SIAM Review
The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms
SIAM Journal on Computing
Lamplighters, Diestel–Leader Graphs, Random Walks, and Harmonic Functions
Combinatorics, Probability and Computing
The Discrete Sine Transform and the Spectrum of the Finite $q$-ary Tree
SIAM Journal on Discrete Mathematics
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Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular, in the case where the underlying graph is the infinite path $ \mathbb{Z} $ . In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C 2-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. If the graph has a transitive isometry group G, we also describe the spectral analysis in terms of the representation theory of the wreath product C 2驴G. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples have already been studied by Häggström and Jonasson by probabilistic methods.