Strong convergence of partial match queries in random quadtrees

TL;DR

This paper proves that the costs of partial match queries in random quadtrees, when properly rescaled, almost surely converge to a random limit identified via martingale theory, using methods related to self-similar fragmentations.

Contribution

It establishes almost sure convergence of query costs in random quadtrees and links the limit to martingale terminal values, advancing understanding of quadtree behavior.

Findings

Rescaled query costs converge almost surely.

Limit identified as martingale terminal value.

Method relates to self-similar fragmentation theory.

Abstract

We prove that the rescaled costs of partial match queries in a random two-dimensional quadtree converge almost surely towards a random limit which is identified as the terminal value of a martingale. Our approach shares many similarities with the theory of self-similar fragmentations.