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

TL;DR

This paper refines the estimate of the critical point for spread-out lattice trees and animals in high dimensions, providing an explicit asymptotic expansion with model-dependent constants.

Contribution

It establishes a precise asymptotic expansion for the critical point in high dimensions, improving previous bounds by using a novel lace expansion approach.

Findings

Critical point $p_c$ approximates $1/e + CL^{-d}$ in high dimensions.

Explicit formulas for model-dependent constants $C_{LT}$ and $C_{LA}$.

Enhanced understanding of phase transition thresholds in lattice models.

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 $d\ge1$. 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.