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Directed polymer transfer matrices as a unified generator of distinct one-point fluctuation laws

Sen Mu1, Abbas Ali Saberi2,1,*, Roderich Moessner1, and Mehran Kardar3

  • *Contact author: asaberi@constructor.university
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Phys. Rev. E 114, 024116 – Published 10 August, 2026

DOI: https://doi.org/10.1103/cf3y-67f4

Abstract

We numerically revisit the transfer-matrix formulation of directed polymers in random media and show that a common finite-dimensional framework organizes the canonical one-point fluctuation laws in (1+1) dimensions. For a fixed realization of the bulk disorder, full-space partition functions are obtained from the same time-ordered product W(t) through endpoint contractions or a Brownian-weighted initial vector, while the half-space construction modifies only the transfer rule at the absorbing boundary. These choices yield distributions consistent with the standard KPZ subclasses: Tracy-Widom GUE for point-to-point geometry, Tracy-Widom GOE for point-to-line geometry, Tracy-Widom GSE for half-space point-to-point geometry, and Baik-Rains for the stationary line-to-point construction. In all four cases, the free-energy fluctuations grow as t1/3, and the low-order cumulants approach the corresponding universal benchmarks. The matrix-product formulation also provides access to intrinsic spectral observables. For the leading eigenvalue λ1(t), the fluctuations of lnλ1(t) exhibit an intermediate t1/3 regime, while the standardized distribution remains distinct from the canonical benchmark laws over the studied time range.

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Phys. Rev. E 114, 024116– Published 10 August, 2026
  • Received 15 March 2026
  • Revised 17 July 2026
  • Accepted 24 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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