论文标题
在高维盒上随机随机步行的两点函数
Two-point functions of random-length random walk on high-dimensional boxes
论文作者
论文摘要
我们将在$ \ zz^d $中使用$ d \ ge3 $的有限盒子上的一般性随机步行一般类随机步行的两点功能，并以其行为提供精确的渐近造型。我们表明，当典型的步行长度为$ O（l^2）$时，有限盒的两点功能是无限时间矩阵两点函数渐近的，但是当典型的步行长度为$ω（l^2）$时会发展出平稳性。我们还在数字上研究步行的长度矩和限制五维托里（Tori）自我避免行走和伊辛模型的分布，并发现它们与在完整图上的自我避免行走的已知结果一致，无论是在临界点还是在广泛的缩放范围/伪智力点上。此外，我们表明，有限盒随机长度随机步行的两点函数，通过完整的图形自我避免步行选择步行长度，在数值上与五维摩托车上的自避免行走和Ising模型的两点函数一致。我们猜想这些观察值在五个维度上也应在所有更高的维度中保持。
We study the two-point functions of a general class of random-length random walks on finite boxes in $\ZZ^d$ with $d\ge3$, and provide precise asymptotics for their behaviour. We show that the finite-box two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is $o(L^2)$, but develops a plateau when the typical walk length is $Ω(L^2)$. We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box random-length random walk, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.
