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

Abstract

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 $d\ge 1$. In this paper, we show that $p_c=1/e+C{L}^{-d}+O(L^{-d-1})$ for all $d>8$, where the model-dependent constant $C$ has the random-walk representation $C_{\mathrm{LT}}={\sum }_{n=2}^{\infty }\frac{n+1}{2e}{U}^{\ast n}(o)$ and $C_{\mathrm{LA}}=C_{\mathrm{LT}}-\frac{1}{2e^2}{\sum }_{n=3}^{\infty }{U}^{\ast n}(o)$, where $U^{\ast n}$ is the $n$-fold convolution of the uniform distribution on the $d$-dimensional ball $\{x\in {\mathbb{R}}^d:\Vert x\Vert \le 1\}$. 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.

arXiv comment36pages, 35figures