Simplicial spanning trees in random Steiner complexes

TL;DR

This paper extends McKay's 1981 result on the number of spanning trees in random regular graphs to high-dimensional simplicial complexes, establishing asymptotic formulas for their weighted counts.

Contribution

It provides the first high-dimensional generalization of McKay's asymptotic result for random $d$-dimensional, $k$-regular simplicial complexes, including explicit constants.

Findings

Weighted number of simplicial spanning trees grows as $(\xi_{d,k}+o(1))^{inom{n}{d}}$

Proves local convergence of random complexes to arboreal complexes

Generalizes McKay's result to higher dimensions and complex structures

Abstract

A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex set as $G$ which is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs. In this paper we prove a high-dimensional generalization of McKay's result for random $d$-dimensional, $k$-regular simplicial complexes on $n$ vertices, showing that the weighted number of simplicial spanning trees is of order $(\xi_{d,k}+o(1))^{\binom{n}{d}}$ as $n\to\infty$, where $\xi_{d,k}$ is an explicit constant, provided $k> 4d^2+d+2$. A key ingredient in our proof is the local convergence of such random complexes to the $d$-dimensional, $k$-regular arboreal complex, which allows us to generalize McKay's result regarding the Kesten-McKay distribution.