Double-scaled SYK from boundary metrics of planar maps

Abstract

The enumeration of planar maps with control on the boundary metric, i.e. the pseudometric induced on the outer face of the map by its bulk graph distance metric, is a difficult problem in general. However, we show that for a family of bipartite planar map models with special q-deformed face weights that arise in the physics context of the double-scaled Sachdev-Ye-Kitaev model (DSSYK) the enumeration admits a very simple answer. Encoding the boundary metric of a bipartite planar map by its so-called geodesic chord diagram, we prove that the weighted enumeration depends only on the crossing number of the chord diagram. At fixed perimeter, the induced law of the geodesic chord diagram in these planar map models coincides exactly with the chord diagram representation of the DSSYK model.