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

The distribution of the hypervolume V and surface partial derivative 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.