# Project Details

### Description

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 on 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 two 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 stuctures, 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 infinitive 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 stucture 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.

In the past few years, random walks on percolative clusters of the Euclidean lattice Z(d) have been investigated with respect to the question of the asymptotic type of their return probability (e.g. Fontes and Mathieu(2004), Barlow(2004)). At the same time, automorphism invariant percolation on transitive graphs and random networks have seen increased research activity (e.g. Benjamini & Schramm (1996), Aldous & Lyons (2006)). For important results initially formulated for Zd (or even amenable groups), concerning the uniqueness of the infinite cluster in the supercritical phase, quite a different situation has been shown to prevail for non-amenable, transitive graphs (e.g. Schonman(1999)), where infinitely many infinite components may occur. Here, unimodularity, i.e. the existence of a transitive, unimodular subgroup of the automorphism group of the graph showed to play an eminent role, since it allows to formulate other classic results of Bernoulli percolation on Z(d) , such as the almost sure finiteness of all clusters in critical Bernoulli percolation for d =2, d > 18, or the exponential cluster-size distribution in the subcritical phase for the non-amenable, however unimodular, transitive graphs (e.g. Benjamini & Lyons & Peres & Schramm (1999)). The question of how amenability arranges with the extra assumption of unimodularity in transitive graphs was settled in 1990 by Soardi and Woess: any transitive group of automorphisms of amenable graphs is unimodular. Moreover, qualities of transitive graphs related to random walks have been investigated with respect to the question of "invariance under random perturbations" (Chen & Peres & Pete (2004)), such as the speed of simple random walk, in connection with the geometrical property of anchored expansion (Virag 2000). The question for the existence of a phase transition (in the sense of Lyons (2000)) for random (possibly percolative) subgraphs is subject of current research (Chen & Peres (2003)). An important criterion in this field is invariance with respect to quasi-isometries (Gromov (1983)). It is known that recurrence/transience, (of simple random walk on the graph) and amenability are such invariants, while unimodularity is not. This is the background in front of which the key impetus for the present project (P18703, FWF) is the investigation of the circumstances needed for invariance of special geometric properties (e.g. amenability, recurrence, positivity of speed, unimodularity) of random walks on transitive graphs under percolative perturbations', such as percolation. The most prominent methodology used and developed further in this project is a comparison technique between Markov chains called interlacing (e.g. Haemers (1995)).

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 on 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 two 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 stuctures, 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 infinitive 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 stucture 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.

In the past few years, random walks on percolative clusters of the Euclidean lattice Z(d) have been investigated with respect to the question of the asymptotic type of their return probability (e.g. Fontes and Mathieu(2004), Barlow(2004)). At the same time, automorphism invariant percolation on transitive graphs and random networks have seen increased research activity (e.g. Benjamini & Schramm (1996), Aldous & Lyons (2006)). For important results initially formulated for Zd (or even amenable groups), concerning the uniqueness of the infinite cluster in the supercritical phase, quite a different situation has been shown to prevail for non-amenable, transitive graphs (e.g. Schonman(1999)), where infinitely many infinite components may occur. Here, unimodularity, i.e. the existence of a transitive, unimodular subgroup of the automorphism group of the graph showed to play an eminent role, since it allows to formulate other classic results of Bernoulli percolation on Z(d) , such as the almost sure finiteness of all clusters in critical Bernoulli percolation for d =2, d > 18, or the exponential cluster-size distribution in the subcritical phase for the non-amenable, however unimodular, transitive graphs (e.g. Benjamini & Lyons & Peres & Schramm (1999)). The question of how amenability arranges with the extra assumption of unimodularity in transitive graphs was settled in 1990 by Soardi and Woess: any transitive group of automorphisms of amenable graphs is unimodular. Moreover, qualities of transitive graphs related to random walks have been investigated with respect to the question of "invariance under random perturbations" (Chen & Peres & Pete (2004)), such as the speed of simple random walk, in connection with the geometrical property of anchored expansion (Virag 2000). The question for the existence of a phase transition (in the sense of Lyons (2000)) for random (possibly percolative) subgraphs is subject of current research (Chen & Peres (2003)). An important criterion in this field is invariance with respect to quasi-isometries (Gromov (1983)). It is known that recurrence/transience, (of simple random walk on the graph) and amenability are such invariants, while unimodularity is not. This is the background in front of which the key impetus for the present project (P18703, FWF) is the investigation of the circumstances needed for invariance of special geometric properties (e.g. amenability, recurrence, positivity of speed, unimodularity) of random walks on transitive graphs under percolative perturbations', such as percolation. The most prominent methodology used and developed further in this project is a comparison technique between Markov chains called interlacing (e.g. Haemers (1995)).

Status | Finished |
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Effective start/end date | 1/10/06 → 30/09/09 |