标题： 临界点的扩展极限在维度d>8时的格点树和格点动物 显示英文标题

标题： Spread-out limit of the critical points for lattice trees and lattice animals in dimensions d>8

摘要： 一个分散的格点动物是$\{ \{x,y\} \subset \mathbb{Z}^d:0<||x-y||\le L \}$中的一个有限连通边集。 一个格点树是一个没有环的格点动物。到目前为止，对临界点$p_c$的最佳估计是由 Penrose (JSP,77(1994):3-15) 获得的：对于所有$d\ge1$，两种模型均为$p_c=1/e+O(L^{-2d/7}\log L)$。 在本文中，我们证明了对于所有$d>8$，$p_c=1/e+CL^{-d}+O(L^{-d-1})$，其中模型相关的常数$C$具有随机游走表示$C_\mathrm{LT}=\sum_{n=2}^\infty\frac{n+1}{2e}U^{*n}(o)$和$C_\mathrm{LA}=C_\mathrm{LT}-\frac1{2e^2}\sum_{n=3}^\infty U^{*n}(o)$，其中$U^{*n}$是单位分布在$d$维球体$\{x\in \mathbb{R}^d:\|x\|\le1\}$上的$n$重卷积。 证明基于对两点函数的勒斯展开的新颖应用，以及在某个设计为使分析极其简单的$p$值处对一点函数的详细分析。 显示英文摘要

摘要： 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.