Random walks, random configurations, and horocyclic products

The heart of this research project are random configurations which are driven by random walks on graphs, resp. groups. 1.) For the simplest model of a "lamplighter"-random walk, imagine that at each vertex of a graph there is a lamp with the two possible states "off" or "on". A "lamplighter" performs a random walk along the graph; at each visited vertex he may (randomly) modify the state of the lamp sitting there. The states of this random process consist of the actual position of the "lamplighter" in the base graph plus the configuration of the lamps that are switched on. The corresponding algebraic construction is that of the wreath product of groups. Within this project, we plan to study specifically lamplighter random walks on trees. 2.) Within the preceding project FWF P15577, for lamplighter random walks on the two-way-infinite path, a detailed understanding of the structure of the state space has lead to several new results. The latter structure is that of the Diestel-Leader graphs. These are horocyclic products of 2 homogeneous trees. In the sequel, also such products of more than two trees have been examined in detail; these are horospheres in a product of trees. Within the present project, we plan to study in detail also horocyclic products of other structures, in particular affine buildings. This comes along with the study of random walks on those products, which constitute a kind of generalization of lamplighter random walks. 3.) "Internal diffusion limited aggregation" is a process where a "source", that is, a root vertex of an infinite graph, emits successive, independent particles. Each one performs a random walk, until it first hits an unoccupied site, which it then occupies. When n particles have occupied their random site, they build a random cluster A n . The basic question is how the geometric structure of the underlying graph determines the asymptotic form of A n . In particular, we plan to study this question for the natural spanning trees if the integer grids ("comb lattices"), and more generally, for trees with finitely many cone types.

The heart of this research project are random configurations which are driven by random walks on graphs, resp. groups. 1. For the simplest model of a "lamplighter"-random walk, imagine that at each vertex of a graph there is a lamp with the two possible states "off" or "on". A "lamplighter" performs a random walk along the graph; at each visited vertex he may (randomly) modify the state of the lamp sitting there. The states of this random process consist of the actual position of the "lamplighter" in the base graph plus the configuration of the lamps that are switched on. The corresponding algebraic construction is that of the wreath product of groups. Within this project, we plan to study specifically lamplighter random walks on trees. 2. Within the preceding project FWF P15577, for lamplighter random walks on the two-way-infinite path, a detailed understanding of the structure of the state space has lead to several new results. The latter structure is that of the Diestel-Leader graphs. These are horocyclic products of 2 homogeneous trees. In the sequel, also such products of more than two trees have been examined in detail; these are horospheres in a product of trees. Within the present project, we plan to study in detail also horocyclic products of other structures, in particular affine buildings. This comes along with the study of random walks on those products, which constitute a kind of generalization of lamplighter random walks. 3. "Internal diffusion limited aggregation" is a process where a "source", that is, a root vertex of an infinite graph, emits successive, independent particles. Each one performs a random walk, until it first hits an unoccupied site, which it then occupies. When n particles have occupied their random site, they build a random cluster An . The basic question is how the geometric structure of the underlying graph determines the asymptotic form of An . In particular, we plan to study this question for the natural spanning trees if the integer grids ("comb lattices"), and more generally, for trees with finitely many cone types.