Dresden 2017 – scientific programme
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DY: Fachverband Dynamik und Statistische Physik
DY 18: Brownian Motion / Noise (joint session DY/TT)
DY 18.1: Talk
Tuesday, March 21, 2017, 10:00–10:15, ZEU 147
Convex Hulls of Random Walks in High Dimensions: A Large-Deviation Study — •Hendrik Schawe1, Alexander K. Hartmann1, and Satya N. Majumdar2 — 1Institut für Physik, Carl von Ossietzky Universität Oldenburg — 2Laboratoire de Physique Théorique et Modèles Statistiques, Université de Paris-Sud
We study the convex hulls of random walks in high dimensions, i.e., the smallest convex polytope enclosing the trajectory of a random walk with T steps. While the convex hulls of two-dimensional random walks are decently studied [1, 2], very little is known about the convex hulls of random walks in d ≥ 3. Using Markov chain Monte Carlo sampling-techniques, we can study a large part of the support of the distributions of the volume V of the convex hulls or its surface ∂ V. This enables us to reach probability densities below P(A)=10−800 and scrutinize large-deviation properties. Similar to two-dimensional random walks, the probability densities show a universal scaling behavior dependent on the exponent ν = 0.5 and the effective dimension of the observable, i.e., deff=d for V and deff=d−1 for ∂ V. Further, we determined the rate function Φ(·) = −1/T logP(·) which shows convergence to a limit shape for T→∞, which seems to be a power law with an exponent only dependent on deff and ν.
[1] G. Claussen, A. K. Hartmann, and S. N. Majumdar, Phys. Rev. E 91, 052104 (2015); [2] T. Dewenter, G. Claussen, A. K. Hartmann, and S. N. Majumdar, Phys. Rev. E 94, 052120 (2016)