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.

In this talk, I will define random recursive metric spaces, and outline the proofs of a law of large numbers and a central limit theorem for the insertion depth; the distance from the master hook to the latch chosen. I will discuss a classic argument for random recursive trees which shows that the insertion depth is distributed as a sum of Bernoulli random variables. This argument is generalized for random recursive metric spaces via a martingale central limit theorem.