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

Yucheng Liu, Gordon Slade

TL;DR

The paper rigorously analyzes spread-out lattice trees and lattice animals in dimensions above the upper critical dimension $d_c=8$, establishing a precise mean-field-like correlation-length exponent of $1/4$ via the mass $m(p)$ and proving a near-critical two-point bound with exponential decay governed by $(p_c-p)^{1/4}$. It develops a tilted-susceptibility framework and employs the lace expansion with exponential tilts to relate the susceptibility to the mass, enabling control of both infinite-lattice and finite-size (torus) behaviors. The main finite-size result shows a plateau phenomenon on the torus: at scaling distance $p_c-p\sim V^{-1/2}$, the torus susceptibility scales as $V^{1/4}$ and the torus two-point function behaves as $|x|^{-(d-2)}+V^{-3/4}$, consistent with universal finite-size-scaling predictions. The methods combine refined diagrammatic lace-expansion bounds, massive infrared bounds, and two bootstrap schemes, extending previous high-dimensional results and strengthening the universality picture for branched polymers in the mean-field regime.

Abstract

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 $(p_c-p)^{-1/4}$ 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 $d > d_{\mathrm c}$) 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.

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

TL;DR

The paper rigorously analyzes spread-out lattice trees and lattice animals in dimensions above the upper critical dimension , establishing a precise mean-field-like correlation-length exponent of via the mass and proving a near-critical two-point bound with exponential decay governed by . It develops a tilted-susceptibility framework and employs the lace expansion with exponential tilts to relate the susceptibility to the mass, enabling control of both infinite-lattice and finite-size (torus) behaviors. The main finite-size result shows a plateau phenomenon on the torus: at scaling distance , the torus susceptibility scales as and the torus two-point function behaves as , consistent with universal finite-size-scaling predictions. The methods combine refined diagrammatic lace-expansion bounds, massive infrared bounds, and two bootstrap schemes, extending previous high-dimensional results and strengthening the universality picture for branched polymers in the mean-field regime.

Abstract

We consider spread-out models of lattice trees and lattice animals on , for above the upper critical dimension . We define a correlation length and prove that it diverges as at the critical point . Using this, we prove that the near-critical two-point function is bounded above by . We apply the near-critical bound to study lattice trees and lattice animals on a discrete -dimensional torus (with ) of volume . For of order , we prove that the torus susceptibility is of order , and that the torus two-point function behaves as and thus has a plateau of size . The proofs require significant extensions of previous results obtained using the lace expansion.

Paper Structure

This paper contains 35 sections, 32 theorems, 250 equations, 4 figures.

Table of Contents

  1. Introduction and results
  2. Introduction
  3. Lattice trees and lattice animals
  4. The correlation length
  5. Results
  6. Results for the infinite lattice
  7. Results for the torus
  8. Organisation
  9. Finite-size scaling: proof of Theorem \ref{['thm:torus']}
  10. Lift from the torus to the infinite lattice
  11. Lower bounds on $G_p$ and $\psi_p$
  12. Upper bound on $E_p$
  13. Proof of Theorem \ref{['thm:torus']}
  14. The tilted susceptibility
  15. Divergence of the tilted susceptibility
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $d>8$ and $L \ge L_0$ with $L_0$ sufficiently large. For lattice trees, there is a constant $a_L>0$ such that $m(p) \sim a_L (p_c-p)^{1/4}$. For lattice animals, there are $L$-dependent constants such that $m(p)\asymp (p_c-p)^{1/4}$.

Figures (4)

  • Figure 1: The lift of a lattice animal from $\mathbb{T}_5^2$ to $\mathbb{Z}^2$. A spanning tree and its lift are shown in black. The two excess edges are shown in grey. Note that the lift of an animal can break torus cycles.
  • Figure 2: $\mathbb{Z}^d$ configurations contributing to $E_{p}(x)$. Lines represent $G_p$, hollow vertices are $0,x'$, box vertices are $y,y'$, and filled vertices are summed over $\mathbb Z^d$. The vertices $y,y'$ are distinct and torus equivalent.
  • Figure 3: Torus configurations contributing to $E_p(x)$. Lines represent $\Gamma_p$, hollow vertices are $0,x$, and filled vertices are summed over the torus. The dashed lines with $\times$ represent $\Gamma_p^{\star 2}-G_p^{*2}\lesssim m(p)^{-4}r^{-d}$, and the last dashed line represents $\Gamma_p - G_{p} \lesssim m(p)^{-2}r^{-d}$.
  • Figure 4: One example of diagrams for $\Pi_p ^{(4)}(x)$ for lattice trees (left) and for lattice animals (right). There are $2^{N-1}$ similar diagrams for $\Pi_p ^{(N)}(x)$ for $N\ge 1$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Conjecture 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 57 more