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Spread-out limit of the critical points for lattice trees and lattice animals in dimensions $\boldsymbol{d}\boldsymbol\gt \textbf{8}$

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  20 November 2023

Noe Kawamoto Open the ORCID record for Noe Kawamoto [Opens in a new window]  and
Akira Sakai Open the ORCID record for Akira Sakai [Opens in a new window]
Noe Kawamoto*
Affiliation:
Graduate School of Science, Hokkaido University, Sapporo, Japan
Akira Sakai
Affiliation:
Faculty of Science, Hokkaido University, Sapporo, Japan
*
Corresponding author: Noe Kawamoto; Email: noe0717kawa@gmail.com
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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.

Keywords

Lattice treeslattice animalscritical phenomena

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B27: Critical phenomena
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 33 , Issue 2 , March 2024 , pp. 238 - 269
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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