We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the initial density are below certain critical values, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Grosskinsky (2004) that these correlations can be described by means of a matrix product representation. In this thesis we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.
|Date of Award||21 Jul 2015|
|Sponsors||Consejo Nacional de Ciencia y Tecnologia|
|Supervisor||Peter Morters (Supervisor) & Georg Zimmer (Supervisor)|
- Large deviations
- exclusion process
- interacting particle systems
- Matrix Product ansatz