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

Source abstract

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 $p \gg n^{-1/m_2(H)}$, 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.