The contact process in an evolving random environment

by Marco Seiler
Doctoral thesis
Date of Examination:2021-10-15
Date of issue:2021-11-26
Advisor:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Dominic Schuhmacher
Persistent Address: http://dx.doi.org/10.53846/goediss-8981

 

 

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Abstract

English

Recently, there has been an increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this thesis a contact process in a time evolving edge random environment (CPERE) on a connected and transitive graph is studied. This thesis is split in two parts. In the first part the graph is assumed to have bounded degrees. In this case the evolving random environment is described by an autonomous ergodic spin system with finite range. This background process determines which edges are open or closed for infections. Sufficient conditions, which are posed on the background dynamics, are provide such that the initial configuration of the system has no influence on the phase transition between extinction and survival and that this phase transition agrees with the phase transition between ergodicity and non-ergodicity of the system. Furthermore, results on the continuity properties of the survival probability and conditions for complete convergence are given. As an application the above results are used on the special case of the contact process on a dynamical percolation (CPDP) on the d-dimensional integer lattice. Hence, continuity of the survival probability and complete convergence for the whole parameter regime are proven. Furthermore, it is shown that the CPDP dies out at criticality almost surely. In the second part we consider a contact process on a dynamical long range percolation (CPLDP). Sufficient conditions are given which ensure existence of this process. Then, the survival probability and the associated phase transition are studied under the assumption of a stationary background. The existence of a immunization phase, i.e. no survival of the infection is possible regardless of the infection rate, is proven and the asymptotic behaviour for fast and slow update speed is studied.
Keywords: contact process; interacting particle systems; evolving random environment; dynamical random graph; infection model
 
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