Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree

Abstract

Let N be the number of triangles in an Erdős–Rényi graph ${\mathcal {G}}(n,p)$ on n vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the upper tail problem for N,  namely, we investigate how fast k must grow, so that ${\mathbb {P}}(N\ge k)$ is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when $k^{1/3} \log k< (\frac{3}{\sqrt{2}} - {o(1)})^{2/3} \log n$ (sub-critical regime) as well as pin down the tail behavior when $k^{1/3} \log k> (\frac{3}{\sqrt{2}} + {o(1)})^{2/3} \log n$ (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately $(6k)^{1/3}$. This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime $p\gg \frac{1}{n}.$ The proofs rely on several novel graph theoretical results which could have other applications.