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Published April 27, 2023 | public
Journal Article

Logarithmic Corrections to Scaling in the Four-dimensional Uniform Spanning Tree


We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice Z⁴. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length n is of order (log n)^(1/3)n⁻¹, that the probability that the past containsat least n vertices is of order (log n)^(1/6)n^(−1/2), and that the probability that the past reaches the boundary of the box [−n, n]⁴ is of order (log n)^(2/3+o(1))n⁻². An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. Our results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.

Additional Information

© 2023 Springer Nature. This work was carried out while TH was a Herchel Smith Postdoctoral Research Fellow at the University of Cambridge and a Junior Research Fellow at Trinity College Cambridge. PS's research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1.

Additional details

August 22, 2023
October 18, 2023