On the lower tail variational problem for random graphs

We study the lower tail large deviation problem for subgraph counts in a random graph. Let $X_H$ denote the number of copies of $H$ in an Erd\H{o}s-R'enyi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $$\mathbb{P}$(X_H \le (1-\delta) $\mathbb{E}$ X_H)$forfixed$0 < \delta < 1$. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for $p\ge n^{-{\alpha }_H}$ (and conjecturally for a larger range of $p$). We study this variational problem and provide a partial characterization of the so-called “replica symmetric” phase. Informally, our main result says that for every $H$, and $0<\delta <{\delta }_H$ for some ${\delta }_H>0$, as $p\to 0$ slowly, the main contribution to the lower tail probability comes from Erd\H{o}s-R'enyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite $H$ and $\delta$ close to 1.