Random matrix statistics in the uniform spanning tree

Nathanael Berestycki
University of Vienna

Consider a wired uniform spanning tree in a simply connected domain such as the unit disc, and condition it so that the branches emanating out of n points close to the origin are distinct until they reach the boundary (the so-called "n arms event"). We show that the position at which these branches hit the boundary is equal in the scaling limit to the eigenvalue distribution of the circular orthogonal ensemble in random matrix theory. Furthermore, the n curves are described by a Loewner evolution driven by Dyson Brownian motion. This gives information on the singular winding of these curves near the origin.

Surprisingly the analysis is very different depending on whether n is odd or even. In the odd case the partition function is given by a celebrated determinantal formula due to Fomin. In the even case however it becomes necessary to analyse a loop measure term carrying “topological” information.

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