Upper tails and independence polynomials in random graphs

Abstract

The upper tail problem in the Erdős–Rényi random graph G∼Gn,p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1+δ. Chatterjee and Dembo showed that in the sparse regime of p→0 as n→∞ with p≥n−α for an explicit α=αH>0, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique.Here we extend the latter work to any fixed graph H and determine a function cH(δ) such that, for p as above and any fixed δ>0, the upper tail probability is exp⁡[−(cH(δ)+o(1))n2pΔlog⁡(1/p)], where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, cH(δ), is governed by the independence polynomial of H, defined as PH(x)=∑iH(k)xk where iH(k) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then cH(δ) is the minimum between 12δ2/m and the unique positive solution of PH(x)=1+δ.