  Strontium titanate is an incipient ferroelectric in which superconductivity
emerges at exceptionally low doping levels. Remarkably, stabilizing the polar
phase through strain or chemical substitution has been shown to significantly
enhance the superconducting critical temperature, and the polar instability
plays a pivotal role in the majority of proposed superconducting pairing
mechanisms. A rigorous understanding of ferroelectricity is therefore essential
to elucidate the electron pairing mechanism in this material. To investigate
the nature of the polar order in strontium titanate, we develop a simplified
free energy model that only includes the degrees of freedom necessary to
capture the relevant physics in a biaxially compressively strained system. Our
model is able to calculate the energies of large, disordered systems with near
DFT-level accuracy. We simulate the ferroelectric and antiferrodistortive phase
transitions using the Monte Carlo method and discuss the coupling between
various order parameters. Finally, we assess the character of the polar
transition, which we find to be neither strictly displacive nor order-disorder.
