Liu and Slade’s Deconvolution Strategy

• Liu and Slade’s deconvolution strategy is a rigorous analytic method that isolates dominant Gaussian contributions in high-dimensional convolution equations to reveal mean-field behavior.
• The approach combines a Gaussian deconvolution theorem with diagrammatic and lace expansion techniques to control the decay of two-point functions in both lattice and continuum settings.
• Its successful application in models like percolation, self-avoiding walks, and critical phenomena simplifies previous methods and unifies the treatment of complex stochastic systems.

The deconvolution strategy of Liu and Slade refers to a class of rigorous analytic methods developed for solving high-dimensional convolution equations, particularly those arising in statistical mechanics and percolation theory, with an emphasis on precise control of the asymptotic decay of correlation or two-point functions. Central to this strategy is the combination of an advanced Gaussian deconvolution theorem with diagrammatic and lace expansion techniques, enabling the explicit extraction of the leading scaling behavior in models above the upper critical dimension in both lattice and continuum settings. This framework yields sharp results for the decay of critical correlation functions and simplifies prior approaches by isolating Gaussian (mean-field) contributions from model-dependent corrections.

1. Convolution Equations and Physical Context

The underlying analytic problem addresses convolution equations of the form

$(\delta - J) * G = g$

on either $\mathbb{Z}^d$ or $\mathbb{R}^d$ for $d > 2$, where $J$ and $g$ are real-valued, even, and satisfy normalization and moment conditions. In physical models, $G(x)$ often represents the critical two-point (connection or correlation) function, $\delta$ denotes the Dirac delta or Kronecker delta, and $J$ encodes the spread-out transition probabilities or interaction kernel. The underlying context includes spread-out self-avoiding walks, Ising models, and continuum percolation at or above the upper critical dimension.

The principal goal is to control the decay of $G(x)$ for large $|x|$, and establish that it exhibits the mean-field behavior, specifically

$$G(x) \sim \frac{a_d}{ (\det \Sigma)^{1/2} (x \cdot \Sigma^{-1} x)^{(d-2)/2}},$$

where $\Sigma$ is a (possibly anisotropic) positive-definite diagonal matrix constructed from the second moments of $J$ and $a_d$ is an explicit dimension-dependent constant (Liu et al., 2023, Liu, 25 Nov 2024, Dickson et al., 25 Jul 2025).

2. Gaussian Deconvolution Theorem

A central technical component is the Gaussian deconvolution theorem. The method proceeds by taking the Fourier transform of the convolution equation:

$$\widehat{G}(k) = \frac{\widehat{g}(k)}{1 - \widehat{J}(k)}.$$

The critical assumption is that $\widehat{J}(0) = 1$ (criticality) and near $k=0$,

$1 - \widehat{J}(k) \asymp K_{IR} |k|^2,$

for some $K_{IR} > 0$, reflecting diffusive scaling. For large $|x|$, this leads (by inverse Fourier analysis) to a decay in $G(x)$ matching the Green function of the Laplacian:

$$G(x) \sim \frac{\text{const.}}{(\det \Sigma)^{1/2}(x \cdot \Sigma^{-1} x)^{(d-2)/2}}.$$

Correction terms, arising from non-Gaussian contributions, are shown to be subleading: the theorem provides control of the error such that $f(x) = o(|x|^{-(d-2)})$ (Liu et al., 2023, Liu, 25 Nov 2024).

The method achieves this by splitting $G(x)$ into a leading "random walk" or Gaussian part, constructed from the second moment matrix $\Sigma$ of $J$, and an explicit remainder, and then systematically estimating the remainder using Fourier regularity and moment bounds.

3. Lace Expansion and Diagrammatic Estimates

The convolution equations relevant in the considered models are typically obtained via the lace expansion, which expresses the two-point function in terms of (possibly) infinite sums of weighted paths or diagrams:

$J(x) = z D(x) + \Pi(x)$

on $\mathbb{Z}^d$ or $$J_\lambda(x) = \lambda[\varphi(x) + \Pi_\lambda(x)]$$ on $\mathbb{R}^d$, where $D$ or $\varphi$ is the step distribution, $\lambda$ is the connection intensity, and $\Pi$ consists of diagrammatic corrections (Liu et al., 2023, Liu, 25 Nov 2024, Dickson et al., 25 Jul 2025).

