Properties of the gradient squared of the discrete Gaussian free field

TL;DR

This paper investigates the properties of the gradient squared of the discrete Gaussian free field, demonstrating its convergence to white noise and analyzing correlation functions, with implications for related models like the Abelian sandpile.

Contribution

It establishes the convergence of the gradient squared of the discrete Gaussian free field to white noise and characterizes its correlation functions in the continuum limit.

Findings

Rescaled gradient squared converges to white noise.

Derived explicit formulas for correlation functions and cumulants.

Connected results to the Abelian sandpile model in two dimensions.

Abstract

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 $d=2$.