a useful quantitative phenomenology for the dielectric properties of both pure [@hemberger95; @hemberger96] and doped[@kleemann00; @kleemann02; @kleemann98_di; @wu03; @guo12] SrTiO$_3$.

The local nature of the Ising pseudospins makes the TIM valuable as a model for inhomogeneous systems, including doped quantum paraelectrics [@kleemann00; @kleemann02; @kleemann98_di; @wu03; @guo12], ferroelectric thin films [@wangcl92; @sun08; @oubelkacem09; @wangCD10; @lu13; @li16], superlattices [@wangCL00; @yao02], and various low-dimensional structures [@xin99; @lang07; @lu14]. However, we show here that the TIM, as it is conventionally formulated, fails to correctly describe SrTiO$_3$ whenever nanoscale inhomogeneity is important. Most egregiously, the TIM fails to predict the formation of a quantized two-dimensional electron gas (2DEG) at LaAlO$_3$/SrTiO$_3$ interfaces, in contradiction with both theory and experiments [@gariglio15]. The goal of this paper is to propose a modification that we believe captures the essential physics of spatial inhomogeneity, and to compare it to the conventional TIM for model SrTiO$_3$ thin films and interfaces. In the TIM, the lattice polarization $P_{i}$ in unit cell $i$ is modelled by a pseudospin. This polarization is given by $$\label{P}
P_{i} = \mu \eta S^{(3)}_{i},$$ where $\mu$ sets the scale of the electric dipole moment, $\eta = a^{-3}$ is the volume density of dipoles, and $a$ is the lattice 