Scaling limits of random planar maps with a unique large face (Q2352748)

This paper considers random planar maps, which have been a subject of much interest in recent years. There is well-known to be a link between planar maps and certain labelled trees (the so-called BFG bijection), which leads to a further connection with so-called simply generated trees. In the evolution of simply generated trees, there is known to be a phase where a unique vertex of high degree (proportional to the number of vertices) emerges as the tree gets bigger -- this phenomenon is sometimes referred to as condensation. The paper under review studies the scaling limit of random bipartite planar maps, with a weight attached to each face, corresponding to the condensation regime for trees: this leads to a unique large face appearing in the planar map. Further topics considered include the limit of the profile of distances to a fixed vertex, after appropriate rescaling, converging to a limit described in terms of a Brownian excursion and the planar maps converging to Aldous' Brownian tree. Techniques, in addition to those implied above, include consideration of certain Galton-Watson trees, further results on which are provided in an appendix.