Random Balanced Cayley Complexes

TL;DR

This paper extends the Alon-Roichman theorem to higher-dimensional complexes, showing that random subsets of a finite group produce complexes with spectral gaps, indicating high-dimensional expansion properties with high probability.

Contribution

It proves a k-dimensional analogue of the Alon-Roichman theorem, establishing spectral gap bounds for random Cayley complexes.

Findings

Spectral gap bounds hold with high probability for random subsets.

The size of the subset depends logarithmically on the sum of degrees of irreducible representations.

The result generalizes known expansion properties to higher-dimensional simplicial complexes.

Abstract

Let $G$ be a finite group of order $n$ and for $1 \leq i \leq k+1$ let $V_i=\{i\} \times G$. Viewing each $V_i$ as a $0$-dimensional complex, let $Y_{G,k}$ denote the simplicial join $V_1*\cdots*V_{k+1}$. For $A \subset G$ let $Y_{A,k}$ be the subcomplex of $Y_{G,k}$ that contains the $(k-1)$-skeleton of $Y_{G,k}$ and whose $k$-simplices are all $\{(1,x_1),\ldots,(k+1,x_{k+1})\} \in Y_{G,k}$ such that $x_1\cdots x_{k+1} \in A$. Let $L_{k-1}$ denote the reduced $(k-1)$-th Laplacian of $Y_{A,k}$, acting on the space $C^{k-1}(Y_{A,k})$ of real valued $(k-1)$-cochains of $Y_{A,k}$. The $(k-1)$-th spectral gap $\mu_{k-1}(Y_{A,k})$ of $Y_{A,k}$ is the minimal eigenvalue of $L_{k-1}$. The following $k$-dimensional analogue of the Alon-Roichman theorem is proved: Let $k \geq 1$ and $\epsilon>0$ be fixed and let $A$ be a random subset of $G$ of size $m= \left\lceil\frac{10 k^2\log D}{\epsilon^2}\right\rceil$ where $D$ is the sum of the degrees of the complex irreducible representations of $G$. Then \[ {\rm Pr}\big[~\mu_{k-1}(Y_{A,k}) < (1-\epsilon)m~\big] =O\left(\frac{1}{n}\right). \]