On breadth-first constructions of scaling limits of random graphs and random unicellular maps
RANDOM STRUCTURES & ALGORITHMS(2022)
摘要
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 Erdos-Renyi 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.
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关键词
breadth-first construction, continuum random trees, critical random graphs, depth-first construction, Erdos-Renyi random graph, Gromov-Hausdorff distance, scaling limit, unicellular maps
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