Spectral expansion of random sum complexes

TL;DR

This paper studies the spectral properties of sum complexes over finite abelian groups, showing that random subsets lead to high minimal eigenvalues of the Laplacian, indicating strong connectivity properties.

Contribution

It establishes probabilistic bounds on the minimal eigenvalue of the Laplacian for random sum complexes, revealing their spectral expansion characteristics.

Findings

Minimal eigenvalue exceeds (1-ε)m with high probability for random subsets of size proportional to log n.

Spectral expansion properties hold asymptotically almost surely for large groups.

Results connect combinatorial structure with spectral properties in algebraic topology.

Abstract

Let $G$ be a finite abelian group of order $n$ and let $\Delta_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex $X_{A,k}$ associated to a subset $A \subset G$ and $k < n$, is the $k$-dimensional simplicial complex obtained by taking the full $(k-1)$-skeleton of $\Delta_{n-1}$ together with all $(k+1)$-subsets $\sigma \subset G$ that satisfy $\sum_{x \in \sigma} x \in A$. Let $C^{k-1}(X_{A,k})$ denote the space of complex valued $(k-1)$-cochains of $X_{A,k}$. Let $L_{k-1}:C^{k-1}(X_{A,k}) \rightarrow C^{k-1}(X_{A,k})$ denote the reduced $(k-1)$-th Laplacian of $X_{A,k}$, and let $\mu_{k-1}(X_{A,k})$ be the minimal eigenvalue of $L_{k-1}$. It is shown that for any $k \geq 1$ and $\epsilon>0$ there exists a constant $c(k,\epsilon)$ such that if $A$ is a random subset of $G$ of size $m=\lceil c(k,\epsilon) \log n \rceil$, then $\mu_{k-1}(X_{A,k}) > (1-\epsilon)m$ asymptotically almost surely.