Harmonic analysis of finite lamplighter random walks

Details Harmonic analysis of finite lamplighter random walks Harmonic analysis of finite lamplighter random walks

2007-01-22

arxiv.org/abs/math/0701603v1

J. Dynam. Control Systems 14 (2008), no. 2, 251--282.

Mathematics

Probability

29 pages

Scientific paper

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when 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. In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.

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