Dresden 2014 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 36: Brownian Motion and Transport
DY 36.6: Vortrag
Donnerstag, 3. April 2014, 16:30–16:45, ZEU 160
Convex Hulls of Random Walks: Large-Deviation Properties — •Gunnar Claußen1, Satya N. Majumdar2, and Alexander K. Hartmann1 — 1Institut für Physik, Carl von Ossietzky Universität Oldenburg — 2Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud
We numerically consider two-dimensional time-discrete random walks of length T represented through sets {δi→} of vectors denoting the steps i ≤ T. Motivated by modeling animal home ranges [1], we are interested in area A and perimeter L of the convex hull over the trajectory x→(t) of this walk. As previous studies determined the analytical averages ⟨ A ⟩ and ⟨ L ⟩ [2], we aim at the according distributions P(A) and P(L) by application of a Monte Carlo method [3] which allows us to sample within ranges of particularly rare values of A and L, leading to probabilities such as 10−300. The resulting distributions can be compared with respect to their scaling behaviour, their rate functions Φ(s) = −T−1 · logP(s) (with s = A/Amax or s = L/Lmax, respectively) and standard analytical distribution functions p(A) and p(L) like the Gumbel distribution. Our analyses of these properties resulted in the obtaining of asymptotic values for the corresponding parameters and exponents for T → ∞. A multitude of walk lengths T, open and closed walks and various types of step displacements δi→ have been covered by our simulations.
L. Giuggioli et al., PLoS Comput. Biol. 7 (2011) e1002008
S.N. Majumdar et al., J. Stat. Phys. 138 (2010) 995-1009
A.K. Hartmann, Eur. Phys. J. B 84 (2011) 627-634