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.
arXiv Mathematics e-prints
- Pub Date:
- April 1999
- Mathematics - Combinatorics;
- Mathematics - Probability;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- 30 pages, revised version corrects an error in the statement of Theorem 4