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

Abstract

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulae for the bulk–boundary and boundary–boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian continuum random tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk–loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.