Universität zu Köln

Gundert, Anna and Wagner, Uli ORCID: 0000-0002-1494-0568 (2016). On eigenvalues of random complexes. Isr. J. Math., 216 (2). S. 545 - 583. JERUSALEM: HEBREW UNIV MAGNES PRESS. ISSN 1565-8511

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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 X (k) (n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = abroken vertical bar(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 aeyen 2 and n a N, there is a k-dimensional complex Y (n) (k) on n vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised k-dimensional Laplacian lie in the interval but whose coboundary expansion is bounded from above by O(log n/n) and so tends to zero as n -> a; moreover, Y (n) (k) can be taken to have vanishing integer homology in dimension less than k.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Gundert, Anna UNSPECIFIED UNSPECIFIED UNSPECIFIED Wagner, Uli UNSPECIFIED orcid.org/0000-0002-1494-0568 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-260097
DOI:	10.1007/s11856-016-1419-1
Journal or Publication Title:	Isr. J. Math.
Volume:	216
Number:	2
Page Range:	S. 545 - 583
Date:	2016
Publisher:	HEBREW UNIV MAGNES PRESS
Place of Publication:	JERUSALEM
ISSN:	1565-8511
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language HOMOLOGICAL CONNECTIVITY; BETTI NUMBERS; TOP HOMOLOGY; GRAPHS; INEQUALITY; SPECTRA Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/26009