  We propose an aggregated random-field model, and investigate the scaling
limits of the aggregated partial-sum random fields. In our model, each copy of
the random field in the aggregation is built from two correlated
one-dimensional random walks, each with a random persistence parameter. When
the persistence parameters are independent, the scaling limit is a fractional
Brownian sheet. When the persistence parameters are dependent, the scaling
limit is more delicate, and in particular depends on the growth rates of the
underlying rectangular region along two directions: at different rates
different operator-scaling Gaussian random fields appear as the region area
tends to infinity. In particular, at the so-called critical speed, a large
family of Gaussian random fields with long-range dependence arise in the limit.
We also identify four different regimes at non-critical speed where fractional
Brownian sheets arise in the limit.

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