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On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words

Craig A. Tracy·Harold Widom
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Abstract

We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N -> infinity limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices of trace zero.

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Craig A. Tracy, Harold Widom. 1999-07-01. On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words. https://doi.org/10.1007/pl00008763

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