The longest increasing subsequence in a random permutation and a unitary random matrix model

If $L_N$ is the expected length of the longest increasing subsequence in a random permutation, then $L_N\sim 2\sqrt{N}$ as $N\to\infty$. We give a new proof of this result using a connection with a certain unitary random matrix model. The asymptotic formula is directly related to a third order phase transition in this model found by Gross and Witten.

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Published 1 January 1998