  Recently, Hammond and Sheffield introduced a model of correlated random walks
that scale to fractional Brownian motions with long-range dependence. In this
paper, we consider a natural generalization of this model to dimension $d\geq
2$. We define a $\mathbb Z^d$-indexed random field with dependence relations
governed by an underlying random graph with vertices $\mathbb Z^d$, and we
study the scaling limits of the partial sums of the random field over
rectangular sets. An interesting phenomenon appears: depending on how fast the
rectangular sets increase along different directions, different random fields
arise in the limit. In particular, there is a critical regime where the limit
random field is operator-scaling and inherits the full dependence structure of
the discrete model, whereas in other regimes the limit random fields have at
least one direction that has either invariant or independent increments, no
longer reflecting the dependence structure in the discrete model. The limit
random fields form a general class of operator-scaling Gaussian random fields.
Their increments and path properties are investigated.
