Moderate deviations in cycle count

Abstract

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a (Figure presented.) random graph. We show that the probability of decreasing triangle density by (Figure presented.), is (Figure presented.) whenever (Figure presented.). These complement results of Goldschmidt, Griffiths, and Scott, who showed that for (Figure presented.), the probability is (Figure presented.). That is, deviations of order smaller than (Figure presented.) behave like small deviations, and deviations of order larger than (Figure presented.) behave like large deviations. We conjecture that a sharp change between the two regimes occurs for deviations of size (Figure presented.), 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 (Figure presented.) -cycle density, for all odd (Figure presented.). Our results can be interpreted as finite size effects in phase transitions in constrained random graphs.