We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.
"Partial match queries in two-dimensional quadtrees: a probabilistic approach." Adv. in Appl. Probab. 43 (1) 178 - 194, March 2011. https://doi.org/10.1239/aap/1300198518