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

Abstract

A spread-out lattice animal is a finite connected set of edges in $\{\{x,y\}\subset \mathbb{Z}^d\;:\;0\lt \|x-y\|\le L\}$ . A lattice tree is a lattice animal with no loops. The best estimate on the critical point $p_{\textrm{c}}$ so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : $p_{\textrm{c}}=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge 1$ . In this paper, we show that $p_{\textrm{c}}=1/e+CL^{-d}+O(L^{-d-1})$ for all $d\gt 8$ , where the model-dependent constant $C$ has the random-walk representation \begin{align*} C_{\textrm{LT}}=\sum _{n=2}^\infty \frac{n+1}{2e}U^{*n}(o),&& C_{\textrm{LA}}=C_{\textrm{LT}}-\frac 1{2e^2}\sum _{n=3}^\infty U^{*n}(o), \end{align*} where $U^{*n}$ is the $n$ -fold convolution of the uniform distribution on the $d$ -dimensional ball $\{x\in{\mathbb R}^d\;: \|x\|\le 1\}$ . The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of $p$ that is designed to make the analysis extremely simple.