On the Limiting Shape of Young Diagrams Associated With Markov Random Words

Details On the Limiting Shape of Young Diagrams Associated With Markov Random Words On the Limiting Shape of Young Diagrams Associated With Markov Random Words

2011-10-20

arxiv.org/abs/1110.4570v1

Mathematics

Probability

77 pages, 6 figures. arXiv admin note: significant text overlap with arXiv:0810.2982

Scientific paper

Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space an ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by showing that, under a cyclic condition, a spectral characterization of the Markov transition matrix delineates precisely when the limiting shape is the spectrum of the traceless GUE. For $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. However, this is no longer true for $m \ge 4$. In arbitrary dimension, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices.

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