Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction

Abstract

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions ${\rho }_1(\mathbf{i})$ and ${\rho }_2(\mathbf{i},\mathbf{j})$ non-negative and symmetric on ${\mathbb{Z}}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\ge 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.

9 pages, 4 figures

References in corpus (2)

• Realizability of point processes
• The Existence of Pair Potential Corresponding to Specified Density and Pair Correlation

Cited by in corpus (4)

• The Truncated Moment Problem on ${\mathbb{N}}_0$
• The truncated moment problem on the union of parallel lines
• The Truncated Moment Problem for Unital Commutative R-Algebras
• Band-limited mimicry of point processes by point processes supported on a lattice