Diagrammatic estimates are essential: one proves that $\Pi(x)$ decays at least as fast as $|x|^{-(d+2+\rho)}$ for some $\rho > 0$, and all relevant moments up to $d-2$ are finite in suitable $L^p$ spaces (Liu, 25 Nov 2024, Dickson et al., 25 Jul 2025). This is verified by expressing $\Pi(x)$ as a sum of convolution diagrams, bounding each using Young's inequality, Hausdorff-Young, and splitting of powers among disjoint paths in the diagram. A central iterative lemma upgrades moment finiteness from lower to higher orders, ensuring the induction proceeds up to $a = d-2$. This establishes the technical inputs needed for the deconvolution theorem to yield sharp asymptotics for $G(x)$.

4. Extension to Continuum and Anisotropic Settings

The method, originally formulated on $\mathbb{Z}^d$ exploiting lattice symmetries, has been generalized to $\mathbb{R}^d$ (Liu, 25 Nov 2024, Dickson et al., 25 Jul 2025). In the continuum, the only requirement is that $J$ and $g$ are even, enabling the theory to accommodate anisotropy. The covariance matrix $$\Sigma = \operatorname{diag} \left( \int J(x) x_1^2 dx, \ldots, \int J(x) x_d^2 dx \right)$$ may differ in each coordinate, and the asymptotic is correspondingly anisotropic. Challenges due to the noncompactness of Fourier dual space are overcome via refined Fourier-analytic and integrability arguments, relying on the moment and $L^p$ bounds derived diagrammatically.

This extension has enabled the analysis of models such as self-repellent Brownian motion and the random connection model in high-dimensional continuum percolation, yielding the same polynomial decay for the two-point connection probability as in the lattice setting (Liu, 25 Nov 2024, Dickson et al., 25 Jul 2025).

5. Applications in Percolation Theory and Critical Phenomena

The method has been applied to several high-dimensional stochastic models:

• Spread-out percolation (lattice and continuum): The critical two-point function (connection probability) $\tau(x)$ satisfies

$$\lambda_c \tau(x) \sim \frac{a_d}{(\lambda_c \det \Sigma)^{1/2}}(x\cdot\Sigma^{-1}x)^{-(d-2)/2}$$

in high-dimensions, confirming mean-field exponents and vanishing anomalous dimension $\eta=0$ (Dickson et al., 25 Jul 2025).

• Self-avoiding walk and Ising model: For $d > 4$, the susceptibility and two-point functions show identical $|x|^{-(d-2)}$ decay (Liu et al., 2023).
• Self-repellent Brownian motion: In $d > 4$, the critical two-point function decays as $|x|^{-(d-2)}$, matching the Laplace Green function (Liu, 25 Nov 2024).

In each case, verification of the moment and decay conditions for $J$ and the diagrammatic expansions is model-specific, but once established, the Gaussian deconvolution theorem produces the universal scaling form. This unifies the treatment of a variety of critical stochastic models and percolative systems.

6. Technical Advances and Simplifications

Compared to previous methods relying on intricate Fourier analysis and bootstrapping, the Liu and Slade deconvolution strategy offers a technically streamlined approach. The key advances include:

• Direct isolation of the dominant Gaussian term via the deconvolution theorem, with transparent error control.
• Reduction of inhomogeneous convolution equations (arising in percolation and Ising models) to simpler impulse equations, ensuring uniform application of the deconvolution argument (Liu et al., 2023).
• Utilization of $L^p$ diagrammatic bounds, facilitating sharper control in continuum settings and simplifying induction schemes over moments (Dickson et al., 25 Jul 2025).

This systematic methodology not only simplifies existing proofs but also extends to new models and settings not tractable by previous approaches.

7. Broader Impact and Further Directions

The Liu and Slade deconvolution strategy synthesizes rigorous techniques from Fourier analysis, probability, and diagrammatic expansions to yield precise asymptotics for key observables in statistical mechanics and percolation. Its flexibility extends to a variety of spread-out models, random connection models in high dimensions, and even settings with anisotropic or spatially inhomogeneous transition kernels. As diagrammatic expansions and deconvolution arguments are ubiquitous in critical phenomena, this framework is poised to play a central role in the analysis of incipient infinite clusters, scaling exponents, and universality classes in high-dimensional stochastic models. Further research may extend these methods to settings with additional structure—for instance, nontrivial boundary conditions, long-range interactions, or spatial constraints.