Dresden 2017 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 60: Posters - Brownian Motion, Noise
DY 60.1: Poster
Donnerstag, 23. März 2017, 17:00–19:30, P1A
Convex Hulls of Self-Avoiding Random Walks: 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 different types of random walks, i.e., the smallest convex polygon enclosing the trajectory of a random walk with T steps. While the convex hulls are of interest from a pure mathematical point of view, they are also considered as a model to estimate animal territories. The convex hulls of normal random walks are decently studied [1, 2], but very little is known about the convex hulls of other important types of random walks as the Self-Avoiding Random Walk (SAW) and the Loop-Erased Random Walk (LERW). Using Markov chain Monte Carlo sampling-techniques, we can study a large part of the support of the distributions of the area A or perimeter L of the convex hulls. This enables us to reach probability densities below P(A)=10−800 and scrutinize large-deviation properties. Similar to normal random walks, the probability densities show a universal scaling behavior dependent on the scaling exponent ν and the dimension of the observable (e.g., d=2 for the area A and d=1 for the perimeter L). Further, we determined the rate function Φ(·) = −1/T logP(·) which shows convergence to a limit shape for T → ∞.
[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)