Topology and the Kardar-Parisi-Zhang universality class

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.