Near-critical and finite-size scaling for high-dimensional lattice trees and animals

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