Upper tails for irregular graphs beyond the mean-field regime

Abstract

Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $\hat{I}”\ge 2$ and $p\ge n^{-1/\hat{I}”-{\epsilon }_H}(\log n{)}^{\ddot{I}‰(1)}$ for an explicit ${\epsilon }_H>0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the naïve mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log \log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

46 pages, 1 figure. Comments are welcome!