Limit Distributions for Euclidean Random Permutations
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We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length L, density \(\rho \), dimension d and jump density \(\varphi \), one samples \(\rho L^d\) particles in a d-dimensional torus of side length L, and a permutation \(\pi \) of the particles, with probability density proportional to the product of values of \(\varphi \) at the differences between a particle and its image under \(\pi \). The distribution may be further weighted by a factor of \(\theta \) to the number of cycles in \(\pi \). Following Matsubara and Feynman, the emergence of macroscopic cycles in \(\pi \) at high density \(\rho \) has been related to the phenomenon of Bose–Einstein condensation. For each dimension \(d\ge 1\), we identify sub-critical, critical and super-critical regimes for \(\rho \) and find the limiting distribution of cycle lengths in these regimes. The results extend the work of Betz and Ueltschi. Our main technical tools are saddle-point and singularity analysis of suitable generating functions following the analysis by Bogachev and Zeindler of a related surrogate-spatial model.
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We thank Daniel Ueltschi for helpful advice and for referring us to the paper of Bogachev and Zeindler . We thank Volker Betz, Gady Kozma, Mikhail Sodin, and Elad Zelingher for useful discussions. We thank Xiaolin Zeng for helpful comments on an earlier version of this work. We thank Omer Angel and Tom Hutchcroft for considering the validity of the statement (17) on the Mallows model and letting us know the conclusion of their calculations. We are grateful to two anonymous referees whose thoughtful comments helped to elucidate the presentation of the results.
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