  Building on the techniques from the breakthrough paper of Harel, Mousset and
Samotij, which solved the upper tail problem for cliques, we compute the
asymptotics of the upper tail for the number of induced copies of the 4-cycle
in the binomial random graph $G_{n,p}$. We observe a new phenomenon in the
theory of large deviations of subgraph counts. This phenomenon is that, in a
certain (large) range of $p$, the upper tail of the induced 4-cycle does not
admit a naive mean-field approximation.
