Abstract

Random planar maps with high degrees are expected to have scaling limits related to the conformal loop ensemble (CLE) equipped with an independent Liouville quantum gravity (LQG). In the dilute case, where informally the degrees have finite expectations, Bertoin, Budd, Curien and Kortchemski established the scaling limit of the distances to the root. However, the scaling limit does not have an interpretation as a distance from the loops to the boundary in terms of LQG . I will focus on the critical case where the probability that a vertex has degree k is of order k^-2. In this case, the distances from the root to the high degree vertices satisfy a scaling limit, and this scaling limit is related to a quantum distance to the boundary on a CLE -decorated critical LQG introduced by Aru, Holden, Powell and Sun. Finally, I will draw a connection with a conformally invariant distance to the boundary on the CLE from Werner and Wu.