Upper tail for homomorphism counts in constrained sparse random graphs

Abstract

AbstractConsider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs (extending the known results for ), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.