Random Lévy looptrees and Lévy maps

What is the analogue of Lévy processes for random surfaces? Motivated by scaling limits of random planar maps in random geometry, we introduce and study Lévy looptrees and Lévy maps. They are defined using excursions of general Lévy 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 Lévy process. The case where the Lévy process is a stable process with a drift naturally appears in the context of stable-Boltzmann planar maps conditioned on having a fixed number of vertices and edges in a near-critical regime.