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The band structure of a model of spatial random permutation

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 \(\beta \) tends to zero; in particular, we show that the mean displacement is of order \(\min \{1/\beta , N\}\). 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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Notes

  1. I.e. there exist \(\alpha _1, \alpha _2 > 0\), \(\alpha _3 \in {\mathbb {R}}\) and \(p_2 > p_1 \ge 1\) such that \(L( u, {\dot{u}} )\ge \alpha _1 |u|^{p_1} + \alpha _2 |{\dot{u}}|^{p_2} - \alpha _3\).

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Correspondence to Stephen Muirhead.

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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.

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Fyodorov, Y.V., Muirhead, S. The band structure of a model of spatial random permutation. Probab. Theory Relat. Fields 179, 543–587 (2021). https://doi.org/10.1007/s00440-020-01019-z

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Keywords

  • Spatial random permutation
  • Band structure
  • Boltzmann weight
  • Gaussian fields

Mathematics Subject Classification

  • 60C05
  • 05A05