Precise asymptotics of longest cycles in random permutations without\n macroscopic cycles

We consider Ewens random permutations of length $n$ conditioned to have no\ncycle longer than $n^\\beta$ with $0<\\beta<1$ and to study the asymptotic\nbehaviour as $n\\to\\infty$. We obtain very precise information on the joint\ndistribution of the lengths of the longest cycles; in particular we prove a\nfunctional limit theorem where the cumulative number of long cycles converges\nto a Poisson process in the suitable scaling. Furthermore, we prove convergence\nof the total variation distance between joint cycle counts and suitable\nindependent Poisson random variables up to a significantly larger maximal cycle\nlength than previously known. Finally, we remove a superfluous assumption from\na central limit theorem for the total number of cycles proved in an earlier\npaper.\n