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W4300019452 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." @default.
W4300019452 created "2022-10-03" @default.
W4300019452 creator A5048261912 @default.
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W4300019452 date "2018-01-19" @default.
W4300019452 modified "2023-09-23" @default.
W4300019452 title "Spectral expansion of random sum complexes" @default.
W4300019452 doi "https://doi.org/10.48550/arxiv.1801.06466" @default.
W4300019452 hasPublicationYear "2018" @default.
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