---
res:
  bibo_abstract:
  - 'We consider higher-dimensional generalizations of the normalized Laplacian and
    the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam
    model Xk(n, p) of random k-dimensional simplicial complexes on n vertices. We
    show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s.
    concentrated around two values. The main tool, which goes back to the work of
    Garland, are arguments that relate the eigenvalues of these matrices to those
    of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns
    the Laplacian; we develop an analogous result for the adjacency matrix. The same
    arguments apply to other models of random complexes which allow for dependencies
    between the choices of k-dimensional simplices. In the second part of the paper,
    we apply this to the question of possible higher-dimensional analogues of the
    discrete Cheeger inequality, which in the classical case of graphs relates the
    eigenvalues of a graph and its edge expansion. It is very natural to ask whether
    this generalizes to higher dimensions and, in particular, whether the eigenvalues
    of the higher-dimensional Laplacian capture the notion of coboundary expansion—a
    higher-dimensional generalization of edge expansion that arose in recent work
    of Linial and Meshulam and of Gromov; this question was raised, for instance,
    by Dotterrer and Kahle. We show that this most straightforward version of a higher-dimensional
    discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and
    n ∈ N, there is a k-dimensional complex Yn k on n vertices that has strong spectral
    expansion properties (all nontrivial eigenvalues of the normalised k-dimensional
    Laplacian lie in the interval [1−O(1/√1), 1+0(1/√1]) but whose coboundary expansion
    is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Yn
    k can be taken to have vanishing integer homology in dimension less than k.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Anna
      foaf_name: Gundert, Anna
      foaf_surname: Gundert
  - foaf_Person:
      foaf_givenName: Uli
      foaf_name: Wagner, Uli
      foaf_surname: Wagner
      foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1494-0568
  bibo_doi: 10.1007/s11856-016-1419-1
  bibo_issue: '2'
  bibo_volume: 216
  dct_date: 2016^xs_gYear
  dct_identifier:
  - UT:000386356400002
  dct_language: eng
  dct_publisher: Springer@
  dct_title: On eigenvalues of random complexes@
...