A first order phase transition for lower tails in sparse random graphs

We study the graphon variational problem that governs lower tail large deviations of the number of copies of a fixed nonbipartite graph $H$ in a sparse Erdős–Rényi random graph. Our main new insight is that any non-constant minimizer of the variational problem exhibits a discontinuous jump away from the constant graphon. We deduce that the replica-symmetric and symmetry-breaking regimes are separated by a single critical parameter and that the phase transition is first order. In the supercritical regime $p\gg n^{-1\mathrm{/}{m}_2(H)}$, we transfer these results to the conditioned random graph, showing that the edge density is a discontinuous order parameter and that the conditioned graph concentrates in cut distance near the full minimizer set.