  We propose a new approach to conjugation-invariant random permutations.
Namely, we explain how to construct uniform permutations in given conjugacy
classes from certain point processes in the plane. This enables the use of
geometric tools to study various statistics of such permutations. For their
longest decreasing subsequences, we prove universality of the $2\sqrt n$
asymptotic. For Robinson--Schensted shapes, we prove universality of the
Vershik--Kerov--Logan--Shepp limit curve, thus solving a conjecture of Kammoun.
For the number of records, we establish a phase transition phenomenon as the
number of fixed points grows. For pattern counts, we obtain an asymptotic
normality result, partially answering a conjecture of Hamaker and Rhoades.
