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

TL;DR

This paper demonstrates that a single transfer-matrix ensemble can generate all major one-point fluctuation laws in (1+1)D directed polymers, unifying them within a matrix framework and revealing new observables.

Contribution

It introduces a unified matrix-based approach to reproduce and analyze all key KPZ fluctuation subclasses and their universal properties in directed polymers.

Findings

Reproduces Tracy-Widom GUE, GOE, GSE, and Baik-Rains distributions from a single matrix ensemble.

Shows $t^{1/3}$ fluctuation scaling and convergence to universal distributions.

Identifies matrix-level observables, like the leading eigenvalue, with distinct fluctuation properties.

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 $\lambda_1(t)$ 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.