\begin{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.
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The plan of this paper is as follows.
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We begin with a non-technical discussion of the historical context of the
project, along with a mention of alternative approaches.
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Relevant previous works are cited, thus annotating the bibliography.
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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.
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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.
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Finite-size scaling techniques are used to obtain information about
infinite-volume quantities from finite-volume computational data.
\end{abstract}

