### Description

This project is devoted to the investigation of two models (random walks and their deterministic counterpart rotor-router walks) on infinite graphs. Namely, we aim at describing the geometric and structural properties of the state spaces (the graphs) which are essential for the behavior of these two models. The methods used within this research are at the confluent of more than one field of specialization (probability, algebra, graph theory, combinatorics). A brief outline of the proposed research problems is given below.

Rotor-router walks (RRW). Informally, a rotor-router walk on a graph can be described as follows. Each vertex of the graph is endowed with a rotor which indicates the direction the particle will follow. After a particle is launched from a vertex, the rotor is updated in a fixed deterministic way. The resulting purely deterministic walk shares many properties with the classical random walk, but there are also subtle differences. Within this theme I want to pursue these issues in detail for particular state spaces: Galton-Watson trees, free products of graphs, lamplighter graphs.

Stochastic abelian networks (SAN). An abelian network can be defined as a network of automata, similarly to a cellular automata with the additional property that the automata can be updated asynchronously, and the final state does not depend on the order in which the automata processed their data. The previously introduced rotor-router walk is an example of an abelian network. The abstract theory of Abelian networks has been recently developed by BOND and LEVINE.

A stochastic abelian network is an abelian network with transition function between automata depending on a probability space. A variety of models in statistical physics can be realised as stochastic abelian networks: Markov chains, branching random walks, internal DLA, activated random walks or stochastic sandpiles. Together with LEVINE, we plan to work out the basic theory of SAN and to tie these models into a common mathematical framework. The starting point here is the analysis of locally Markov walks.

Rotor-router walks (RRW). Informally, a rotor-router walk on a graph can be described as follows. Each vertex of the graph is endowed with a rotor which indicates the direction the particle will follow. After a particle is launched from a vertex, the rotor is updated in a fixed deterministic way. The resulting purely deterministic walk shares many properties with the classical random walk, but there are also subtle differences. Within this theme I want to pursue these issues in detail for particular state spaces: Galton-Watson trees, free products of graphs, lamplighter graphs.

Stochastic abelian networks (SAN). An abelian network can be defined as a network of automata, similarly to a cellular automata with the additional property that the automata can be updated asynchronously, and the final state does not depend on the order in which the automata processed their data. The previously introduced rotor-router walk is an example of an abelian network. The abstract theory of Abelian networks has been recently developed by BOND and LEVINE.

A stochastic abelian network is an abelian network with transition function between automata depending on a probability space. A variety of models in statistical physics can be realised as stochastic abelian networks: Markov chains, branching random walks, internal DLA, activated random walks or stochastic sandpiles. Together with LEVINE, we plan to work out the basic theory of SAN and to tie these models into a common mathematical framework. The starting point here is the analysis of locally Markov walks.

Status | Finished |
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Effective start/end date | 1/04/16 → 30/09/16 |