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The uniform spanning tree in 4 dimensions


Date: Tuesday, March 23, 2021 16:30 - 17:15
Speaker: Perla Sousi (University of Cambridge)
Location: Online via Zoom
Series: Mathematics and CS Seminar
Host: M. Beiglböck, N. Berestycki, L. Erdös, J. Maas, F. Toninelli

A uniform spanning tree of Z^4 can be thought of as the ‘’uniform measure’’ on trees  of Z^4. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length n, that it has volume at least n and that it reaches the boundary of the box of side length n around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is 
joint work with Tom Hutchcroft.

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