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On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words | | Craig A. Tracy
; Harold Widom
; | | Date: |
10 Apr 1999 | | Journal: | Probab. Theory Relat. Fields 119 (2001), 350-380 | | Subject: | Combinatorics; Probability; Exactly Solvable and Integrable Systems MSC-class: 05A15; 47B35; 60C05; 82B23 | math.CO math.PR nlin.SI solv-int | | 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. | | Source: | arXiv, math.CO/9904042 | | Services: | Forum | Review | PDF | Favorites | |
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