## Electronic Communications in Probability

### A large deviation principle for the Erdős–Rényi uniform random graph

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 79, 13 pp.

Dates
Accepted: 9 October 2018
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/euclid.ecp/1540346603

Digital Object Identifier
doi:10.1214/18-ECP181

Zentralblatt MATH identifier
1398.05179

Subjects

#### Citation

Dembo, Amir; Lubetzky, Eyal. A large deviation principle for the Erdős–Rényi uniform random graph. Electron. Commun. Probab. 23 (2018), paper no. 79, 13 pp. doi:10.1214/18-ECP181. https://projecteuclid.org/euclid.ecp/1540346603

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