Upper tail behavior of the number of triangles in random graphs with constant average degree

Let $N$ be the number of triangles in an Erd\H{o}s-R\'enyi 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 the probability of $\{N\ge k\}$ 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 $k^{1/3} \log k< (\frac{3}{\sqrt{2}})^{2/3} \log n$ (sub-critical regime) as well as pin down the tail behavior when $k^{1/3} \log k> (\frac{3}{\sqrt{2}})^{2/3} \log n$ (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, culminating in (Harel, Moussat, Samotij,'19) 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.