Gaussian deconvolution and the lace expansion
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912213548138496 |
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| author | Liu, Yucheng Slade, Gordon |
| author_facet | Liu, Yucheng Slade, Gordon |
| contents | We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = δ_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results were obtained in the 2000s, using intricate Fourier analysis. In 2022, a very simple deconvolution theorem was proved, but its applicability was limited. We extend the 2022 theorem to remove its limitations while maintaining its simplicity -- our main tools are Hölder's inequality, weak derivatives, and basic Fourier theory in $L^p$ space. Our motivation comes from critical phenomena in equilibrium statistical mechanics, where the convolution equation is provided by the lace expansion and $G$ is a critical two-point function. Our results significantly simplify existing proofs of critical $|x|^{-(d-2)}$ decay in high dimensions for self-avoiding walk, Ising and $φ^4$ models, percolation, and lattice trees and lattice animals. We also improve previous error estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07635 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gaussian deconvolution and the lace expansion Liu, Yucheng Slade, Gordon Probability Mathematical Physics 42B05, 60K35, 82B27, 82B41, 82B43 We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = δ_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results were obtained in the 2000s, using intricate Fourier analysis. In 2022, a very simple deconvolution theorem was proved, but its applicability was limited. We extend the 2022 theorem to remove its limitations while maintaining its simplicity -- our main tools are Hölder's inequality, weak derivatives, and basic Fourier theory in $L^p$ space. Our motivation comes from critical phenomena in equilibrium statistical mechanics, where the convolution equation is provided by the lace expansion and $G$ is a critical two-point function. Our results significantly simplify existing proofs of critical $|x|^{-(d-2)}$ decay in high dimensions for self-avoiding walk, Ising and $φ^4$ models, percolation, and lattice trees and lattice animals. We also improve previous error estimates. |
| title | Gaussian deconvolution and the lace expansion |
| topic | Probability Mathematical Physics 42B05, 60K35, 82B27, 82B41, 82B43 |
| url | https://arxiv.org/abs/2310.07635 |