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On breadth‐first constructions of scaling limits of random graphs and random unicellular maps
Author(s) -
Miermont Grégory,
Sen Sanchayan
Publication year - 2022
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21076
Subject(s) - scaling limit , mathematics , random graph , scaling , random tree , brownian motion , limit (mathematics) , combinatorics , graph , point process , planar graph , discrete mathematics , statistical physics , mathematical analysis , geometry , physics , computer science , statistics , artificial intelligence , robot , motion planning
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.