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Description
The discrete Gaussian free field (DGFF) is a famous model in statistical mechanics. In this work we study the scaling limit of a non-linear transformation of its gradient. More precisely, we study the (centered) square of the norm of the gradient DGFF at every point of a square lattice. One of the reasons for studying this object stems from the so-called Abelian sandpile, which is an example of a dynamical system displaying self-organized criticality. Surprisingly, our real-valued model is connected to the height-one field in the sandpile, which only assumes the values 0 or 1. With different methods we are able to obtain the same scaling limits: on the one hand, we show an identity relating the cumulants of our model to those of the height-one field. Besides, we show our field converges to white noise in the limit, as it happens for the height-one field. Joint work with Alessandra Cipriani (UCL), Rajat Subhra Hazra (Leiden) and Wioletta Ruszel (Utrecht).
