  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.
