The distribution of the lengths of longest increasing subsequences in random permutations has attracted much attention especially in the last five years.  In 1999, Baik, Deift, and Johansson determined the (nonstandard) central limit behavior of these random lengths.  They use deep methods from complex analysis and integrable systems, especially noncommutative Riemann Hilbert theory.    The large deviation behavior of the same random variables was determined before by Seppaelaeinen, Deuschel, and Zeitouni, using more classical large deviation techniques.

In the talk, some recent results on the moderate deviations of the length of longest increasing subsequences are presented.  This concerns the domain between the central limit regime and the large deviation regime.

The proof is based on a Gaussian saddle point approximation around the stationary points of the transition function matrix elements of a certain noncommutative Riemann Hilbert problem of rank 2.

Joint work with Matthias Loewe and (for the lower tail deviations) with Silke Rolles.