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Shift in Critical Temperature for Random Spatial Permutations with Cycle Weights

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Abstract

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter α. For weak interactions, the shift in critical temperature is expected to be linear in α with constant of linearity c. Using Markov chain Monte Carlo methods and finite-size scaling, we find c=0.618±0.086. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations.

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    John Kerl

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  1. John Kerl
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Kerl, J. Shift in Critical Temperature for Random Spatial Permutations with Cycle Weights. J Stat Phys 140, 56–75 (2010). https://doi.org/10.1007/s10955-010-9988-6

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  • DOI: https://doi.org/10.1007/s10955-010-9988-6

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