Consider a big finite rescaled piece of $\mathbb{Z}^2$, i.e. consider a unit square with a thick grid. Famous Wilson's method allows to generate a colored spanning tree of such a graph in a uniform way by popping out the cycles: if the meshsize tends to 0, then the interesting fractal structure occurs: one can read about this in R.Lyons, Y. Peres Probability on Trees and Networks. It is said there, that this fractal structure is not yet explained. Could someone point to me some works connected to this question, if there are any?
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Is this the fractal structure you have in mind?


