Near-Critical and Finite-Size Scaling for High-Dimensional Lattice Trees and Animals

Abstract

We consider spread-out models of lattice trees and lattice animals on \({\mathbb {Z}}^d\), for d above the upper critical dimension \(d_{\textrm{c}}=8\). We define a correlation length and prove that it diverges as \((p_c-p)^{-1/4}\) at the critical point \(p_c\). Using this, we prove that the near-critical two-point function is bounded above by \(C|x|^{-(d-2)}\exp [-c(p_c-p)^{1/4}|x|]\). We apply the near-critical bound to study lattice trees and lattice animals on a discrete d-dimensional torus (with \(d > d_{\textrm{c}}\)) of volume V. For \(p_c-p\) of order \(V^{-1/2}\), we prove that the torus susceptibility is of order \(V^{1/4}\), and that the torus two-point function behaves as \(|x|^{-(d-2)} + V^{-3/4}\) and thus has a plateau of size \(V^{-3/4}\). The proofs require significant extensions of previous results obtained using the lace expansion.