B-urns

TL;DR

This paper analyzes the asymptotic behavior of the fringe of B-trees using Pólya urn models, revealing a phase transition at m=60 where fluctuations shift from Gaussian to oscillatory with a complex limit distribution.

Contribution

It establishes the asymptotic distribution of B-tree fringe composition vectors, including the phase transition and properties of the limiting complex-valued random variable W.

Findings

For m ≤ 59, fluctuations are Gaussian.

For m ≥ 60, fluctuations oscillate and converge to a complex random variable W.

W has exponential moments and a density on the complex plane.

Abstract

The fringe of a B-tree with parameter $m$ is considered as a particular P\'olya urn with $m$ colors. More precisely, the asymptotic behaviour of this fringe, when the number of stored keys tends to infinity, is studied through the composition vector of the fringe nodes. We establish its typical behaviour together with the fluctuations around it. The well known phase transition in P\'olya urns has the following effect on B-trees: for $m\leq 59$, the fluctuations are asymptotically Gaussian, though for $m\geq 60$, the composition vector is oscillating; after scaling, the fluctuations of such an urn strongly converge to a random variable $W$. This limit is $\mathbb C$-valued and it does not seem to follow any classical law. Several properties of $W$ are shown: existence of exponential moments, characterization of its distribution as the solution of a smoothing equation, existence of a density relatively to the Lebesgue measure on $\mathbb C$, support of $W$. Moreover, a few representations of the composition vector for various values of $m$ illustrate the different kinds of convergence.