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Moderate Deviations of Triangle Counts in the Erdős-Rényi Random Graph $G(n,m)$: The Lower Tail

José Alvarado, Gabriel Dias, Simon Griffiths

TL;DR

This work resolves the missing-range lower-tail behavior for triangle counts in $G(n,m)$ by introducing a conditioning framework built around the $p$-synergy $S^p_G(u,w)$ and a carefully crafted event $E_\alpha$ that biases the remaining edge additions toward low-synergy non-edges. By proving that the vector of normalized synergies is typically near-normal, and relating synergy structure to codegree sums under quasirandomness constraints, the authors construct an event with probability at least $\exp(-C\delta^2 n^3)$ that yields a substantial negative drift in $N_{\triangle}$, achieving the conjectured rate $\exp(-\Theta(\delta^2 n^3))$ in the intermediate regime $n^{-1}\ll \delta\ll n^{-3/4}$. The analysis decomposes triangle counts into four building blocks corresponding to how many edges come from the pre-existing graph vs. the remainder, and establishes precise bounds for each block under the conditioned event. The results advance the understanding of moderate deviations for subgraph counts in dense random graphs and hint at connections to spectral properties and eigenvalue behavior, with potential implications for broader classes of subgraph counts.

Abstract

Let $N_{\triangle}(G)$ be the number of triangles in a graph $G$. In [14] and [25] (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs $G_m\sim G(n,m)$: \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ\left(δ^2n^3\right)\right) \qquad \text{if $n^{-3/2}\ll δ\ll n^{-1}$} \] and \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ(δ^{2/3}n^2) \right) \qquad \text{if $n^{-3/4} \ll δ\ll 1$.} \] Neeman, Radin and Sadun [25] also conjectured that the probability should be of the form $\exp\left(-Θ\left(δ^2n^3\right)\right)$ in the "missing interval" $n^{-1}\ll δ\ll n^{-3/4}$. We prove this conjecture. As part of our proof we also prove that some random graph statistics, related to degrees and codegrees, are normally distributed with high probability.

Moderate Deviations of Triangle Counts in the Erdős-Rényi Random Graph $G(n,m)$: The Lower Tail

TL;DR

This work resolves the missing-range lower-tail behavior for triangle counts in by introducing a conditioning framework built around the -synergy and a carefully crafted event that biases the remaining edge additions toward low-synergy non-edges. By proving that the vector of normalized synergies is typically near-normal, and relating synergy structure to codegree sums under quasirandomness constraints, the authors construct an event with probability at least that yields a substantial negative drift in , achieving the conjectured rate in the intermediate regime . The analysis decomposes triangle counts into four building blocks corresponding to how many edges come from the pre-existing graph vs. the remainder, and establishes precise bounds for each block under the conditioned event. The results advance the understanding of moderate deviations for subgraph counts in dense random graphs and hint at connections to spectral properties and eigenvalue behavior, with potential implications for broader classes of subgraph counts.

Abstract

Let be the number of triangles in a graph . In [14] and [25] (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs : \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ\left(δ^2n^3\right)\right) \qquad \text{if } \] and \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ(δ^{2/3}n^2) \right) \qquad \text{if .} \] Neeman, Radin and Sadun [25] also conjectured that the probability should be of the form in the "missing interval" . We prove this conjecture. As part of our proof we also prove that some random graph statistics, related to degrees and codegrees, are normally distributed with high probability.
Paper Structure (16 sections, 25 theorems, 157 equations, 2 figures, 2 tables)

This paper contains 16 sections, 25 theorems, 157 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An overview of the proof
  3. The Synergies are normally normal
  4. Sequences of iid random variables
  5. Sampling
  6. Proof of Proposition \ref{['prop:syngp']}
  7. Relating Codegrees and Synergies
  8. Defining the event $E_{\alpha}$
  9. Completing the proof of Theorem \ref{['thm:main']}
  10. Proof of Lemma \ref{['lem:3,0']}
  11. Proof of Lemma \ref{['lem:2,1']}
  12. Proof of Lemma \ref{['lem:1,2']}
  13. Proof of Lemma \ref{['lem:0,3']}
  14. Concluding Remarks
  15. Standard Deviations
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\lambda \in (0,1)$. There exists a constant $C>0$ such that the following holds. Let $\delta=\delta_n$ be such that $n^{-1}\leqslant \delta\leqslant C^{-1}n^{-3/4}$ and let $m=m_n$ be such that $\lambda \binom{n}{2}\leqslant m\leqslant (1-\lambda)\binom{n}{2}$, then

Figures (2)

  • Figure 1: In this figure, the $\theta$-axis parameterises $\delta$ as $\delta=n^{\theta}$, and the $\gamma$-axis parameterises the log-probability $-\log{\mathbb{P}\left(N_{\triangle}(G_{(m)})<(1-\delta)\mu_{n,m}\right)}=n^{\gamma}$. The black lines show the results proved in GGS and NRS22 respectively. The green and red lines show the upper and lower bounds of NRS22 in the interval $-1\leqslant \theta\leqslant -3/4$. Our main theorem shows that the log-probability follows the green line in this interval.
  • Figure 2: In this figure we illustrate the fact that, conditioned on the neighbourhood of $u$, $N_{G_p}(u)$, each edge from $w$ adds $1-p$ to the synergy $S_{G_p}^p(u,w)$ if the other end is in $N_{G_p}(u)$ and $-p$ if not.

Theorems & Definitions (50)

  • Theorem 1.1
  • Conjecture 1.2
  • Proposition 2.0
  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Remark
  • proof : Proof of Proposition \ref{['prop:syngm']}
  • Lemma 3.3
  • proof
  • ...and 40 more