Spectral computations on lamplighter groups and Diestel-Leader graphs

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 of the cyclic group of order q with the infinite cyclic group) with respect to a natural generating set. For the "Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and Dicks & 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 l^2-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 behaviour 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 Foelner approximations; in the specific case of DL(q,r), the answer is positive only when r=q

Polyharmonic functions for finite graphs and Markov chains

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a $\lambda$-polyharmonic function is a complex function $f$ on the vertex set which satisfies $(\lambda \cdot I - P)^n f(x) = 0$ at each interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more generally, we consider the transition matrix $P$ of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these `global' polyharmonic functions, we turn to solving the Riquier problem, where $n$ boundary functions are preassigned and a corresponding `tower' of $n$ successive Dirichlet type problems are solved. The resulting unique solution will be polyharmonic only at those points which have distance at least $n$ from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and Singman, and more recently, by Picardello and Woess

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where $q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-graphs and, in particular, lamplighter groups. This completes and provides a better approach to previous results of Woess, who has determined all minimal positive harmonic functions.Comment: 26 page