Jamming transition of kinetically-constrained models in rectangular systems

We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: for long and wide channels it decreases algebraically $v_{c}\sim W^{-1/2}$ with the system's width W, while in square systems it decreases logarithmically $v_{c}\sim1/\ln L$ with length L. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as $W\approx10$.