Logarithmic Corrections to Scaling in the Four-dimensional Uniform Spanning Tree
Logarithmic Corrections to Scaling in the Four-dimensional Uniform Spanning Tree
复制标题
DOI：
10.1007/s00220-023-04686-w
复制
发表时间：
2023-04-04
影响因子：
2.4
通讯作者：
Sousi，Perla
中科院分区：
文献类型：
--
作者：
Hutchcroft，Tom;Sousi，Perla
我们计算精确的对数修正的平均场标度的各种数量描述的均匀生成树的四维超立方晶格。我们特别感兴趣的是起源的过去的分布，也就是说，树的有限部分被起源与无限分开。我们证明了过去包含一条长度为n的路的概率是有序的，过去包含至少n个顶点的概率是有序的，过去到达盒子边界的概率是有序的。我们的证明的一个重要部分是证明浓度估计的能力的四维环擦除随机游走，这可能是独立的利益。我们的研究结果意味着阿贝尔沙堆模型也表现出非平凡的多对数校正的平均场标度在四维，虽然它仍然是开放的，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. We are particularly interested in the distribution of thepastof 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 lengthnis of order, that the probability that the past contains at leastnvertices is of order, and that the probability that the past reaches the boundary of the boxis of order. 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.