How to judge a Ph.D. applicant that arrives "out of thin air"
I would tell them what you expect from a PhD student: prove theorems/run empirical tests/conduct human studies or all the above. If their profile is a clear mismatch, I wouldn’t bother. 

I would next ask them to review two papers of their choice out of your recent publications. One page review each. The review should include at least one idea for future directions.  It tells you whether they’re serious, what they want to work on, and whether they can reason about ideas in your field. It also guarantees that the problem is at least somewhat tractable (you’ve solved it after all).

If you’re happy with the write ups, schedule a video interview. You both are committing to 4+ years together, best to make sure you can discuss things together.

Good luck!