Large deviations for dense random graphs beyond mean field

In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erd}os-Renyi random graph, viewed as a random graphon. This directly pro- vides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained. In this talk, we will explore large deviations for dense random graphs, be- yond the \mean-eld" setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdos-Renyi random graphs. Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.