Logarithmic corrections to scaling in the fourdimensional uniform spanning tree
Abstract
We compute the precise logarithmic corrections to meanfield scaling for various quantities describing the uniform spanning tree of the fourdimensional hypercubic lattice $\mathbb{Z}^4$. 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 contains at 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 fourdimensional looperased random walk which may be of independent interest. Our results imply that the Abelian sandpile model also exhibits nontrivial polylogarithmic corrections to meanfield scaling in four dimensions, although it remains open to compute the precise order of these corrections.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.15830
 arXiv:
 arXiv:2010.15830
 Bibcode:
 2020arXiv201015830H
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 80 pages. V2: Minor revisions. Accepted version, to appear in CMP