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High-dimensional long-range statistical mechanical models have random walk correlation functions

Yucheng Liu

TL;DR

This work analyzes long-range statistical mechanical models on $\mathbb{Z}^d$ with couplings decaying as $|x|^{-(d+α)}$, proving an exact random-walk representation for the two-point function in the spread-out regime above the upper critical dimension: $G_p(x)=A_p\,S_{\mu_p}(x)$. Using this representation, the authors derive near-critical decay bounds for the subcritical two-point function that interpolate between mean-field behaviour and long-range decay, with explicit forms for $0<α<2$ and logarithmic corrections at $α=2$. The lace expansion is shown to converge via a bootstrap argument, and the two-point function analysis is reduced to controlled random-walk estimates with a perturbed one-step distribution $P$. These results enable a streamlined approach to mean-field analyses for long-range models and provide a simple route to related torus-plateau phenomena on high-dimensional tori. Overall, the paper significantly clarifies the link between interacting long-range models and an exact random-walk surrogate in high dimensions.

Abstract

We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+α)}$ where $0 < α\le 2$, above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for $0<α< 2$, we prove upper and lower bounds of the form $|x|^{-(d-α)} \min\{ 1, (p_c - p)^{-2} |x|^{-2α} \}$ for the two-point function near the critical point $p_c$. For $α=2$, we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates.

High-dimensional long-range statistical mechanical models have random walk correlation functions

TL;DR

This work analyzes long-range statistical mechanical models on with couplings decaying as , proving an exact random-walk representation for the two-point function in the spread-out regime above the upper critical dimension: . Using this representation, the authors derive near-critical decay bounds for the subcritical two-point function that interpolate between mean-field behaviour and long-range decay, with explicit forms for and logarithmic corrections at . The lace expansion is shown to converge via a bootstrap argument, and the two-point function analysis is reduced to controlled random-walk estimates with a perturbed one-step distribution . These results enable a streamlined approach to mean-field analyses for long-range models and provide a simple route to related torus-plateau phenomena on high-dimensional tori. Overall, the paper significantly clarifies the link between interacting long-range models and an exact random-walk surrogate in high dimensions.

Abstract

We consider long-range percolation, Ising model, and self-avoiding walk on , with couplings decaying like where , above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for , we prove upper and lower bounds of the form for the two-point function near the critical point . For , we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates.
Paper Structure (11 sections, 11 theorems, 98 equations)

This paper contains 11 sections, 11 theorems, 98 equations.

Table of Contents

  1. Introduction and results
  2. Introduction
  3. The models
  4. Main results
  5. Lace expansion
  6. Strategy of proof
  7. Random walk two-point function
  8. Proof of Theorem \ref{['thm:G=S']}
  9. Bootstrap argument
  10. Proof of Proposition \ref{['prop:lace']}
  11. Characterisation of critical point

Key Result

Theorem 1.2

Let $\alpha, d > 0$ be such that $d > d_c$ and $\alpha \le 2 + (\ell - 1)(d - d_c)$. If $\alpha = 2$, we furthur allow $d = d_c$. Let $D$ (for percolation or SAW) or $J$ (for Ising) obey Assumption ass:D, with $L \ge L_0$ where $L_0$ is sufficiently large. Then for all $p\in [\frac{1}{2}, p_c]$, the where $S_\mu$ denotes the random walk two-point function generated by $P$. Moreover, we have unifo

Theorems & Definitions (23)

  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof : Proof of Lemma \ref{['lem:heat_kernel']}
  • proof : Proof of Proposition \ref{['prop:S']}
  • ...and 13 more