Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous
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| Format: | Preprint |
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2023
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| _version_ | 1866915064156520448 |
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| author | Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor |
| author_facet | Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor |
| contents | For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus $\mathbb{T}^n_L = \frac{1}{L} \mathbb{Z}^{n} / \mathbb{Z}^{n}$, that is the random field whose law on $\mathbb{R}^{\mathbb{T}^{n}_{L}}$ given by \begin{equation*}
c_n\, e^{-b_n\|(-Δ_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $Δ_{L}$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on $\mathbb{T}^{n}_{L}$ \begin{equation*}μ_{L}(dz) = \exp \Big( γh_L(z) - \frac{γ^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz,\end{equation*} where $γ$ is a regularity parameter. As $L\to\infty$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus $\mathbb{T}^n = \mathbb{R}^{n} / \mathbb{Z}^{n}$, as well as convergence of the random measures $μ_L$ to the LQG measure $μ$ on $\mathbb{T}^n$, for all $|γ| < \sqrt{2n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_02963 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor Probability 60J65, 60K37, 60J67 For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus $\mathbb{T}^n_L = \frac{1}{L} \mathbb{Z}^{n} / \mathbb{Z}^{n}$, that is the random field whose law on $\mathbb{R}^{\mathbb{T}^{n}_{L}}$ given by \begin{equation*} c_n\, e^{-b_n\|(-Δ_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $Δ_{L}$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on $\mathbb{T}^{n}_{L}$ \begin{equation*}μ_{L}(dz) = \exp \Big( γh_L(z) - \frac{γ^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz,\end{equation*} where $γ$ is a regularity parameter. As $L\to\infty$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus $\mathbb{T}^n = \mathbb{R}^{n} / \mathbb{Z}^{n}$, as well as convergence of the random measures $μ_L$ to the LQG measure $μ$ on $\mathbb{T}^n$, for all $|γ| < \sqrt{2n}$. |
| title | Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous |
| topic | Probability 60J65, 60K37, 60J67 |
| url | https://arxiv.org/abs/2302.02963 |