Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$
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| Format: | Preprint |
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2023
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| _version_ | 1866917887897239552 |
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| author | Alvarado, José D. de Oliveira, Leonardo Gonçalves Griffiths, Simon |
| author_facet | Alvarado, José D. de Oliveira, Leonardo Gonçalves Griffiths, Simon |
| contents | We consider the question of determining the probability of triangle count deviations in the Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$ with densities larger than $n^{-1/2}(\log{n})^{1/2}$. In particular, we determine the log probability $\log\mathbb{P}(N_{\triangle}(G)\, >\, (1+δ)p^3n^3)$ up to a constant factor across essentially the entire range of possible deviations, in both the $G(n,m)$ and $G(n,p)$ model. For the $G(n,p)$ model we also prove a stronger result, up to a $(1+o(1))$ factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length $2$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04326 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$ Alvarado, José D. de Oliveira, Leonardo Gonçalves Griffiths, Simon Combinatorics Probability We consider the question of determining the probability of triangle count deviations in the Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$ with densities larger than $n^{-1/2}(\log{n})^{1/2}$. In particular, we determine the log probability $\log\mathbb{P}(N_{\triangle}(G)\, >\, (1+δ)p^3n^3)$ up to a constant factor across essentially the entire range of possible deviations, in both the $G(n,m)$ and $G(n,p)$ model. For the $G(n,p)$ model we also prove a stronger result, up to a $(1+o(1))$ factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length $2$). |
| title | Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$ |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2305.04326 |