A local central limit theorem for triangles in a random graph

What is the distribution of the number of triangles in the random graph $G(n, 1/2)$? It was known for a long time that this distribution obeys a central limit theorem: from the point of view of large intervals (~ standard-deviation length), the distribution looks like a Gaussian random variable. We show that it even obeys a LOCAL central limit theorem: the distribution is pointwise close to a suitable discrete Gaussian random variable. Joint work with Justin Gilmer.