Temperature and Disorder Chaos in Low Dimensional Directed Paths

TL;DR

This paper provides an exact analysis of how low-dimensional directed paths respond to temperature and potential variations, revealing scaling behaviors and physical mechanisms behind decorrelation and residual correlations.

Contribution

It offers an exact calculation of response behaviors in 1+ε dimensional directed paths, clarifying the physical mechanisms and confirming heuristic predictions.

Findings

Overlap length matches heuristic predictions

Temperature dependence aligns with scaling and numerical results

Residual correlations decay algebraically beyond overlap length

Abstract

The responses of a $1+\epsilon $ dimensional directed path to temperature and to potential variations are calculated exactly, and are governed by the same scaling form. The short scale decorrelation (strong correlation regime) leads to the overlap length predicted by heuristic approaches; its temperature dependence and large absolute value agree with scaling and numerical observations. Beyond the overlap length (weak correlation regime), the correlation decays algebraically. A clear physical mechanism explains the behavior in each case: the initial decorrelation is due to `fragile droplets,' which contribute to the entropy fluctuations as $\sqrt{T}$, while the residual correlation results from accidental intersections of otherwise uncorrelated configurations.