On breadth‐first constructions of scaling limits of random graphs and random unicellular maps. Issue 4 (7th January 2022) - Miermont, Grégory; Sen, Sanchayan
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.