Random Steiner complexes and simplical spanning trees

Abstract: A spanning tree of $G$ is a subgraph of $G$ with the same vertex set as $G$ that is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs, showing that the number of spanning trees $\kappa_1(G_n)$ in a random $k$-regular graph on $n$ vertices satisfies $\lim_{n\to\infty}\Big( \kappa_{1}(G_n) \Big)^{1/n}=c_{1,k}$ in probability, where $c_{1,k}= \frac{(k-1)^{k-1}}{(k^2-2k)^{\frac{k-2}{2}}}$.

In this talk we will discuss a high-dimensional of the matching model for simplicial complexes, known as random Steiner complexes. In particular, we will prove a high-dimensional counterpart of McKay's result and discuss the local limit of such random complexes.

Based on a joint work with Lior Tenenbaum.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry