Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study

Abstract

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 $T$.

9 pages, 8 figures, 3 tables

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• Maps of random walks on complex networks reveal community structure
• The large deviation approach to statistical mechanics
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• Exact distribution of the maximal height of p vicious walkers
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