Take two copies of a given graph and connect their vertices by a perfect matching. What are the properties of the graph obtained by recursively repeating this procedure? If you start with a single vertex, and all the matchings are “the identity”, then the result is just your standard-issue hypercube. But what if they are random?
In this talk, we will investigate this “random duplicube” graph, and show that it shares structural properties with both the hypercube (which is a bastion of structure) and random regular graphs (which are almost as random as you can get). We will discuss its diameter, expansion, eigenvalues and symmetries, while still leaving behind many open questions (including derandomization and Benjamini-Schramm convergence).

(Joint work with Itai Benjamini, Yotam Dikstein and Maksim Zhukovskii; see https://arxiv.org/abs/2211.06988 for the paper and https://sarcasticresonance.wordpress.com/2022/11/15/random-graphs-the-duplicube-graph-new-paper-on-arxiv/ for a more popular report)