Loose laplacian spectra of random hypergraphs

• Authors: Linyuan Lu;Xing Peng
• Venue: Random Structures & Algorithms
• Year: 2012

Let H = (V, E) be an r-uniform hypergraph with the vertex set V and the edge set E. For 1 ≤ s ≤ r/2, we define a weighted graph G(s) on the vertex set \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${V\choose s}$\end{document}**image** as follows. Every pair of s-sets I and J is associated with a weight w(I, J), which is the number of edges in H containing I and J if I ∩ J = ∅︁, and 0 if I ∩ J ≠ ∅︁. The s-th Laplacian \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\mathcal L^{(s)}$\end{document} **image** of H is defined to be the normalized Laplacian of G(s). The eigenvalues of \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\mathcal L^{(s)}$\end{document} **image** are listed as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\lambda^{(s)}_0, \lambda^{(s)}_1, \ldots, \lambda^{(s)}_{{n\choose s}-1}$\end{document} **image** in non-decreasing order. Let \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\bar\lambda^{(s)}(H)=\max_{i\neq 0}\{|1-\lambda^{(s)}_i|\}$\end{document} **image** . The parameters \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\bar\lambda^{(s)}(H)$\end{document} **image** and λ**math-image**(H), which were introduced in our previous paper, have a number of connections to the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions. For 0 p Hr(n, p) be a random r-uniform hypergraph over [n] := {1, 2, …, n}, where each r-set of [n] has probability p to be an edge independently. For 1 ≤ s ≤ r/2, \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$p(1-p)\gg \frac{\log^4 n}{n^{r-s}}$\end{document} **image** , and \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$1-p\gg \frac{\log n}{n^2}$\end{document} **image** , we prove that almost surely \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*} \bar\lambda^{(s)}(H^r(n,p))\leq \frac{s}{n-s}+ (3+o(1))\sqrt{\frac{1-p}{{n-s\choose r-s}p}}.\end{align*}\end{document} We also prove that the empirical distribution of the eigenvalues of \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$\mathcal L^{(s)}$\end{document} **image** for Hr(n, p) follows the Semicircle Law if \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$p(1-p)\gg \frac{\log^{1/3} n}{n^{r-s}}$\end{document} **image** and \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$1-p\gg \frac{\log n}{n^{2+2r-2s}}$\end{document} **image** . © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.