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 <italic>c</italic>.  Using Markov
chain Monte Carlo methods and finite-size scaling, we find
<italic>c</italic>&nbsp;=&nbsp;0.618&nbsp;\plusmn&nbsp;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 &mdash; the swap-only,
swap-and-reverse, band-update, and worm algorithms &mdash;
are presented, compared, and contrasted.  Finite-size scaling
techniques are used to obtain information about infinite-volume
quantities from finite-volume computational data.