The random spanning tree on ladder-like graphs

Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two, respectively. Up to a trivial thinning with additional coin flips, for each of the first two examples the renewal processes exhaust the whole class of stationary regenerative (of order one) determinantal point processes. We also give an example of a regenerative process of order two that has no representation in terms of a random spanning tree. Our examples illustrate a theorem of Lyons and Steif (2003) which characterizes regenerative determinantal point processes in terms of their Fourier transform. For the regenerative process, we also establish a Markov chain description in the spirit of H\"aggstr\"om (1994). On the technical side, a systematic counting scheme for random spanning trees is developed that allows to compute explicitly the probabilities. In some cases an electrical network point of view simplifies matters.