Shift in critical temperature for random spatial permutations with cycle weights

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 $\alpha$. For weak interactions, the shift in critical temperature is expected to be linear in $\alpha$ with constant of linearity $c$. Using Markov chain Monte Carlo methods and finite-size scaling, we find $c=0.618\pm 0.086$. 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.

v2 incorporated reviewer comments. v3 removed two extraneous figures which appeared at the end of the PDF.

References in corpus (5)

• Worm Algorithm and Diagrammatic Monte Carlo: A New Approach to Continuous-Space Path Integral Monte Carlo Simulations
• Spatial random permutations and infinite cycles
• Random permutations with cycle weights
• Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases
• Finite-Size Scaling in the Driven Lattice Gas

Cited by in corpus (3)

• Spatial random permutations and Poisson-Dirichlet law of cycle lengths
• Lattice permutations and Poisson-Dirichlet distribution of cycle lengths
• Random permutations of a regular lattice