Topology and the Kardar-Parisi-Zhang universality class

TL;DR

This paper investigates how the topology of the background space influences the KPZ universality class by analyzing growth on disordered 2D manifolds, revealing a connection between topology and Tracy-Widom fluctuation distributions.

Contribution

It demonstrates that the background topology determines the Tracy-Widom distribution type (GOE or GUE) governing fluctuations in KPZ growth models.

Findings

Radial fluctuations follow TW-GOE on cylindrical geometries.

Fluctuations follow TW-GUE on conical geometries with non-zero aperture.

Topological argument links background topology to fluctuation statistics.

Abstract

We study the role of the topology of the background space on the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class. To do so, we study the growth of balls on disordered 2D manifolds with random Riemannian metrics, generated by introducing random perturbations to a base manifold. As base manifolds we consider cones of different aperture angles $\theta$, including the limiting cases of a cylinder ($\theta=0$, which corresponds to an interface with periodic boundary conditions) and a plane ($\theta=\pi/2$, which corresponds to an interface with circular geometry). We obtain that in the former case the radial fluctuations of the ball boundaries follow the Tracy-Widom (TW) distribution of the largest eigenvalue of random matrices in the Gaussian orthogonal ensemble (TW-GOE), while on cones with any aperture angle $\theta\neq 0$ fluctuations correspond to the TW-GUE distribution related with the Gaussian unitary ensemble. We provide a topological argument to justify the relevance of TW-GUE statistics for cones, and state a conjecture which relates the KPZ universality subclass with the background topology.