Localization transition of stiff directed lines in random media

TL;DR

This paper studies the localization transition of stiff directed lines in random media, revealing a disorder-induced phase change in certain dimensions and linking it to broader models like the KPZ equation.

Contribution

It introduces a theoretical framework combining perturbative, Flory, and replica methods to analyze localization transitions in stiff directed lines, supported by numerical simulations.

Findings

Localization transition occurs for d>2/3 in 1+d dimensions.

Disorder reduces the persistence length of stiff directed lines.

A relation between stiff directed lines and tensioned directed lines in higher dimensions is proposed.

Abstract

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. Using perturbative arguments, Flory arguments, and a replica calculation, we show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for $d>2/3$. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang (KPZ) equation. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.