  Stimulated by experimental advances in electrolyte gating methods, we
investigate theoretically percolation in thin films of inhomogenous complex
oxides, such as La$_{1-x}$Sr$_{x}$CoO$_{3}$ (LSCO), induced by a combination of
bulk chemical and surface electrostatic doping. Using numerical and analytical
methods, we identify two mechanisms that describe how bulk dopants reduce the
amount of electrostatic surface charge required to reach percolation: (i)
bulk-assisted surface percolation, and (ii) surface-assisted bulk percolation.
We show that the critical surface charge strongly depends on the film thickness
when the film is close to the chemical percolation threshold. In particular,
thin films can be driven across the percolation transition by modest surface
charge densities \emph{via} surface-assisted bulk percolation. If percolation
is associated with the onset of ferromagnetism, as in LSCO, we further
demonstrate that the presence of critical magnetic clusters extending from the
film surface into the bulk results in considerable volume enhancement of the
saturation magnetization, with pronounced experimental consequences. These
results should significantly guide experimental work seeking to verify
gate-induced percolation transitions in such materials.
