  A global picture of a random particle movement is given by the convex hull of
the visited points. We obtained numerically the probability distributions of
the volume and surface of the convex hulls of a selection of three types of
self-avoiding random walks, namely the classical Self-Avoiding Walk, the
Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a
comprehensive description of the measured random quantities, we applied
sophisticated large-deviation techniques, which allowed us to obtain the
distributions over a large range of the support down to probabilities far
smaller than $P = 10^{-100}$ . We give an approximate closed form of the
so-called large-deviation rate function $\Phi$ which generalizes above the
upper critical dimension to the previously studied case of the standard random
walk. Further we show correlations between the two observables also in the
limits of atypical large or small values.
