Simplicial Spanning Trees in Random Steiner Complexes

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 $(\xi _{d,k}+o(1))^{\left( {\begin{array}{c}n\\ d\end{array}}\right) }$ as $n\rightarrow \infty$, 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.

Appendices

Appendices

Appendix A: Proof of Theorem 5.4

Let us start by stating a result of Friedmann regarding the spectral gap in the matching model.

Theorem A.1

([6]) Fix $k\in \mathbb {N}$ and $\varepsilon >0$. Then there exists a constant $C_{k,\varepsilon }\in (0,\infty )$ such that a random graph G on n vertices sampled according to the matching model satisfies

$$\begin{aligned} \mathbb {P}\Big (\vert \lambda _i(G)\vert \le 2\sqrt{k-1} +\varepsilon \Big )\ge 1- \frac{C_{k,\varepsilon }}{n^{\tau (k)}},\qquad \forall 2\le i\le n, \end{aligned}$$

(A.1)

where $\lambda _1(G)\ge ...\ge \lambda _n(G)$ are the eigenvalues of A(G), the adjacency matrix of G, and $\tau (k)= \lceil \sqrt{k-1} \rceil -1$. Furthermore, there exists a constant $C_k>0$, such that

$$\begin{aligned} \mathbb {P}\Big (\lambda _2(G) > 2\sqrt{k-1} \Big )\ge \frac{C_k}{n^{s(k)}}, \end{aligned}$$

where $s(k)= \lfloor \sqrt{k-1} \rfloor$.

Let $\epsilon >0$. As stated before in Theorem A.1, a random graph G on n vertices sampled according to the matching model satisfies (A.1). Recall that for every $\sigma \in X_i^{d-2}$, the link of $\sigma$, denoted $\textrm{lk}(X_i,\sigma )$ is a random graph on $n_i-d+1$ vertices, distributed according to the matching model with parameter k and therefore, the event

$$\begin{aligned} E_{i,\sigma ,\varepsilon } = \{\lambda _2(\textrm{lk}(X_i,\sigma ))>2\sqrt{k-1}+\varepsilon \} \end{aligned}$$

satisfies

$$\begin{aligned} {\mathbb {P}}(E_{i,\sigma ,\varepsilon })\le C_{k,\varepsilon }(n_i-d+1)^{-(\lceil \sqrt{k-1}\rceil -1)}. \end{aligned}$$

A union bound, thus gives

$$\begin{aligned} {\mathbb {P}}\bigg (\bigcup _{\sigma \in X^{d-2}_i} E_{i,\sigma ,\varepsilon }\bigg ) \le |X^{d-2}_i|\cdot C_{k,\varepsilon }(n_i-d+1)^{-(\lceil \sqrt{k-1}\rceil -1)}\le C_{k,\varepsilon }n_i^{d-\lceil \sqrt{k-1}\rceil }, \end{aligned}$$

where in the last bound we used the fact that $|X^{d-2}_i|=\left( {\begin{array}{c}n_i\\ d-1\end{array}}\right)$.

By Garland’s method (cf. [7]), on the event $\bigcup _{\sigma \in X^{d-2}_i} E_{i,\sigma ,\varepsilon }$, all non-trivial eigenvalues of the adjacency matrix of $X_i$ are within $(-\infty ,2d\sqrt{k-1}+\varepsilon )$.

Hence, whenever $\lceil \sqrt{k-1} \rceil >d+1$, by the Borel-Cantelli lemma, only finitely many of the random complexes $X_i$ do not satisfy

$$\begin{aligned} \text {spec}(A_{X_i}) \subset (-\infty ,2d\sqrt{k-1}+\varepsilon ). \end{aligned}$$

Since $\lceil \sqrt{k-1}\rceil >d+1$ whenever $k>(d+1)^2+1$, the result follows.