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

TL;DR

The paper presents an explicit method to determine if certain correlation functions can originate from a translation invariant point process on a lattice, providing new bounds for the maximum realizable density in higher dimensions.

Contribution

It introduces an explicit construction for realizing correlation functions on the lattice and derives a lower bound for the maximal realizable density in any dimension.

Findings

Constructed a translation invariant point process for given correlation functions.

Derived a lower bound for the maximal realizable density.

Compared new bounds with existing ones.

Abstract

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and $\rho_2({\bf i},{\bf j})$ 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 $d\geq 2$ 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.