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. The plan of this paper is as follows. We begin with a non-technical discussion of the historical context of the project, along with a mention of alternative approaches. Relevant previous works are cited, thus annotating the bibliography. The random-cycle approach to the BEC problem requires a model of spatial permutations. This model it is of its own probabilistic interest; it is developed mathematically, without reference to the Bose gas. Our Markov-chain Monte Carlo algorithms for sampling from the random-cycle distribution—the swap-only, swap-and-reverse, band-update, and worm algorithms—are presented, compared, and contrasted. Finite-size scaling techniques are used to obtain information about infinite-volume quantities from finite-volume computational data.
|Commitee:||Bhattacharya, Rabindra, Lin, Kevin, Ueltschi, Daniel|
|School:||The University of Arizona|
|School Location:||United States -- Arizona|
|Source:||DAI-B 71/05, Dissertation Abstracts International|
|Subjects:||Mathematics, Condensed matter physics|
|Keywords:||Bose gases, Bose-Einstein condensation, Integrated autocorrelation time, Markov-chain Monte Carlo, Random spatial permutations|
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