  We study a model of spatial random permutations over a discrete set of
points. Formally, a permutation $\sigma$ is sampled proportionally to the
weight $\exp\{-\alpha \sum_x V(\sigma(x)-x)\},$ where $\alpha>0$ is the
temperature and $V$ is a non-negative and continuous potential. The most
relevant case for physics is when $V(x)=\|x\|^2$, since it is related to
Bose-Einstein condensation through a representation introduced by Feynman in
1953. In the context of statistical mechanics, the weights define a probability
when the set of points is finite, but the construction associated to an
infinite set is not trivial and may fail without appropriate hypotheses. The
first problem is to establish conditions for the existence of such a measure at
infinite volume when the set of points is infinite. Once existence is derived,
we are interested in establishing its uniqueness and the cycle structure of a
typical permutation. We here consider the large temperature regime when the set
of points is a Poisson point process in $\mathbb{Z}^d$ with intensity $\rho
\in(0,1/2)$, and the potential verifies some regularity conditions. In
particular, we prove that if $\alpha$ is large enough, for almost every
realization of the point process, there exists a unique Gibbs measure that
concentrates on finite cycle permutations. We then extend these results to the
continuous setting, when the set of points is given by a Poisson point process
in $\mathbb{R}^d$ with low enough intensity.
