Kardar-Parisi-Zhang Universality in First-Passage Percolation: the Role of Geodesic Degeneracy

TL;DR

This study demonstrates that first-passage percolation models on different lattices exhibit KPZ universality in their fluctuation behavior, with geodesic degeneracy influencing pre-asymptotic dynamics.

Contribution

It reveals how lattice topology and geodesic degeneracy affect the scaling and fluctuation properties in FPP, extending KPZ universality understanding.

Findings

Fluctuations follow KPZ universality regardless of lattice or distribution.

Pre-asymptotic behavior depends on geodesic degeneracy.

Geodesic degeneracy impacts anisotropic growth in regular lattices.

Abstract

We have characterized the scaling behavior of the first-passage percolation (FPP) model on two types of discrete networks, the regular square lattice and the disordered Delaunay lattice, thereby addressing the effect of the underlying topology. Several distribution functions for the link-times were considered. The asymptotic behavior of the fluctuations for both the minimal arrival time and the lateral deviation of the geodesic path are in perfect agreement with the Kardar-Parisi-Zhang (KPZ) universality class regardless of the type of the link-time distribution and of the lattice topology. Pre-asymptotic behavior, on the other hand, is found to depend on the uniqueness of geodesics in absence of disorder in the local crossing times, a topological property of lattice directions that we term geodesic degeneracy. This property has important consequences on the model, as for example the well-known anisotropic growth in regular lattices. In this work we provide a framework to understand its effect as well as to characterize its extent.