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 Erdős–Rényi 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 < δ < 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 ≥ n
−α
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 < δ < δ
H
for some δ
H
> 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.