Component sizes for large quantum erdos renyi graph near criticality

TL;DR

This paper analyzes the size distribution of connected components in a large quantum Erdős-Rényi graph near criticality, showing convergence to a reflected Brownian motion with drift, and establishing universality class results.

Contribution

It introduces a new quantum random graph model at criticality and proves its component size distribution converges to a known stochastic process, extending universality results.

Findings

Component sizes scaled by N^{2/3} converge to excursions of reflected Brownian motion.

First example of an inhomogeneous random graph beyond rank-1 models in this universality class.

Rigorous proof connecting quantum graph component sizes to classical stochastic processes.

Abstract

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 re-scaled 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 Erd\H{o}s-R\'{e}nyi graphs universality class in terms of Aldous's results.