Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime

Let $N$ be the number of copies of a small subgraph $H$ in an Erdős-Rényi graph $G\sim \mathcal{G}(n,p_n)$ where $p_n\to 0$ is chosen so that $\mathbb{E}N=c$, a constant. Results of Bollobás show that for regular graphs $H$, the count $N$ weakly converges to a Poisson random variable. For large but finite $n$, and for the specific case of the triangle, investigations of the upper tail $\mathbb{P}(N\ge k_n)$ by Ganguly, Hiesmayr and Nam (2022) revealed that there is a phase transition in the tail behavior and the associated mechanism. Smaller values of $k_n$ correspond to disjoint occurrences of $H$, leading to Poisson tails, with a different behavior emerging when $k_n$ is large, guided by the appearance of an almost clique. We show that a similar phase transition also occurs when $H$ is any regular graph, at the point where $k_n^{1-2\mathrm{/}q}\log k_n=\log n$ ($q$ is the number of vertices in $H$). This establishes universality of this transition, previously known only for the case of the triangle.