KPZ modes in $d$-dimensional directed polymers

TL;DR

This paper introduces a stochastic lattice model for directed polymers in higher dimensions, demonstrating that their stationary fluctuations are governed by KPZ or diffusive behavior, with modes that are largely independent and follow specific coupling conditions.

Contribution

It establishes a new lattice model for directed polymers in multiple dimensions and characterizes the nature of their stationary fluctuations using mode coupling theory.

Findings

Stationary fluctuations are KPZ type or diffusive in any dimension.

Modes are pure with subleading couplings, excluding modified KPZ or Lévy fluctuations.

Mode-coupling matrices satisfy the trilinear condition.

Abstract

We define a stochastic lattice model for a fluctuating directed polymer in $d\geq 2$ dimensions. This model can be alternatively interpreted as a fluctuating random path in 2 dimensions, or a one-dimensional asymmetric simple exclusion process with $d-1$ conserved species of particles. The deterministic large dynamics of the directed polymer are shown to be given by a system of coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using non-linear fluctuating hydrodynamics and mode coupling theory we argue that stationary fluctuations in any dimension $d$ can only be of KPZ type or diffusive. The modes are pure in the sense that there are only subleading couplings to other modes, thus excluding the occurrence of modified KPZ-fluctuations or L\'evy-type fluctuations which are common for more than one conservation law. The mode-coupling matrices are shown to satisfy the so-called trilinear condition.