Moderate deviations in cycle count

Abstract

We 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.

Existence of a symmetric bipodal phase in the edge-triangle model

“…In [11], and again in [21], we studied small perturbations of the Erdős-Rényi graph G(n, p) and attempted to get the greatest possible change in triangle count for the smallest possible entropy cost. The results are closely related to moderate deviation estimates [20]; depending on the sizes of n and e 3 − t, a finite graph with triangle density slightly less than e 3 can be viewed either as a typical (e, t) graph or as a deviation of an Erdős-Rényi graph. When e > 1/2, moderate deviations estimates that apply when n −1 ≪ e 3 − t ≪ 1 agree to leading order with large deviations estimates.…”

“…Does it resemble the example given in section 6, or is its structure still wilder? How does the behavior of this graphon as σ → 0 compare to the moderate deviations results for G(n, m) in [20]? (2) Is there a succession of phases as e → 0 and t remains close to e 3 , with tripodal graphons giving way to 4-podal, 5-podal, and so on?…”