The final size of the C_4-free process

We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C_4. We show that, with probability tending to 1 as $n \to \infty$, the final graph produced by this process has maximum degree O((n \log n)^{1/3}) and consequently size O(n^{4/3}\log(n)^{1/3}), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollob\'as and Riordan and Osthus and Taraz.