Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
- Creators
- Hutchcroft, Tom
- Sousi, Perla
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.
Additional Information
Perla Sousi's research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1.Attached Files
Submitted - 2010.15830.pdf
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Additional details
- Eprint ID
- 111036
- Resolver ID
- CaltechAUTHORS:20210924-202133236
- EP/R022615/1
- Engineering and Physical Sciences Research Council (EPSRC)
- Created
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2021-09-27Created from EPrint's datestamp field
- Updated
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2023-06-02Created from EPrint's last_modified field