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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 3: Financial Markets, Risk Management and Stochastic Processes

SOE 3.3: Vortrag

Montag, 16. März 2020, 11:45–12:00, GÖR 226

The north pole problem – Explaining asymmetry in repeated random rotations — Malte Schröder and •Marc Timme — Chair for Network Dynamics, Institute for Theoretical Physics and Center for Advancing Electronics Dresden (cfaed), TU Dresden

Imagine rotating a globe uniformly at random by applying an isotropic rotation R, resulting equally likely in any orientation of the globe. By definition, a given point like the north pole is mapped to any other point on the sphere with equal probability. Repeating the same random rotation twice (applying R²), the north pole ends up on the northern hemisphere with 71% probability. This north-pole-problem was originally observed numerically [Marzetta et al., IEEE Trans. Inform. Theory 48 (2002)] and subsequently mathematically analyzed using measure theory [Eaton and Muirhead, Stat. Probab. Lett. 79 (2009)]. Here we provide an intuitive, geometric explanation for this phenomenon [Schröder and Timme, Phys. Rev. Res. 1 (2019)] by decomposing the isotropic rotation into two sequential elementary rotations and explicitly following the action of the rotation. The same geometric argument generalizes to higher dimensions d>3 and also explains why the asymmetry is absent in d=2 dimensions, is strongest in d=3 dimensions and disappears again as d → ∞.