Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220675736