Skip to content
DraconiticResearch
Search Sources API
203,948 studies31 sourcesUpdated 11 min ago
PreprintarXiv licencearXiv

Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle

  • Jiro Akahori
  • Riki Kitano
  • Norio Konno
  • Iwao Sato
Open at sourcePDFDOICite

Abstract

We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.

Related

  1. Measuring a Quantum System Thanks to the Correspondence Principle
    Marco Falconi, Clotilde Fermanian Kammerer, Raphaël Gautier
    arXivPreprintarXiv licenceFull textMathematical physicsarXiv
  2. Local theta correspondence and generalized Shalika models
    Chong Zhang
    arXivPreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv
  3. Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current
    Hassan Jolany
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  4. A geometric Jacquet--Langlands correspondence for Shimura varieties
    Pol van Hoften, Jack Sempliner
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Generalized reflection complexity
    Adam Blažek, Lubomíra Dvořáková, Edita Pelantová
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{akahori2026walkzeta,
  title = {{Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle}},
  author = {Jiro Akahori and Riki Kitano and Norio Konno and Iwao Sato},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05393},
  eprint = {2610.05393},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05393},
}
RIS
TY  - UNPB
TI  - Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle
AU  - Jiro Akahori
AU  - Riki Kitano
AU  - Norio Konno
AU  - Iwao Sato
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05393
UR  - https://arxiv.org/abs/2610.05393
AB  - We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.
ER  -