论文标题
晶格树和晶格动物的临界点的扩散极限d> 8
Spread-out limit of the critical points for lattice trees and lattice animals in dimensions d>8
论文作者
论文摘要
扩展的晶格动物是$ \ {\ {x，y \} \ subset \ mathbb {z}^d：0 <|| x-y | | | | le l \ \} $中的有限连接的边缘。晶格树是一种晶格动物，没有回路。在本文中，我们表明$ p_c = 1/e+cl^{ - d}+o（l^{ - d-1}）$对于所有$ d> 8 $ $ c_ \ mathrm {lt} = \ sum_ {n = 2}^\ infty \ frac {n+1} {2e} u^{*n}（n}（o）$和$ c_ \ mathrm {la} = c_ \ mathrm {c_ \ mathrm {lt} - {lt} - {lt} - \ frac frac}} u^{*n}（o）$，其中$ u^{*n} $是$ n $ -fold卷积$ d $ -dimensional Ball $ \ \ {x \ in \ Mathbb {r}^d：该证明是基于蕾丝扩展对两点函数的新颖使用，并以一定值为$ p $的一定值对1点函数进行了详细分析，该函数旨在使分析非常简单。
A spread-out lattice animal is a finite connected set of edges in $\{ \{x,y\} \subset \mathbb{Z}^d:0<||x-y||\le L \}$. A lattice tree is a lattice animal with no loops.The best estimate on the critical point $p_c$ so far was achieved by Penrose(JSP,77(1994):3-15): $p_c=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge1$. In this paper, we show that $p_c=1/e+CL^{-d}+O(L^{-d-1})$ for all $d>8$, where the model-dependent constant $C$ has the random-walk representation $C_\mathrm{LT}=\sum_{n=2}^\infty\frac{n+1}{2e}U^{*n}(o)$ and $C_\mathrm{LA}=C_\mathrm{LT}-\frac1{2e^2}\sum_{n=3}^\infty U^{*n}(o)$, where $U^{*n}$ is the $n$-fold convolution of the uniform distribution on the $d$-dimensional ball $\{x\in \mathbb{R}^d:\|x\|\le1\}$. The proof is based on a novel use of the lace expansion for the two-point function and detailed analysis of the 1-point function at a certain value of $p$ that is designed to make the analysis extreamly simple.
