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

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 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 λ_{1}(t), the fluctuations of lnλ_{1}(t) exhibit an intermediate t^{1/3} regime, while the standardized distribution remains distinct from the canonical benchmark laws over the studied time range.