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 \(U_{\varepsilon }=U/\varepsilon \cap \mathbb {Z}^d\), \(U\subset \mathbb {R}^d\) and \(d\ge 2\). 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 \(U_{\varepsilon }\) and in the continuum limit as \(\varepsilon \rightarrow 0\). 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 \(d=2\).