Affordable Access

Access to the full text

Spectral Computations on Lamplighter Groups and Diestel-Leader Graphs

Authors
  • Bartholdi, Laurent1
  • Woess, Wolfgang2
  • 1 IGAT, Bâtiment BCH, École Polytechnique Fédérale, CH-1015 Lausanne, Switzerland
  • 2 Institut für Mathematik C, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria
Type
Published Article
Journal
Journal of Fourier Analysis and Applications
Publisher
Birkhäuser-Verlag
Publication Date
Mar 17, 2005
Volume
11
Issue
2
Pages
175–202
Identifiers
DOI: 10.1007/s00041-005-3079-0
Source
Springer Nature
Keywords
License
Yellow

Abstract

The Diestel-Leader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q + 1 and r + 1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) ℤq ≀ ℤ with respect to a natural generating set. For the “Simple random walk” (SRW) operator on the latter group, Grigorchuk and Zuk, and Dicks and Schick have determined the spectrum and the (on-diagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DL-graphs by directly computing an ℓ2-complete orthonormal system of finitely supported eigenfunctions of the SRW. This allows computation of all matrix elements of the spectral resolution, including the Plancherel measure. As one application, we determine the sharp asymptotic behavior of the N-step return probabilities of SRW. The spectral computations involve a natural approximating sequence of finite subgraphs, and we study the question whether the cumulative spectral distributions of the latter converge weakly to the Plancherel measure. To this end, we provide a general result regarding Følner approximations; in the specific case of DL(q, r), the answer is positive only when r = q.

Report this publication

Statistics

Seen <100 times