Cipriani, A; Hazra, RS; Rapoport, A; Ruszel, WM; (2023) Properties of the Gradient Squared of the Discrete Gaussian Free Field. Journal of Statistical Physics , 190 (11) , Article 171. 10.1007/s10955-023-03187-3.

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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-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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