# Optimal paths on the road network as directed polymers

## TL;DR

This paper studies the statistical properties of shortest and fastest paths on road networks, revealing their directed nature at large scales and differences from directed polymers in random media due to correlations and local deviations.

## Contribution

It introduces a novel analysis of road network paths, comparing their scaling behaviors to directed polymers and highlighting the impact of correlations and local path deviations.

## Key findings

- Optimal paths are approximately directed at large scales.
- Scaling behaviors of fluctuations resemble directed polymers but differ in exponents.
- Short-scale paths exhibit non-directed, circuitous excursions with fat-tailed distributions.

## Abstract

We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. To a good approximation, these optimal paths are found to be directed in that their lengths (at large scales) are linearly proportional to the absolute distance between them. This motivates comparisons to universal features of directed polymers in random media. There are similarities in scalings of fluctuations in length/time and transverse wanderings, but also important distinctions in the scaling exponents, likely due to long-range correlations in geographic and man-made features. At short scales the optimal paths are not directed due to circuitous excursions governed by a fat-tailed (power-law) probability distribution.