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.
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