  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 $\mathbb{P}(X_H \le (1-\delta) \mathbb{E}
X_H)$ for fixed $0 < \delta < 1$.
  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.
