Moderate deviations for longest increasing subsequences: The upper tail

Abstract

AbstractWe derive the upper‐tail moderate deviations for the length of a longest increasing subsequence in a random permutation. This concerns the regime between the upper‐tail large‐deviation regime and the central limit regime. Our proof uses a formula to describe the relevant probabilities in terms of the solution of the rank 2 Riemann‐Hilbert problem (RHP); this formula was invented by Baik, Deift, and Johansson [3] to find the central limit asymptotics of the same quantities. In contrast to the work of these authors, who apply a third‐order (nonstandard) steepest‐descent approximation at an inflection point of the transition matrix elements of the RHP, our approach is based on a (more classical) second‐order (Gaussian) saddle point approximation at the stationary points of the transition function matrix elements. © 2001 John Wiley & Sons, Inc.