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March 2019 Component sizes for large quantum Erdős–Rényi graph near criticality
Amir Dembo, Anna Levit, Sreekar Vadlamani
Ann. Probab. 47(2): 1185-1219 (March 2019). DOI: 10.1214/17-AOP1209


The $N$ vertices of a quantum random graph are each a circle independently punctured at Poisson points of arrivals, with parallel connections derived through for each pair of these punctured circles by yet another independent Poisson process. Considering these graphs at their critical parameters, we show that the joint law of the rescaled by $N^{2/3}$ and ordered sizes of their connected components, converges to that of the ordered lengths of excursions above zero for a reflected Brownian motion with drift. Thereby, this work forms the first example of an inhomogeneous random graph, beyond the case of effectively rank-1 models, which is rigorously shown to be in the Erdős–Rényi graphs universality class in terms of Aldous’s results.


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Amir Dembo. Anna Levit. Sreekar Vadlamani. "Component sizes for large quantum Erdős–Rényi graph near criticality." Ann. Probab. 47 (2) 1185 - 1219, March 2019.


Received: 1 June 2015; Revised: 1 April 2017; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053568
MathSciNet: MR3916946
Digital Object Identifier: 10.1214/17-AOP1209

Primary: 05C80 , 60F17 , 82B10

Keywords: Brownian excursions , critical point , Quantum random graphs , scaling limits , weak convergence

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 2 • March 2019
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