  In this thesis, we consider one of the most popular models of non-equilibrium
statistical physics: the Asymmetric Simple Exclusion Process, in which
particles jump stochastically on a one-dimensional lattice, between two
reservoirs at fixed densities, with the constraint that each site can hold at
most one particle at a given time. This model has the mathematical property of
being integrable, which makes it a good candidate for exact calculations. What
interests us in particular is the current of particles that flows through the
system (which is a sign of it being out of equilibrium), and how it fluctuates
with time. We present a method, based on the "matrix Ansatz" devised by
Derrida, Evans, Hakim and Pasquier, that allows to access the exact cumulants
of that current, for any finite size of the system and any value of its
parameters. We also analyse the large size asymptotics of our result, and make
a conjecture for the phase diagram of the system in the so-called "s-ensemble".
Finally, we show how our method relates to the algebraic Bethe Ansatz, which
was thought not to be applicable to this situation.
