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 $(\xi_{d,k}+o(1))^{\binom{n}{d}}$ 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.

• The paper extends McKay's spanning tree result to d-dimensional random Steiner complexes, establishing an asymptotic formula for weighted spanning trees.
• It demonstrates local convergence to k-regular arboreal complexes and uses spectral distribution of the Laplacian to validate the asymptotic behavior.
• The findings offer practical guidelines for sampling Steiner systems and performing spectral analysis for efficient computation of combinatorial topologies.

Simplicial Spanning Trees in Random Steiner Complexes

Introduction

The paper presents a significant advancement by generalizing McKay's 1981 result regarding spanning trees in $k$-regular graphs to the context of $d$-dimensional simplicial complexes. Specifically, it establishes an asymptotic formula to calculate the weighted number of simplicial spanning trees ($\kappa_d$) in random $d$-dimensional, $k$-regular simplicial complexes. This formula converges as $n \rightarrow \infty$, provided that the number of complexes meets a regularity condition given by $k>4d^2+d+2$.

Key Results

1. Generalization to High Dimensions: The core result is an extension of Theorem 1 in graph theory to higher dimensions for simplicial complexes, showing that the number of weighted simplicial spanning trees $\kappa_d(X_i)$ for random Steiner complexes is of asymptotic order $(\xi_{d,k}+o(1))^{\binom{n}{d}}$, where $\xi_{d,k}$ is a constant derived in the paper.
2. Convergence to Arboreal Complexes: A crucial part of the proof involves demonstrating that random simplicial complexes locally converge to a $d$-dimensional, $k$-regular arboreal complex. This convergence is central to generalizing McKay's classical result for spanning trees in random $k$-regular graphs to the simplicial setting.
3. Spectral Distribution Analysis: The paper employs spectral analysis, showing that the eigenvalues of the Laplacian of the random complexes converge to a certain spectrum, which is crucial for establishing the main asymptotic result.

Implementation

For practitioners seeking to implement these results, the paper outlines a structured approach involving:

• Sampling Steiner Complexes: Begin by constructing random $(n,d)$-Steiner systems, from which the random $d$-complexes are built. Assuming access to a uniform sampling algorithm for Steiner systems, this step requires efficient management of associated combinatorial structures.
• Spectral Analysis: Utilize algorithms for spectral analysis to compute eigenvalues of the up-Laplacian for these complex structures. This would typically involve leveraging existing high-performance numerical libraries to handle large-scale matrix computations effectively.
• Statistical Asymptotics: Implement statistical techniques to observe the behavior of quantities of interest (e.g., weighted spanning trees) across numerous sample complexes to empirically verify the theoretical asymptotic results.

Conclusion

This work provides a robust theoretical foundation for generalized spanning tree enumeration in high-dimensional simplicial complexes. The asymptotic convergence results, underpinned by spectral methods, open avenues for exploring random constructs beyond graphs. The presence of explicit constants such as $\xi_{d,k}$ paves the way for practical computational applications and empirical investigations within the combinatorial and computational topology domains.