Spectral expansion of random sum complexes

Let $G$ be a finite abelian group of order $n$ and let ${\mathrm{\Delta }}_{n-1}$ 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 ${\mathrm{\Delta }}_{n-1}$ together with all $(k+1)$-subsets $\sigma \subset G$ that satisfy $\sum_{x \in \sigma} x \in A$\mathrm{.}Let$C^{k-1}(X_{A,k})$denotethespaceofcomplexvalued$(k-1)$-cochainsof$X_{A,k}$\mathrm{.}Let$L_{k-1}:C^{k-1}(X_{A,k}) \rightarrow C^{k-1}(X_{A,k})$denotethereduced$(k-1)$-thLaplacianof$X_{A,k}$,andlet$\mu_{k-1}(X_{A,k})$betheminimaleigenvalueof$L_{k-1}$. It is shown that for any $k\ge 1$ and $ϵ>0$ there exists a constant $c(k,ϵ)$ 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.