  In the design and analysis of political redistricting maps, it is often
useful to be able to sample from the space of all partitions of the graph of
census blocks into connected subgraphs of equal population. There are
influential Markov chain Monte Carlo methods for doing so that are based on
sampling and splitting random spanning trees. Empirical evidence suggests that
the distributions such algorithms sample from place higher weight on more
"compact" redistricting plans, which is a practically useful and desirable
property. In this paper, we confirm these observations analytically,
establishing an inverse exponential relationship between the total length of
the boundaries separating districts and the probability that such a map will be
sampled. This result provides theoretical underpinnings for algorithms that are
already making a significant real-world impact.
