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Random permutations without long cycles: total variation approximations and limit shapes of cycles

Random permutations without long cycles: total variation approximations and limit shapes of cycles

We consider uniform random permutations in Sn conditioned to have no cycles of length larger or equal to a(n) with a(n)~n^b with b\in (0, 1). We study the behaviour of the a typical permutation and compared it to the behaviour on the full symmetric group. In particular, we determine the total variation distance between the small cycles and a family of independent Poisson random variables. Furthermore, we establish a limit shape for the cycles of order a(n).