Full counting statistics in the not-so-long-time limit

The full counting statistics of charge transport is the probability distribution ${p}_{n}({t}_{m})$ that n electrons have flown through the system in measuring time tm. The cumulant generating function (CGF) of this distribution $F(\chi ,{t}_{m})$ has been well studied in the long time limit ${t}_{m}\to \infty $, however there are relatively few results on the finite measuring time corrections to this. In this work, we study the leading finite time corrections to the CGF of interacting Fermi systems with a single transmission channel at zero temperature but driven out of equilibrium by a bias voltage. We conjecture that the leading finite time corrections are logarithmic in tm with a coefficient universally related to the long time limit. We provide detailed numerical evidence for this with reference to the self-dual interacting resonant level model. This model further contains a phase transition associated with the fractionalization of charge at a critical bias voltage. This transition manifests itself technically as branch points in the CGF. We provide numerical results of the dependence of the CGF on measuring time for model parameters in the vicinity of this transition, and thus identify features in the time evolution associated with the phase transition itself.