On breadth‐first constructions of scaling limits of random graphs and random unicellular maps

Abstract

We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map of a given genus that start with a suitably tilted Brownian continuum random tree and make “horizontal” point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth-first construction of a finite connected graph. In particular, this yields a breadth-first construction of the scaling limit of the critical Erdős–Rényi random graph which answers a question posed by Addario-Berry, Broutin, and Goldschmidt. As a consequence of this breadth-first construction, we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions.