Moderate deviations in cycle count

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a G?(n,m) random graph. We show that the probability of decreasing triangle density by t3, is exp(-Theta(n2t2)) whenever n-3/4MUCH LESS-THANt3MUCH LESS-THAN1. These complement results of Goldschmidt, Griffiths, and Scott, who showed that for n-3/2MUCH LESS-THANt3MUCH LESS-THANn-1, the probability is exp(-Theta(n3t6)). That is, deviations of order smaller than n-1 behave like small deviations, and deviations of order larger than n-3/4 behave like large deviations. We conjecture that a sharp change between the two regimes occurs for deviations of size n-3/4, which we associate with a single large negative eigenvalue of the adjacency matrix becoming responsible for almost all of the cycle deficit. We give analogous results for the k-cycle density, for all odd k. Our results can be interpreted as finite size effects in phase transitions in constrained random graphs.