Moderate deviations in cycle count

Abstract

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