Directed Polymer Transfer Matrices as a Unified Generator of Distinct One-Point Fluctuation Laws

Abstract

We revisit the transfer-matrix approach to directed polymers in random media and show that a single ensemble of random transfer-matrix products provides a unified realization of the canonical one-point fluctuation laws in $(1+1)$ dimensions. For a fixed disorder realization, the polymer partition function is obtained as a contraction of the same product matrix $W(t)$, and different contractions reproduce the standard KPZ subclasses: Tracy-Widom GUE (point-to-point), GOE (point-to-line), GSE (half-space point-to-point), and Baik-Rains (stationary line-to-point). In each case, we observe $t^{1/3}$ free-energy fluctuation growth and convergence of standardized distributions with low-order cumulants close to the corresponding universal benchmarks. Viewing geometry-dependent subclasses as projections of a single matrix-product ensemble naturally suggests additional observables intrinsic to $W(t)$. As an example, we examine the leading eigenvalue $λ_1(t)$ whose logarithm exhibits $t^{1/3}$ scaling, while its standardized statistics remain distinct from the canonical Tracy-Widom laws within the accessible range. This transfer-matrix perspective thus organizes known KPZ one-point subclasses within a finite-dimensional matrix framework and highlights matrix-level fluctuation observables beyond geometry-selected universality classes.

Figures (8)

• Figure 1: Illustration of the random energies on the green bonds in the 6-vertex model representation considered for the directed polymer in random media.
• Figure 2: Standard deviation of the free energy, $\sigma[F(t)]$, as a function of time $t$ for the Brownian-weighted line-to-point, half-space point-to-point, point-to-point, and point-to-line boundary conditions, plotted on log-log scales. The black solid line indicates the power law $t^{1/3}$. The data are consistent with KPZ scaling of the free-energy fluctuations in all four cases. Model parameters are $N=128$, $\mu=0$, and $\sigma=3$, with evolution up to $t=1024$ over $10^6$ disorder realizations.
• Figure 3: Statistics of the standardized free energy for the point-to-point boundary condition. Main: Distribution of the standardized free energy $\tilde{F}_{\rm pt\text{-}pt}$ at different times ($t=512,768,1024$), together with the standardized TW-GUE distribution. Inset: Time evolution of the skewness and excess kurtosis of the standardized free energy. The black dashed and red dot-dashed lines denote the corresponding TW-GUE values, 0.22 and 0.09, respectively. Model parameters are $N=128$, $\mu=0$, $\sigma=3$, and $x_0=64$ with evolution up to $t=1024$ over $10^6$ disorder realizations.
• Figure 4: Statistics of the standardized free energy for the point-to-line boundary condition. Main: Distribution of the standardized free energy $\tilde{F}_{\rm pt\text{-}line}$ at different times ($t=512,768,1024$), together with the standardized TW-GOE distribution. Inset: Time evolution of the skewness and excess kurtosis of the standardized free energy. The black dashed and red dot-dashed lines denote the corresponding TW-GOE values, 0.29 and 0.17, respectively. Model parameters are $N=128$, $\mu=0$, $\sigma=3$ and $x_0=64$ with evolution up to $t=1024$ over $10^6$ disorder realizations.
• Figure 5: Statistics of the standardized free energy for the point-to-point boundary condition in half space. Main: Distribution of the standardized free energy $\tilde{F}_{\rm hs\, pt\text{-}pt}$ at different times ($t=512,768,1024$), together with the standardized TW-GSE distribution. Inset: Time evolution of the skewness and excess kurtosis of the standardized free energy. The black dashed and red dot-dashed lines denote the corresponding TW-GSE values, 0.16 and 0.04, respectively. Model parameters are $N=128$, $\mu=0$, $\sigma=3$ and $x_0=127$ with evolution up to $t=1024$ over $10^6$ disorder realizations.