Random Lévy Looptrees and Lévy Maps

Abstract: What is the analogue of L\'evy processes for random surfaces? Motivated by scaling limits of random planar maps in random geometry, we introduce and study L\'evy looptrees and L\'evy maps. They are defined using excursions of general L\'evy processes with no negative jump and extend the known stable looptrees and stable maps, associated with stable processes. We compute in particular their fractal dimensions in terms of the upper and lower Blumenthal-Getoor exponents of the coding L\'evy process. In a second part, we consider excursions of stable processes with a drift and prove that the corresponding looptrees interpolate continuously as the drift varies from $-\infty$ to $\infty$, or as the self-similarity index varies from $1$ to $2$, between a circle and the Brownian tree, whereas the corresponding maps interpolate between the Brownian tree and the Brownian sphere.