## The Annals of Probability

### Component sizes for large quantum Erdős–Rényi graph near criticality

#### 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 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 1185-1219.

Dates
Received: June 2015
Revised: April 2017
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171650

Digital Object Identifier
doi:10.1214/17-AOP1209

Mathematical Reviews number (MathSciNet)
MR3916946

Zentralblatt MATH identifier
07053568

#### Citation

Dembo, Amir; Levit, Anna; Vadlamani, Sreekar. Component sizes for large quantum Erdős–Rényi graph near criticality. Ann. Probab. 47 (2019), no. 2, 1185--1219. doi:10.1214/17-AOP1209. https://projecteuclid.org/euclid.aop/1551171650

#### References

• [1] Aizenman, M., Klein, A. and Newman, C. M. (1993). Percolation methods for disordered quantum Ising models, phase transitions. In Mathematics, Physics, Biology,... (R. Kotecky, ed.). 124 1–26. World Scientific, Singapore.
• [2] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.
• [3] Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2010). Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Probab. 15 1682–1703.
• [4] Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2012). Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab. 40 2299–2361.
• [5] Bollobás, B. (1984). The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.
• [6] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
• [7] Campanino, M., Klein, A. and Perez, J. F. (1991). Localization in the ground state of the Ising model with a random transverse field. Comm. Math. Phys. 135 499–515.
• [8] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 5 17–61.
• [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
• [10] Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford Univ. Press, New York.
• [11] Hartley, H. O. and David, H. A. (1954). Universal bounds for mean range and extreme observation. Ann. Math. Stat. 25 85–99.
• [12] Ioffe, D. (2009). Stochastic geometry of classical and quantum Ising models. In Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Math. 1970 87–127. Springer, Berlin.
• [13] Ioffe, D. and Levit, A. (2007). Long range order and giant components of quantum random graphs. Markov Process. Related Fields 13 469–492.
• [14] Janson, S. (2007). On a random graph related to quantum theory. Combin. Probab. Comput. 16 757–766.
• [15] Joseph, A. (2014). The component sizes of a critical random graph with given degree sequence. Ann. Appl. Probab. 24 2560–2594.
• [16] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
• [17] Łuczak, T. (1990). Component behavior near the critical point of the random graph process. Random Structures Algorithms 1 287–310.
• [18] Nachmias, A. and Peres, Y. (2010). The critical random graph, with martingales. Israel J. Math. 176 29–41.
• [19] Nachmias, A. and Peres, Y. (2010). Critical percolation on random regular graphs. Random Structures Algorithms 36 111–148.
• [20] Riordan, O. (2012). The phase transition in the configuration model. Combin. Probab. Comput. 21 265–299.
• [21] Turova, T. S. (2011). Survey of scalings for the largest connected component in inhomogeneous random graphs. In Random Walks, Boundaries and Spectra. Progress in Probability 64 259–275. Birkhäuser, Basel.
• [22] Turova, T. S. (2013). Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1. Random Structures Algorithms 43 486–539.