  We study the length of cycles in the model of spatial random permutations in
Euclidean space. In this model, for given length $L$, density $\rho$, dimension
$d$ and jump density $\varphi$, one samples $\rho L^d$ particles in a
$d$-dimensional torus of side length $L$, and a permutation $\pi$ of the
particles, with probability density proportional to the product of values of
$\varphi$ at the differences between a particle and its image under $\pi$. The
distribution may be further weighted by a factor of $\theta$ to the number of
cycles in $\pi$. Following Matsubara and Feynman, the emergence of macroscopic
cycles in $\pi$ at high density $\rho$ has been related to the phenomenon of
Bose-Einstein condensation. For each dimension $d\ge 1$, we identify
sub-critical, critical and super-critical regimes for $\rho$ and find the
limiting distribution of cycle lengths in these regimes. The results extend the
work of Betz and Ueltschi. Our main technical tools are saddle-point and
singularity analysis of suitable generating functions following the analysis by
Bogachev and Zeindler of a related surrogate-spatial model.
