Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions

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 prev…

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 di…

Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs

We determine all positive harmonic functions for a large class of "semi-isotropic" random walks on the lamplighter group, i.e., the wreath product of the cyclic group of order q with the infinite cyclic group. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q,q). More generally, DL(q,r) is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$, and our result applies to all DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.

以Diestel-Leader图为重点的理想Poisson-Voronoi剖分的收敛性

Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs

研究在适当点态度量空间上理想Poisson-Voronoi剖分（IPVT）收敛的充要条件，应用于高秩对称空间并解决Diestel-Leader图的开放问题。