Markovian Random Words, a Gaussian Random Matrix Model and Beyond
Abstract
After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian 3× 3 Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.
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BibTeX
@misc{houdre2026markovian,
title = {{Markovian Random Words, a Gaussian Random Matrix Model and Beyond}},
author = {Christian Houdré and Dorian Kidonakis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12950},
eprint = {2610.12950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12950},
}
RIS
TY - UNPB
TI - Markovian Random Words, a Gaussian Random Matrix Model and Beyond
AU - Christian Houdré
AU - Dorian Kidonakis
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12950
UR - https://arxiv.org/abs/2610.12950
AB - After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian 3× 3 Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.
ER -