Wolfgang Woess

Vertex-Transitive Graphs and Accessibility

We call an infinite graph G accessible if there exists a natural number k such that any two ends of G can be separated by k edges. C. T. C. Wall?s accessibility conjecture for finitely generated groups has a simple and attractive graph version: Every locally finite Cayley graph is accessible. Wall?s conjecture has recently been disproved by M.J. Dunwoody. In this paper we show that all locally finite, 2-transitive graphs and all 1-transitive graphs of prime degree are accessible. We prove that every locally finite, vertex-transitive graph with at least one thick end has a thick end with a 2-way infinite geodesic, while no thin end has a 2-way infinite geodesic. Furthermore, those ends in a locally finite, accessible vertex-transitive graph which have a 2-way infinite geodesic are precisely the thick ends. In addition, there are only finitely many non-isomorphic thick ends. We obtain these and other results from a precise description of the end structure of every locally finite, accessible, vertex-transitive graph. We also investigate the ends in inaccessible vertex-transitive graphs.

SPECTRAL COMPUTATIONS ON LAMPLIGHTER GROUPS AND DIESTEL-LEADER GRAPHS

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.

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Zn} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Zn} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Zn} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Zn} converges to this end. If {Zn} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.

Random Walks on Trees with Finitely Many Cone Types

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.

Lamplighters, Diestel–Leader Graphs, Random Walks, and Harmonic Functions

The lamplighter group over

Topological groups and infinite graphs

\Z

Martin boundaries of random walks: ends of trees and groups

is the wreath product

Local limits and harmonic functions for nonisotropic random walks on free groups

\Z_q \wr \Z

Martin and end compactifications for non-locally finite graphs

. With respect to a natural generating set, its Cayley graph is the Diestel–Leader graph

Uniqueness of currents in infinite resistive networks

\mbox{\sl DL}(q,q)