Convex hulls of random walks in higher dimensions: A large-deviation study

Hendrik Schawe ¹ Alexander K. Hartmann ¹ Satya N. Majumdar ²

Physical Review E , American Physical Society (APS), 2017, 96 (6), 〈10.1103/PhysRevE.96.062101〉

The distribution of the hypervolume $V$ and surface $\partial V$ of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than $P = 10^{-1000}$ to estimate large deviation properties. For arbitrary dimensions and large walk lengths $T$, we suggest a scaling behavior of the distribution with the length of the walk $T$ similar to the two-dimensional case, and behavior of the distributions in the tails. We underpin both with numerical data in $d=3$ and $d=4$ dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large $T$.

• 1. University of Oldenburg
• 2. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques