Shift in critical temperature for random spatial permutations with cycle weights

TL;DR

This paper investigates how interactions affect the critical temperature for phase transitions in a model of random spatial permutations related to Bose-Einstein condensation, using Monte Carlo simulations to quantify the temperature shift.

Contribution

It provides the first numerical estimate of the linear coefficient for the critical temperature shift due to interactions in a spatial permutation model, confirming analytical predictions.

Findings

Critical temperature shift coefficient c = 0.618 ± 0.086

Linear dependence of critical temperature shift on interaction strength α

Recovery of known results for non-spatial permutation cycle lengths

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.