  We study the fluctuations of the current J(t) of the totally asymmetric
exclusion process with open boundaries. Using a density matrix renormalization
group approach, we calculate the cumulant generating function of the current.
This function can be interpreted as a free energy for an ensemble in which
histories are weighted by exp(-sJ(t)). We show that in this ensemble the model
has a first order space-time phase transition at s=0. We numerically determine
the finite size scaling of the cumulant generating function near this phase
transition, both in the non-equilibrium steady state and for large times.
