Universality and time-scale invariance for the shape of planar L\'evy processes

For a broad class of planar Markov processes, viz. L\'evy processes satisfying certain conditions (valid \textit{eg} in the case of Brownian motion and L\'evy flights), we establish an exact, universal formula describing the shape of the convex hull of sample paths. We show indeed that the average number of edges joining paths' points separated by a time-lapse $\Delta \tau \in \left[\Delta \tau _1, \Delta \tau_2\right]$ is equal to $2\ln \left(\Delta \tau_2 / \Delta \tau_1 \right)$, regardless of the specific distribution of the process's increments and regardless of its total duration $T$. The formula also exhibits invariance when the time scale is multiplied by any constant. Apart from its theoretical importance, our result provides new insights regarding the shape of two-dimensional objects modelled by stochastic processes' sample paths (\textit{eg} polymer chains): in particular for a total time (or parameter) duration $T$, the average number of edges on the convex hull ("cut off" to discard edges joining points separated by a time-lapse shorter than some $\Delta \tau < T$) will be given by $2 \ln \left(\frac{T}{\Delta \tau}\right)$. Thus it will only grow logarithmically, rather than at some higher pace.