calculation shows that $$\left . \frac{\partial^2 I(g_c)}{\partial c^2} \right |_{c=1/2} \ge 16e^2$$ for small $t$, indicating that (nearly) bipartite graphs with pieces of unequal size have a higher rate function than $g$. This provides strong evidence that our perturbative solution is in fact a global minimizer for sufficiently small $t$.

\[cor1\] Assuming our perturbative solution is the global optimizer, there is a phase transition near the boundary point $(1/2,0)$ between the first and second scallop.

Our perturbative solution yields a formula for the entropy:

$$\label{entropy} s_{e,t} = -\frac{1}{2} [I_0(\epsilon)
  + I_0(2e-\epsilon)].$$

This formula for the entropy cannot be extended analytically beyond $e=(1+\epsilon)/2$, as $\partial^2 s/\partial
  e^2$ diverges as $e \to (1+\epsilon)/2$. However, $e=(1+
  \epsilon)/2$ corresponds to $t= (\epsilon^3 + 3\epsilon)/4$, or, using the more basic variable $e$,

$$t= [(2e-1)^3+3(2e-1)]/4,$$

which is in the interior of $(e,t)$ space. (Since the graphon $g(x,y)$ is nowhere zero, it differs in form from the graphons describing graphs with minimal $t$.) Thus $s_{e,t}$ must fail to be analytic in some neighborhood of the first scallop.

Of course there must also be a phase transition, presumably from this bipartite phase to a homogeneous phase, if one fixes $e$ and raises $t$, which we see as follows.

