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Article contents
Depths in random recursive metric spaces
Part of:
Graph theory
Combinatorial probability
Limit theorems
Geometric probability and stochastic geometry
Published online by Cambridge University Press: 20 May 2024
Abstract
As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.
MSC classification
Primary:
60C05: Combinatorial probability
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust
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