Moderate Deviations in Cycle Count

TL;DR

This paper establishes bounds on the probability of moderate deviations in the number of odd cycles in random graphs, revealing phase transition behaviors and eigenvalue influences.

Contribution

It provides new moderate deviation bounds for cycle counts in random graphs, extending understanding of phase transitions and eigenvalue effects in these probabilistic models.

Findings

Probability of decreasing triangle density by t^3 is exp(-Theta(n^2 t^2)) for certain t ranges.

For k ≥ 5, similar estimates hold for cycle density deviations with different t ranges.

Identifies a potential sharp transition point at deviations of size n^{-3/4} for triangles.

Abstract

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a $\calG(n, m)$ random graph. We show that the probability of decreasing triangle density by $t^3$, is $\exp(-\Theta(n^2 t^2))$ whenever $n^{-3/4} \ll t^3 \ll 1$, while for $k \ge 5$ we give the same estimate for the probability of decreasing the $k$-cycle density by $t^k$, but for the larger range $n^{-1} \ll t^k \ll 1$. When $m \ge \frac 12 \binom n2$, we also find the leading coefficient in the exponent. This complements results of Goldschmidt et al., who showed that for $n^{-3/2} \ll t^k \ll n^{-1}$, the probability is $\exp(-\Theta(n^3 t^{2k}))$. That is, deviations of order smaller than $n^{-1}$ behave like small deviations, and deviations of order larger than $n^{-3/4}$ (for triangles) or $n^{-1}$ (for $k$-cycles with $k \ge 5$) behave like large deviations. For triangles, 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. Our results can be interpreted as finite size effects in phase transitions in constrained random graphs.