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

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

1999-04-09

arxiv.org/abs/math/9904042v3

Probab. Theory Relat. Fields 119 (2001), 350-380

Mathematics

Combinatorics

30 pages, revised version corrects an error in the statement of Theorem 4

Scientific paper

10.1007/PL00008763

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 Painlev\'e 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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