Description
Imagine the infinite square lattice each site of which is either vacant (state 0) or occupied (state 1). An interacting particle system is a Markov chain on configurations of zeros and ones, in which the dynamics at each site depends on the states of neighbouring sites.The simplest example is Richardson's growth model. Here the dynamics of sites is very simple:
- Once a site gets occupied, it stays occupied forewer.
- A vacant site becomes occupied at the rate proportional to the number of its occupied neighbours (at most four).
Let now the rate for a vacant site to become occupied be still
times the number of its occupied neighbours, while occupied sites become vacant at rate 1, independently of state of any other sites. This model is known as the contact process, and is used to model phenomena such as infection spread on the lattice.
Suppose again that initially only the origin is occupied. Some of the key questions are: will the process die out (ie., all sites eventually become vacant) for
small enough? will the process continue forever (ie., there always will be occupied sites) for large enough?
Other classical examples include the voter model, in which individual sites flip their state (to the opposite) at the rate depending on the number of neighbours in that state, or the exclusion process, in which pairs of neighbouring sites, if in the opposite states, simultaneously flip (thus imitating a jump of a particle from an occupied site to a vacant neighbour).
To get some intuition one can play with the simulation applet from Interactivate (you will need to modify your parameters accordingly!) or with the voter model simulation applet by Bryan Gillespie.
The aim of the project is to explore some of these models and to get some experience with the exciting world of stochastic geometry.
Prerequisites
2H Probability and 4H Stochastic Processes are essential; 3H Probability could be helpful.Resources
If interested, feel free to play with the applets above and/or search online for further properties of these models (see. eg., Wiki pages for contact process, voter model or Google some of the terms mentioned above). The Lecture notes by R. Durrett provide a taster of mathematical tools and results in the area. Some further references might be suggested once the project is underway.Get in touch, have any questions and/or if you would be interested in doing some simulations!