Spectral expansion of random sum complexes

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.