Georgia Institute of Technology College of Sciences

We study the graphon variational problem that governs lower tail large deviations of the triangle count in a sparse Erdős--Rényi random graph G(n,p). We deduce that the replica-symmetric and symmetry-breaking regimes are separated by a single critical parameter q_c and that the phase transition is first-order. I’ll also discuss tools to transfer these results to the conditioned random graph, showing for example that the edge density is a discontinuous order parameter.