Expected worst-case partial match in random quadtries

We consider random multivariate quadtries obtained from n points independently and uniformly distributed on the unit cube of Rd. Let Nn(y) be the complexity of the standard partial match algorithm for fixed vector y, where y is a vector in Rs, 0 < s < d. We study Nn = supy Nn(y), the worst-case time for partial match. Among other things, we show that partial match is very stable, in the sense that supy Nn(y)/ infy Nn(y) → 1 in probability.