4-cycles at the triangle-free process

We consider the triangle-free process: given an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving graph - unless its addition creates a triangle. We study the evolving graph at around the time where Θ(n3/2+ϵ) edges of Kn have been traversed for any fixed ϵ∈(0,10−10). At that time, we give a tight concentration result for the number of copies of the 4-cycle in the evolving graph. Our analysis combines Spencer's original branching process approach for analysing the triangle-free process with the semi-random method.