How random is the graduate admission process in the United States? I wonder how random the graduate admission process is in the United States. I am mostly interested in the fields of computer science, and PhD programs (i.e., not Master programs). 

One way to quantify it would be following a similar methodology as in this study  (mirror), which showed that half the papers appearing at NIPS would be rejected if the review process were rerun. Of course it may not be flawless, but might give some decent approximations.

I am looking for referenced numbers, not guesses.
There is without a doubt some randomness in the process, but I find your question impossible to answer. In the question, you implicitly assume that the "quality" of an application is a well-defined concept that could be used to replicate an experiment.

But this isn't the case. The only way you could do this is by submitting N "equally good" applications to the same graduate programs in the same year, and expect the same outcome for all. But this is clearly not a workable concept. Rather, different graduate programs may look for different candidates (for example because they have different areas of strength), and possibly may want to balance areas in successive years. In other words, the "quality" of an application is a concept that can only be measured against the current needs of individual graduate programs. You can't repeat the experiment at a different place or different time and expect the same outcome.