The Symbiotic Contact Process on Non-Lattice Structures

点击翻译按钮获取中文摘要

The symbiotic contact process (SCP) was introduced by de Oliviera, Dos Santos, and Dickman (2012) and is an interacting particle system for modelling two different species. The study of the SCP on the lattice was extended by Durrett and Yao (2020) who gave results for the critical value for the infection parameter between extinction and survival. Our aim will be to further this research by studying the process on non-lattice structures. Structures such as random graphs with power law behavior are frequently being used to model human interactions and relationships such as modelling connections people have on social media sites. Surprising results have recently been found for the contact process on these random structures (Chatterjee and Durrett (2009), Huang and Durrett (2020)) and we wish to extend this area of research.The process can be seen as an adaptation of the contact process with two particle types 'A' and 'B'. Both particle types infect their neighbours at a rate lambda. The particles of both types die at rate one unless both types are present at the same site and then they die at rate mu, strictly less than one, giving the symbiotic nature of the model. This model is motivated by symbiotic relationships that naturally occur with the two most useful motivational examples being the symbiotic survival of two species, and the worse recovery rate of a patient with two diseases. Our short term aim is to study the critical values for infection parameter, as a function of mu, for this process on a Galton-Watson tree, the starting point for the research on the contact process on random structures. More specifically, we will be aiming to compare these values with the corresponding critical values for the standard contact process; we hope that the symbiotic nature of our model will lead to critical values that are smaller than those for the corresponding contact process. We will aim to extend our study to other random structures including Erdos-Renyi random graphs and dynamic random graphs with power law behavior. The overall goal is to fully characterise phase transitions of the process on these non-lattice structures. The mathematical theory that we will use to help analyse the process on these structures frequently hail from percolation theory and martingale theory.