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 $\hat{I}{»}_1(t)$, the fluctuations of $\ln \hat{I}{»}_1(t)$ exhibit an intermediate $t^{1/3}$ regime, while the standardized distribution remains distinct from the canonical benchmark laws over the studied time range.

10 pages, 9 figures,