Moderate Deviations of Triangle Counts in the Erdos-R\'enyi Random Graph G (n, m): The Lower Tail

Let N_ (G) be the number of triangles in a graph G. In 14 and 25 (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdos-R\'enyi random graphs Gₘ G (n, m): \ P (N_ (Gₘ) \, < \, (1-) E[N_ (Gₘ) ) \, =\, (- (²n³) ) if n^-3/2 n^{-1} \] and \ P (N_ (Gₘ) \, < \, (1-) E[N_ (Gₘ) ) \, =\, (- (^2/3n²) ) if n^-3/4 1. \] Neeman, Radin and Sadun 25 also conjectured that the probability should be of the form (- (²n³) ) in the "missing interval" n^-1 n^-3/4. We prove this conjecture. As part of our proof we also prove that some random graph statistics, related to degrees and codegrees, are normally distributed with high probability.