Universal properties of shortest paths in isotropically correlated random potentials

TL;DR

This study investigates how isotropic correlations in random potentials affect the universal scaling properties of shortest paths in lattices, revealing the relevance of correlations and confirming key exponent relations.

Contribution

It demonstrates that correlations decaying slower than 1/r influence the scaling exponents and confirms the exponent relation 2nu-omega=1 in 2D, extending understanding of path universality.

Findings

Correlations decay slower than 1/r are relevant in 2D and 3D.

Exponent relation 2nu-omega=1 holds in 2D with correlations.

Overhangs are irrelevant even with strong correlations.

Abstract

We consider the optimal paths in a $d$-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential. We study how the universal scaling exponents, the roughness and the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results using Dijkstra's algorithm to determine the optimal path in directed as well as undirected lattices indicate that the correlations become relevant if they decay with distance slower than 1/r in d=2 and 3. We show that the exponent relation 2nu-omega=1 holds at least in d=2 even in case of correlations. Both in two and three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations.