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 halfspace 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 t 1/3, 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 ln λ1(t) exhibit an intermediate t 1/3 regime, while the standardized distribution remains distinct from the canonical benchmark laws over the studied time range.
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Mehran Kardar |