the so-called spatial random permutations recently studied by Betz and Ueltschi (hence the name “surrogate-spatial”), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical, and supercritical domains. In particular, in the supercritical regime there is a parametric “phase transition” from the Poisson–Dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Pólya’s Enumeration Theorem and complex variable methods.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2015, Springer Verlag. This is an author produced version of a paper published in Communications in Mathematical Physics . Uploaded in accordance with the publisher's self-archiving policy. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-014-2110-1 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 12 Mar 2015 10:30 |
Last Modified: | 15 Apr 2021 11:36 |
Published Version: | http://dx.doi.org/10.1007/s00220-014-2110-1 |
Status: | Published |
Publisher: | Springer Verlag |
Refereed: | Yes |
Identification Number: | 10.1007/s00220-014-2110-1 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:83404 |