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Asymptotics for the Length of the Longest Increasing Subsequence of a Binary Markov Random Word

  • Christian HoudréEmail author
  • Trevis J. Litherland
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

Let (X n ) n ≥ 0 be an irreducible, aperiodic, and homogeneous binary Markov chain and let LI n be the length of the longest (weakly) increasing subsequence of (X k )1 ≤ k ≤ n . Using combinatorial constructions and weak invariance principles, we present elementary arguments leading to a new proof that (after proper centering and scaling) the limiting law of LI n is the maximal eigenvalue of a 2 ×2 Gaussian random matrix. In fact, the limiting shape of the RSK Young diagrams associated with the binary Markov random word is the spectrum of this random matrix.

Keywords

Longest increasing subsequence Markov chains Functional central limit theorem Random matrices Young diagrams 

Notes

Acknowledgements

Research supported in part by the NSA Grant H98230-09-1-0017.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics, Georgia Institute of TechnologyAtlantaUSA

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