Geodesics on random surfaces

Aula 110/111. Date: 2019-07-12 02:15 PM – 02:30 PM

Consider finding the shortest walking path between two fixed points of a given terrain, and then drawing this path on a map. When the inhomogeneities of the terrain can be considered to be random and presenting only short range correlations, the obtained geodesic presents fractal properties, with transverse fluctuations scaling as L^2/3, with L the Euclidean distance between the two points [Silvia N Santalla et al 2015 New J. Phys. 17 033018].

In this work we consider the statistical properties of geodesics and balls (isochrone curves) on random surfaces, relating them to the problem of first passage percolation (FPP) and the celebrated universality class of Kardar, Parisi and Zhang (KPZ). Balls are no longer circumferences, and their roughness scales as R^1/3. Moreover, the full histogram of radial fluctuations can be found to correspond to the Tracy-Widom distribution associated to the gaussian unitary ensemble (TW-GUE) of random matrix theory. We will describe how the underlying topology of the manifold can change this distribution, and the effects of regular and random discretizations (e.g. into a Delaunay lattice) [Pedro Córdoba-Torres et al J. Stat. Mech. (2018) 063212; Silvia N Santalla et al J. Stat. Mech. (2017) 023201].