  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 is below a critical value, 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 Gro{\ss}kinsky
(2004) that these correlations can be described by means of a matrix product
representation. In this paper 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.
