论文标题
在四维统一树中缩放的对数校正
Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
论文作者
论文摘要
我们将精确的对数校正计算为各种数量的均值缩放，描述了四维超同质晶格的均匀跨度树$ \ mathbb {z}^4 $。我们对原点过去的分布特别感兴趣，即与原产地与无限分开的树的有限块。我们证明，过去包含长度$ n $的概率是顺序$（\ log n）^{1/3} n^{ - 1} $，即过去至少包含$ n $顶点的概率是$（\ log n）^{1/6} n^{1/2} n^{ - 1/2} $，并且bose the counce the counce tocors tocuns tocuns tocuns tocuns tocuns tocuns tocor tos tos toce tos toce tos toce tos toce tos of toce of formation of toce of forage tos of。 $ [ - n，n]^4 $是顺序$（\ log n）^{2/3+o（1）} n^{ - 2} $。我们证明的一个重要部分是证明可能具有独立感兴趣的四维循环随机步行的能力估计。 我们的结果表明，阿贝尔·沙珀模型还表现出非平凡的多层次校正以在四个维度上进行平均场缩放，尽管它仍然开放以计算这些校正的精确顺序。
We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional 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 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.
