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Loose laplacian spectra of random hypergraphs

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Abstract

Let H = (V, E) be an r-uniform hypergraph with the vertex set V and the edge set E. For 1 ≤ sr/2, we define a weighted graph G(s) on the vertex set equation 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 IJ = ∅, and 0 if IJ ≠ ∅. The s-th Laplacian equation image of H is defined to be the normalized Laplacian of G(s). The eigenvalues of equation image are listed as equation image in non-decreasing order. Let equation image. The parameters equation image and λmath 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 < 1, let 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 ≤ sr/2, equation image, and equation image, we prove that almost surely

equation image

We also prove that the empirical distribution of the eigenvalues of equation image for Hr(n, p) follows the Semicircle Law if equation image and equation image. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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