Correct bounds on the Ising lace-expansion coefficients

Lace expansion for the Ising model

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-poi…

量子 Ising 磁化率的平均场行为及经典 Ising 模型的一种新花边展开

Mean-field behavior of the quantum Ising susceptibility and a new lace expansion for the classical Ising model

本文利用随机几何表示证明 d>4 维横场 Ising 模型磁化率在临界逆温度处呈平均场型 $(β_c-β)^{-1}$ 发散，并导出经典 Ising 模型的新花边展开。

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)

Application of the lace expansion to the $φ^4$ model

Using the Griffiths-Simon construction of the $φ^4$ model and the lace expansion for the Ising model, we prove that, if the strength $λ\ge0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, then the critical $φ^4$ two-point function $\langleφ_oφ_x\rangle_{μ_c}$ is asymptotically $|x|^{2-d}$ times a model-dependent constant, and the critical point is estimated as $μ_c=\mathscr{\hat J}-\frac\lambda2\langleφ_o^2\rangle_{μ_c}+O(λ^2)$, where $\mathscr{\hat J}$ is the massless point for the Gaussian model.

高斯反卷积与 Lace 展开在稀疏模型中的应用

Gaussian deconvolution and the lace expansion for spread-out models

本文通过高斯反卷积定理和 Lace 展开，证明了稀疏模型在高于上临界维数时临界两点函数的衰减性质，并提供了更简洁的证明方法。

高斯反卷积与花边展开

Gaussian deconvolution and the lace expansion

本文扩展了2022年的反卷积定理，利用 Hölder 不等式等基础工具证明了卷积方程解的高斯衰减，简化了统计力学中多种高维模型临界衰减的证明并改进了误差估计。