  Ionic liquid gating has become a popular tool for tuning the charge carrier
densities of complex oxides. Among these the band insulator SrTiO$_3$ is one of
the most extensively studied materials. While experiments have succeeded in
inducing (super)conductivity, the process by which ionic liquid gating turns
this insulator into a conductor is still under scrutiny. Recent experiments
have suggested an electrochemical rather than electrostatic origin of the
induced charge carriers. Here, we report experiments probing the time evolution
of conduction of SrTiO$_3$ near the glass transition temperature of the ionic
liquid. By cooling down to temperatures near the glass transition of the ionic
liquid the process develops slowly and can be seen to evolve in time. The
experiments reveal a process characterized by waiting times that can be as long
as several minutes preceding a sudden appearance of conduction. For the
conditions applied in our experiments we find a consistent interpretation in
terms of an electrostatic mechanism for the formation of a conducting path at
the surface of SrTiO$_3$. The mechanism by which the conducting surface channel
develops relies on a nearly homogeneous lowering of the surface potential until
the conduction band edge of SrTiO$_3$ reaches the Fermi level of the
electrodes.
