Hutchcroft, Tom and Sousi, Perla
(2020)
*Logarithmic corrections to scaling in the four-dimensional uniform spanning tree.*
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(Unpublished)
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236

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## Abstract

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 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 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.

Item Type: | Report or Paper (Discussion Paper) | ||||||
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Additional Information: | Perla Sousi’s research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1. | ||||||

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Record Number: | CaltechAUTHORS:20210924-202133236 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236 | ||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 111036 | ||||||

Collection: | CaltechAUTHORS | ||||||

Deposited By: | George Porter | ||||||

Deposited On: | 27 Sep 2021 17:02 | ||||||

Last Modified: | 27 Sep 2021 17:02 |

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