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Cycle affinity and winding localize eigenvalues of Markov generators

  • Artemy Kolchinsky
  • Naruo Ohga
  • Sosuke Ito
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Abstract

The complex eigenvalues of Markov generators govern oscillatory properties of relaxation, autocorrelation, and linear response. We show that these eigenvalues are localized by nonequilibrium cycles of the generator, thus revealing a fundamental tradeoff between thermodynamic driving, oscillation, and decay of eigenmodes. Specifically, we prove that each complex eigenvalue is confined to a region determined by the cycle affinity and the eigenvector ``winding number'' of some nonequilibrium cycle. In discrete and continuous unicyclic systems, we also demonstrate that the winding number is determined by the ordered eigenvalue index, yielding new thermodynamic bounds on the slowest and fastest relaxation modes. In discrete multicyclic systems, our approach unifies and extends several previous inequalities and proves the Uhl--Seifert ellipse conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{kolchinsky2026cycle,
  title = {{Cycle affinity and winding localize eigenvalues of Markov generators}},
  author = {Artemy Kolchinsky and Naruo Ohga and Sosuke Ito},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.15884},
  eprint = {2605.15884},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.15884},
}
RIS
TY  - UNPB
TI  - Cycle affinity and winding localize eigenvalues of Markov generators
AU  - Artemy Kolchinsky
AU  - Naruo Ohga
AU  - Sosuke Ito
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2605.15884
UR  - https://arxiv.org/abs/2605.15884
AB  - The complex eigenvalues of Markov generators govern oscillatory properties of relaxation, autocorrelation, and linear response. We show that these eigenvalues are localized by nonequilibrium cycles of the generator, thus revealing a fundamental tradeoff between thermodynamic driving, oscillation, and decay of eigenmodes. Specifically, we prove that each complex eigenvalue is confined to a region determined by the cycle affinity and the eigenvector ``winding number'' of some nonequilibrium cycle. In discrete and continuous unicyclic systems, we also demonstrate that the winding number is determined by the ordered eigenvalue index, yielding new thermodynamic bounds on the slowest and fastest relaxation modes. In discrete multicyclic systems, our approach unifies and extends several previous inequalities and proves the Uhl--Seifert ellipse conjecture.
ER  -