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It is known from the work of Baik, Deift, and Johansson [1999] that we haveTracy-Widom fluctuations for the longest increasing subsequence of uniformpermutations. In this paper, we prove that this result holds also in the caseof the Ewens distribution and more generally for a class of random permutationwith distribution invariant under conjugation. Moreover, we obtain theconvergence of the first components of the associated Young tableaux to theAiry Ensemble as well as the global convergence to theVershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove thatthe limiting descent process of a large class of random permutations isstationary, one-dependent and determinantal.

Monotonous subsequences and the descent process of invariant random permutations