Lower tails via relative entropy

We show that the naive mean-field approximation correctly predicts leading term of logarithmic lower tail probabilities for number copies a given subgraph in G(n,p) and arithmetic progressions length random subsets integers entire range densities where is viable. Our main technical result provides sufficient conditions on maximum degrees uniform hypergraph H guarantee edges, induced by binomial subset vertices H, can be well approximated considering only product distributions. This may interpreted as weak, probabilistic version container lemma applicable to all sparser-than-average (and not independent) sets.