Open Access
2018 A large deviation principle for the Erdős–Rényi uniform random graph
Amir Dembo, Eyal Lubetzky
Electron. Commun. Probab. 23: 1-13 (2018). DOI: 10.1214/18-ECP181


Starting with the large deviation principle (LDP) for the Erdős–Rényi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.


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Amir Dembo. Eyal Lubetzky. "A large deviation principle for the Erdős–Rényi uniform random graph." Electron. Commun. Probab. 23 1 - 13, 2018.


Received: 30 April 2018; Accepted: 9 October 2018; Published: 2018
First available in Project Euclid: 24 October 2018

zbMATH: 1398.05179
MathSciNet: MR3873786
Digital Object Identifier: 10.1214/18-ECP181

Primary: 05C80 , 60F10

Keywords: constrained random graphs , Erdős–Rényi graphs , large deviations

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