### Abstract

What does an Erdo{double acute}s-Rényi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovász and coauthors and Szemerédi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

Original language | English (US) |
---|---|

Pages (from-to) | 1000-1017 |

Number of pages | 18 |

Journal | European Journal of Combinatorics |

Volume | 32 |

Issue number | 7 |

DOIs | |

State | Published - Oct 2011 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*European Journal of Combinatorics*,

*32*(7), 1000-1017. https://doi.org/10.1016/j.ejc.2011.03.014

**The large deviation principle for the Erdo{double acute}s-Rényi random graph.** / Chatterjee, Sourav; Varadhan, Srinivasa.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 32, no. 7, pp. 1000-1017. https://doi.org/10.1016/j.ejc.2011.03.014

}

TY - JOUR

T1 - The large deviation principle for the Erdo{double acute}s-Rényi random graph

AU - Chatterjee, Sourav

AU - Varadhan, Srinivasa

PY - 2011/10

Y1 - 2011/10

N2 - What does an Erdo{double acute}s-Rényi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovász and coauthors and Szemerédi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

AB - What does an Erdo{double acute}s-Rényi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovász and coauthors and Szemerédi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

UR - http://www.scopus.com/inward/record.url?scp=79960894509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960894509&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2011.03.014

DO - 10.1016/j.ejc.2011.03.014

M3 - Article

VL - 32

SP - 1000

EP - 1017

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 7

ER -