Stiff directed lines in random media

TL;DR

This paper explores the localization transition of stiff directed lines in random media, combining theoretical analysis and numerical simulations to understand the critical behavior and its relation to directed lines under tension.

Contribution

It introduces a novel relation between the localization of stiff directed lines and directed lines under tension, supported by multifractal analysis and free energy distribution comparisons.

Findings

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

Numerical transfer matrix calculations confirm the transition in 1+1 dimensions.

Disorder reduces the persistence length of stiff directed lines.

Abstract

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. We apply perturbative arguments, Flory scaling arguments, a variational replica calculation, and functional renormalization to 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 in detail. 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. We support the proposed relation to directed lines via multifractal analysis revealing an analogous Anderson transition-like scenario and a matching correlation length exponent. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.