open access

https://hdl.handle.net/2115/91166

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

英語

critical phenomena

2023-03

Cambridge University Press

Combinatorics, Probability and Computing

33(2)

238-269

A spread-out lattice animal is a finite connected set of edges in {{𝑥,𝑦}⊂ℤ𝑑:0<‖𝑥−𝑦‖≤𝐿}. A lattice tree is a lattice animal with no loops. The best estimate on the critical point 𝑝c so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : 𝑝c=1/𝑒+𝑂(𝐿−2𝑑/7log𝐿) for both models for all 𝑑≥1. In this paper, we show that 𝑝c=1/𝑒+𝐶𝐿−𝑑+𝑂(𝐿−𝑑−1) for all 𝑑>8, where the model-dependent constant 𝐶 has the random-walk representation 𝐶LT=∑𝑛=2∞𝑛+12𝑒𝑈∗𝑛(𝑜),𝐶LA=𝐶LT−12𝑒2∑𝑛=3∞𝑈∗𝑛(𝑜), where 𝑈∗𝑛 is the 𝑛-fold convolution of the uniform distribution on the 𝑑-dimensional ball {𝑥∈ℝ𝑑:‖𝑥‖≤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.

学術雑誌論文

Accepted Manuscript

0963-5483

1469-2163