Properties of the gradient squared of the discrete Gaussian free field

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

Published 2022-07-19Version 1

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in $U_{\epsilon}=U/\epsilon\cap \mathbb{Z}^d$, $U\subset \mathbb{R}^d$ and $d\geq 2$. 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\"older space. Moreover, under a different rescaling, we determine the $k$-point correlation function and cumulants on $U_{\epsilon}$ and in the continuum limit as $\epsilon\to 0$. This result is related to the analogue limit for the height-one field of the Abelian sandpile (\citet{durre}), with the same conformally covariant property in $d=2$.