Hammond and Sheffield (2013) recently introduced a model of correlated random walk of which the partial sums scale to a fractional Brownian motion with long-range dependence. In this talk, we consider a natural generalization of this model to higher dimensions. We define a $\mathbb{Z}^{d}$-indexed random field with dependence determined by an underlying random graph, and we study the scaling limit of its partial sums over increasing rectangles. An interesting phenomenon occurs: when the rectangles increase at different rates, different limiting fields may arise. In particular, there is a critical regime where the limiting field is operator-scaling, while this is not the case in other regimes.
Joint work with Hermine Biermé and Olivier Durieu.