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The band structure of a model of spatial random permutation

Research output: Contribution to journalArticlepeer-review

Original languageEnglish
Number of pages33
JournalPROBABILITY THEORY AND RELATED FIELDS
Volume2021
DOIs
Accepted/In press30 Nov 2020
Published7 Feb 2021

Bibliographical note

Funding Information: This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/N009436/1 “The many faces of random characteristic polynomials” and the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467. The authors would like to thank Jeremiah Buckley, Naomi Feldheim and Daniel Ueltschi for enlightening discussions, and in particular Ron Peled for helpful discussions at an early stage. The authors would also like to thank an anonymous referee for detailed comments which improved the presentation of the paper, and also for pointing out corrections to an earlier version. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

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Abstract

We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size N tends to infinity and the inverse temperature β tends to zero; in particular, we show that the mean displacement is of order min { 1 / β, N}. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac–Murdock–Szegő matrices.

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