Abstract:
In this thesis, we study simple stochastic models that show rich and interesting behavior. The plan of the thesis is as follows: after a brief introduction and overview of the work in Chapter 1, Chapter 2 begins with a discussion on the simplest stochastic interacting particle system − the exclusion process. After briefly motivating the process, we will describe a two-species exclusion problem that shows a remarkable, counter-intuitive feature, called the TASEP Speed Process. Our aim in this chapter will be to provide a simple approximate description for this phenomena using what we here call the effective medium approach. After addressing the issue of the TASEP Speed Process, we will extend our approach to different variations of the multi-species exclusion process. Following the discussing in this chapter on a system of stochastically interacting particle system, in Chapter 3, we will discuss the stochastic evolution of a surface, taking the example of the Eden Model. We start with an small exposition of the process and its variants following which, we will motivate the question of finding upper bounds to the shape of growing clusters using examples from day- to-day life. In this chapter, we will obtain upper bounds of increasing accuracy by using the independent branching process and its variants. In Chapter 4, we will move on to the study of the stochastic evolution of a model with two interacting surfaces, called Chase-Escape. This process, defined as a prey-predator model, shows a very interesting feature that the prey can survive even when it can run only half as fast as the predators. After studying the critical properties of this process, we will compare the dynamics of the front of Chase- Escape process with the Eden front. Our analysis of this problem will be partly inspired by our analysis of the Eden process in the previous chapter. In Chapter 5, we will present a summary of all our results and will outline future directions.
