Markov chains at the onset of non-reversibility
Abstract
For a one-dimensional path graph and a lifted path graph constructed from a duplication of each of its sites, we study how a reversible Markov chain can be perturbed and gradually driven into non-reversibility. The reversible Markov chain has a transition matrix that is diagonalizable and features real-valued eigenvalues and eigenvectors. The left and right eigenvectors form a biorthogonal system. We discuss in concrete examples how the transition matrix of a non-reversible Markov chain may be diagonalizable or non-diagonalizable, and it may have real eigenvalues and complex-conjugate pairs. For a number of steady states (flat, square-wave, wedge, V-shape), we compute eigenvalue spectra on both graphs and discuss the speedup that can be achieved through lifting. We develop a Green's matrix formalism, which we use to compute Kemeny times and mean first-passage times, and which provides valuable information and allows us to interpret the results for the characteristic times.
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BibTeX
@misc{robichon2026markov,
title = {{Markov chains at the onset of non-reversibility}},
author = {Gustave Robichon and Cecile Monthus and Werner Krauth},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05183},
eprint = {2609.05183},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05183},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Markov chains at the onset of non-reversibility
AU - Gustave Robichon
AU - Cecile Monthus
AU - Werner Krauth
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05183
UR - https://arxiv.org/abs/2609.05183
AB - For a one-dimensional path graph and a lifted path graph constructed from a duplication of each of its sites, we study how a reversible Markov chain can be perturbed and gradually driven into non-reversibility. The reversible Markov chain has a transition matrix that is diagonalizable and features real-valued eigenvalues and eigenvectors. The left and right eigenvectors form a biorthogonal system. We discuss in concrete examples how the transition matrix of a non-reversible Markov chain may be diagonalizable or non-diagonalizable, and it may have real eigenvalues and complex-conjugate pairs. For a number of steady states (flat, square-wave, wedge, V-shape), we compute eigenvalue spectra on both graphs and discuss the speedup that can be achieved through lifting. We develop a Green's matrix formalism, which we use to compute Kemeny times and mean first-passage times, and which provides valuable information and allows us to interpret the results for the characteristic times.
ER -