Topology and the Kardar–Parisi–Zhang universality class

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 θ, including the limiting cases of a cylinder (, which corresponds to an interface with periodic boundary conditions) and a plane (, which corresponds to an interface with circular geometry). We obtain that in the former case the radial fluctuations of the ball boundaries approach 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 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.