  We examine optimal matchings or transport between two stationary point
processes and in particular, from a point process to the (integer) lattice or
the Lebesgue measure respectively. The main focus of the article is the
implication of hyperuniformity (reduced variance fluctuations in point
processes) to optimal transport: in dimension $2$, we show that the typical
matching cost has finite second moment under a mild logarithmic integrability
condition on the reduced pair correlation measure, showing that most planar
hyperuniform point processes are $ L^2$-perturbed lattices. Our method does not
formally require assumptions on the correlation measure or the variance
behaviour and it retrieves known sharp bounds for neutral integrable systems
such as Poisson processes, and also applies to hyperfluctuating systems. The
proof relies on the estimation of the optimal transport cost between point
processes restricted to large windows for a well-chosen cost through their
Fourier-Stieljes transforms, related to their structure factor. The existence
of an infinite matching is obtained through a compactness argument on the space
of random measures.
