تقرير
Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous
| العنوان: | Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous |
|---|---|
| المؤلفون: | Schiavo, Lorenzo Dello, Herry, Ronan, Kopfer, Eva, Sturm, Karl-Theodor |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, 60J65, 60K37, 60J67 |
| الوصف: | 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\|(-\Delta_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $\Delta_{L}$ 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*}\mu_{L}(dz) = \exp \Big( \gamma h_L(z) - \frac{\gamma^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz,\end{equation*} where $\gamma$ 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 $\mu_L$ to the LQG measure $\mu$ on $\mathbb{T}^n$, for all $|\gamma| < \sqrt{2n}$. Comment: 33 pages, 5 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2302.02963 |
| رقم الأكسشن: | edsarx.2302.02963 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |