Properties of the Gradient Squared of the Discrete Gaussian Free Field

Alessandra Cipriani; Rajat S. Hazra; Alan Rapoport; Wioletta M. Ruszel

doi:10.1007/s10955-023-03187-3

Properties of the Gradient Squared of the Discrete Gaussian Free Field

Alessandra Cipriani, Rajat S. Hazra, Alan Rapoport, Wioletta M. Ruszel^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in U_ε= U/ ε∩ Z^d , U⊂ R^d and d≥ 2 . The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the k-point correlation function and joint cumulants on U_ε and in the continuum limit as ε→ 0 . This result is related to the analogue limit for the height-one field of the Abelian sandpile (Dürre in Stoch Process Appl 119(9):2725–2743, 2009), with the same conformally covariant property in d= 2 .

Original language	English
Article number	171
Number of pages	33
Journal	Journal of Statistical Physics
Volume	190
Issue number	11
DOIs	https://doi.org/10.1007/s10955-023-03187-3
Publication status	Published - 4 Nov 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s).

Funding

AC initiated this work at TU Delft funded by Grant 613.009.102 of the Dutch Organisation for Scientific Research (NWO). RSH was supported by a STAR cluster visitor grant during a visit to TU Delft where part of this work was carried out. AR is supported by Klein-2 Grant OCENW.KLEIN.083 and did part of the work at TU Delft.

Funders	Funder number
STAR	OCENW.KLEIN.083
Technische Universiteit Delft	613.009.102
Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Keywords

Abelian sandpile model
Besov–Hölder spaces
Cumulants
Fock spaces
Gaussian free field
K-point correlation functions
Point processes
Scaling limit

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Cite this

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title = "Properties of the Gradient Squared of the Discrete Gaussian Free Field",

abstract = "In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in Uε= U/ ε∩ Zd , U⊂ Rd and d≥ 2 . The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-H{\"o}lder space. Moreover, under a different rescaling, we determine the k-point correlation function and joint cumulants on Uε and in the continuum limit as ε→ 0 . This result is related to the analogue limit for the height-one field of the Abelian sandpile (D{\"u}rre in Stoch Process Appl 119(9):2725–2743, 2009), with the same conformally covariant property in d= 2 .",

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AB - In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in Uε= U/ ε∩ Zd , U⊂ Rd and d≥ 2 . The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the k-point correlation function and joint cumulants on Uε and in the continuum limit as ε→ 0 . This result is related to the analogue limit for the height-one field of the Abelian sandpile (Dürre in Stoch Process Appl 119(9):2725–2743, 2009), with the same conformally covariant property in d= 2 .

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KW - Besov–Hölder spaces

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KW - Fock spaces

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Properties of the Gradient Squared of the Discrete Gaussian Free Field

Abstract

Bibliographical note

Funding

Keywords

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