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On the distributions of the lengths of the longest monotone subsequences in random words
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  • Published: March 2001

On the distributions of the lengths of the longest monotone subsequences in random words

  • Craig A. Tracy1 &
  • Harold Widom2 

Probability Theory and Related Fields volume 119, pages 350–380 (2001)Cite this article

  • 197 Accesses

  • 57 Citations

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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. (In the limit as k→∞ these become the corresponding distributions for permutations on N 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 unction gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N→∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k×k hermitian matrices of trace zero.

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Authors and Affiliations

  1. Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA. e-mail: tracy@itd.ucdavis.edu, , , , , , US

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, CA 95064, USA. e-mail: widom@math.ucsc.edu, , , , , , US

    Harold Widom

Authors
  1. Craig A. Tracy
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  2. Harold Widom
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Received: 9 September 1999 / Revised version: 24 May 2000 / Published online: 24 January 2001

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Tracy, C., Widom, H. On the distributions of the lengths of the longest monotone subsequences in random words. Probab Theory Relat Fields 119, 350–380 (2001). https://doi.org/10.1007/PL00008763

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  • Issue Date: March 2001

  • DOI: https://doi.org/10.1007/PL00008763

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  • Mathematics Subject Classification (2000): 60C05, 60F05, 05A16
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