The Symbiotic Contact Process on Non-Lattice Structures

共生接触过程（SCP）是由de Oliviera， Dos Santos和Dickman（2012）提出的，是一个相互作用的粒子系统，用于模拟两个不同的物种。Durrett和Yao（2020）对晶格上SCP的研究进行了扩展，给出了灭绝与存活之间感染参数的临界值。我们的目标是通过研究非晶格结构的过程来进一步研究这一问题。具有幂律行为的随机图等结构经常被用来模拟人类的互动和关系，比如模拟人们在社交媒体网站上的联系。最近在这些随机结构的接触过程中发现了令人惊讶的结果(Chatterjee和Durrett (2009)， Huang和Durrett(2020))，我们希望扩展这一研究领域。这一过程可以看作是对“A”和“B”两种粒子类型的接触过程的适应。两种粒子都以λ的速率感染它们的邻居。两种类型的粒子都以1的速率死亡除非两种类型都出现在同一位置然后它们以mu的速率死亡，严格小于1，给出了模型的共生性质。这个模型是由自然发生的共生关系驱动的，其中两个最有用的动机例子是两个物种的共生生存，以及患有两种疾病的病人的较差的康复率。我们的短期目标是在Galton-Watson树（研究随机结构上接触过程的起点）上研究该过程的感染参数临界值作为mu的函数。更具体地说，我们的目标是将这些值与标准接触过程的相应临界值进行比较；我们希望我们模型的共生性质将导致临界值小于相应接触过程的临界值。我们的目标是将我们的研究扩展到其他随机结构，包括Erdos-Renyi随机图和具有幂律行为的动态随机图。总体目标是充分表征这些非晶格结构上的相变过程。我们将用来帮助分析这些结构上的过程的数学理论通常来自渗透理论和鞅理论。

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.