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

Dello Schiavo, Lorenzo; Herry, Ronan; Kopfer, Eva; Sturm, Karl Theodor

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

Dello Schiavo L, Herry R, Kopfer E, Sturm KT. 2025. Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous. Mathematische Nachrichten. 298(1), 244–281.

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2025_MathNachrichten_DelloSchiavo.pdf 1.73 MB [Published Version]

DOI

10.1002/mana.202400169

Journal Article | Published | English

Scopus indexed

Author

Dello Schiavo, Lorenzo^ISTA ; Herry, Ronan; Kopfer, Eva; Sturm, Karl Theodor

Department

Maas Group

Grant

Optimal Transport and Stochastic Dynamics
Taming Complexity in Partial Differential Systems
Configuration Spaces over Non-Smooth Spaces

Abstract

For an arbitrary dimension (Formula presented.), we study: the polyharmonic Gaussian field (Formula presented.) on the discrete torus (Formula presented.), that is the random field whose law on (Formula presented.) given by (Formula presented.) where (Formula presented.) is the Lebesgue measure and (Formula presented.) is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on (Formula presented.) (Formula presented.) where (Formula presented.) is a regularity parameter. As (Formula presented.), we prove convergence of the fields (Formula presented.) to the polyharmonic Gaussian field (Formula presented.) on the continuous torus (Formula presented.), as well as convergence of the random measures (Formula presented.) to the LQG measure (Formula presented.) on (Formula presented.), for all (Formula presented.).

Publishing Year

2025

Date Published

2025-01-01

Journal Title

Mathematische Nachrichten

Publisher

Wiley

Acknowledgement

KTS is grateful to Christoph Thiele for valuable discussions and helpful references. LDS is grateful to Nathanaël Berestycki for valuable discussions on Gaussian Multiplicative Chaoses. The authors are grateful to an anonymous reviewer for suggestions which improved the presentation. The authors gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft through the project ‘Random Riemannian Geometry’ within the SPP 2265 ‘Random Geometric Systems.' LDS gratefully acknowledges financial support from the European Research Council (grant agreement No. 716117, awarded to J. Maas) and from the Austrian Science Fund (FWF). His research was funded by the Austrian Science Fund (FWF) project 10.55776/F65 and project 10.55776/ESP208. RH, EK, and KTS gratefully acknowledge funding by the Hausdorff Center for Mathematics (project ID 390685813), and through project B03 within the CRC 1060 (project ID 211504053). RH and KTS also gratefully acknowledges financial support from the European Research Council through the ERC AdG ‘RicciBounds’ (grant agreement 694405). Open access funding enabled and organized by Projekt DEAL.

Volume

298

Issue

Page

244-281

ISSN

0025-584X

eISSN

1522-2616

IST-REx-ID

18632

Cite this

Dello Schiavo L, Herry R, Kopfer E, Sturm KT. Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous. Mathematische Nachrichten. 2025;298(1):244-281. doi:10.1002/mana.202400169

Dello Schiavo, L., Herry, R., Kopfer, E., & Sturm, K. T. (2025). Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous. Mathematische Nachrichten. Wiley. https://doi.org/10.1002/mana.202400169

Dello Schiavo, Lorenzo, Ronan Herry, Eva Kopfer, and Karl Theodor Sturm. “Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: From Discrete to Continuous.” Mathematische Nachrichten. Wiley, 2025. https://doi.org/10.1002/mana.202400169.

L. Dello Schiavo, R. Herry, E. Kopfer, and K. T. Sturm, “Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous,” Mathematische Nachrichten, vol. 298, no. 1. Wiley, pp. 244–281, 2025.

Dello Schiavo, Lorenzo, et al. “Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: From Discrete to Continuous.” Mathematische Nachrichten, vol. 298, no. 1, Wiley, 2025, pp. 244–81, doi:10.1002/mana.202400169.

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