Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

Lorenzo Dello Schiavo, Ronan Herry, Eva Kopfer and Karl‐Theodor Sturm

Mathematische Nachrichten, 2025, vol. 298, issue 1, 244-281

Abstract: For an arbitrary dimension n$n$, we study: the polyharmonic Gaussian field hL$h_L$ on the discrete torus TLn=1LZn/Zn$\mathbb {T}^n_L = \frac{1}{L} \mathbb {Z}^{n} / \mathbb {Z}^{n}$, that is the random field whose law on RTLn$\mathbb {R}^{\mathbb {T}^{n}_{L}}$ given by cne−bn(−ΔL)n/4h2dh,$$\begin{equation*} \hspace*{-4.5pc}c_n\, \text{e}^{-b_n{\left\Vert (-\Delta _L)^{n/4}h\right\Vert} ^2} dh, \end{equation*}$$where dh$dh$ is the Lebesgue measure and ΔL$\Delta _{L}$ is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on TLn$\mathbb {T}^{n}_{L}$ μL(dz)=expγhL(z)−γ22EhL(z)dz,$$\begin{equation*} \hspace*{-7.5pc}\mu _{L}(dz) = \exp {\left(\gamma h_L(z) - \frac{\gamma ^{2}}{2} \mathbf {E} h_{L}(z) \right)} dz, \end{equation*}$$where γ$\gamma$ is a regularity parameter. As L→∞$L\rightarrow \infty$, we prove convergence of the fields hL$h_L$ to the polyharmonic Gaussian field h$h$ on the continuous torus Tn=Rn/Zn$\mathbb {T}^n = \mathbb {R}^{n} / \mathbb {Z}^{n}$, as well as convergence of the random measures μL$\mu _L$ to the LQG measure μ$\mu$ on Tn$\mathbb {T}^n$, for all γ

Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202400169

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:298:y:2025:i:1:p:244-281

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().