The Dirichlet process has, on expectation, $\alpha \log n$ "restaurants", when borrowing the Chinese restaurant process terminology, with $\alpha$ being the concentration parameter.
Pitman-Yor, on the other hand, has $\alpha n^d$ "tables", with $\alpha$ being the concentration parameter, and $d$ being the discount parameter.
How does that relate to power-law? I understand PY has power-law behavior, but DP does not.
"The parameters governing the Pitman–Yor process are: 0 ≤ d < 1 a discount parameter, a strength parameter θ > −d and a base distribution G0 over a probability space X. When d = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman–Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman–Yor process useful for modeling data with power-law tails "
This, unfortunately, is not clear enough for me.