Persistence of Asymptotic Variance under Transport

Kurzfassung

We introduce p-uniformity to characterize the scaling of density fluctuations in spatial random systems in R^d, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve p-uniformity under transport. First, a finite (d+p)-th moment of the transport distance allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus answer a previously stated open problem; indeed we extend it, since our result applies to a general p-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and p-uniform with arbitrarily high p, and that can be simulated in linear time. We conclude with an outlook on a possible converse statement.