## Electronic Communications in Probability

### 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 $\rho _1({\bf i})$ 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.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 45, 9 pp.

Dates
Accepted: 11 April 2016
First available in Project Euclid: 26 May 2016

https://projecteuclid.org/euclid.ecp/1464281070

Digital Object Identifier
doi:10.1214/16-ECP4620

Mathematical Reviews number (MathSciNet)
MR3510253

Zentralblatt MATH identifier
1346.44003

#### Citation

Caglioti, Emanuele; Infusino, Maria; Kuna, Tobias. Translation invariant realizability problem on the $d$-dimensional lattice: an explicit construction. Electron. Commun. Probab. 21 (2016), paper no. 45, 9 pp. doi:10.1214/16-ECP4620. https://projecteuclid.org/euclid.ecp/1464281070

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