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We consider quadrangulations with a boundary and derive explicit expressionsfor the generating functions of these maps with either a marked vertex at aprescribed distance from the boundary, or two boundary vertices at a prescribedmutual distance in the map. For large maps, this yields explicit formulas forthe bulk-boundary and boundary-boundary correlators in the various encounteredscaling regimes: a small boundary, a dense boundary and a critical boundaryregime. The critical boundary regime is characterized by a one-parameter familyof scaling functions interpolating between the Brownian map and the BrownianContinuum Random Tree. We discuss the cases of both generic and self-avoidingboundaries, which are shown to share the same universal scaling limit. Wefinally address the question of the bulk-loop distance statistics in thecontext of planar quadrangulations equipped with a self-avoiding loop. Hereagain, a new family of scaling functions describing critical loops isdiscovered.

Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop