Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

TL;DR

This paper analyzes the asymptotic eigenvalue behavior of the Laplacian on a flag complex of subspaces over finite fields, confirming a conjecture for the 0-dimensional case as the field size grows.

Contribution

It determines the eigenvalue distribution of the Laplacian on flag complexes as the field size increases, solving a specific case of Papikian's conjecture.

Findings

Number of distinct eigenvalues stabilizes for large q

Non-zero eigenvalues tend to n-2 as q increases

Confirmed the 0-dimensional case of Papikian's conjecture

Abstract

Let $\text{Fl}_{n,q}$ be the simplicial complex whose vertices are the non-trivial subspaces of $\mathbb{F}_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let $\Delta^{+}_k(\text{Fl}_{n,q})$ be the $k$-dimensional weighted upper Laplacian on $ \text{Fl}_{n,q}$. The spectrum of $\Delta^{+}_k(\text{Fl}_{n,q})$ was first studied by Garland, who obtained a lower bound on its non-zero eigenvalues. Here, we focus on the $k=0$ case. We determine the asymptotic behavior of the eigenvalues of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ as $q$ tends to infinity. In particular, we show that for large enough $q$, $\Delta_{0}^{+}(\text{Fl}_{n,q})$ has exactly $\left\lfloor n^2/4\right\rfloor+2$ distinct eigenvalues, and that every eigenvalue $\lambda\neq 0,n-1$ of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ tends to $n-2$ as $q$ goes to infinity. This solves the $0$-dimensional case of a conjecture of Papikian.