Spectral Computations on Lamplighter Groups and Diestel-Leader Graphs

By Laurent Bartholdi, Wolfgang Woess. 2004

71 citations recorded by the source corpus.

AbstractThe 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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/15772843d31e9ca097ab78d2df40270214a57aff

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In Geometry and Topology

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Spectra of Large Random Trees (2009, background): “One might therefore hope to use the representation theory of the lamplighter group (see, for example, [53–55]) to compute the relevant quantities.”