I'm in electronic engineering, but most of the experience is from computer science departments. There are programs like Caltech where students do rotations and

I'm in electronic engineering, but most of the experience is from computer science departments. There are programs like Caltech where students do rotations and aren't admitted into a lab. But for example, at the MIT Media Lab, a professor can explicitly pull in a student for their group (and it's bloody hard to switch, if there's an advisor-advisee conflict). And some departments are a hybrid, where a professor can pull someone in by writing an appropriate recommendation note to the admissions committee. If a professor says in a rec "I've worked with Alice for the past few months. If you admit Alice, we have a project defined together, and I have funding to pay for it," that's more-or-less an automatic admit, barring exceptional circumstances.