Open Access
2016 Translation invariant realizability problem on the $d$-dimensional lattice: an explicit construction
Emanuele Caglioti, Maria Infusino, Tobias Kuna
Electron. Commun. Probab. 21: 1-9 (2016). DOI: 10.1214/16-ECP4620


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.


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Emanuele Caglioti. Maria Infusino. Tobias Kuna. "Translation invariant realizability problem on the $d$-dimensional lattice: an explicit construction." Electron. Commun. Probab. 21 1 - 9, 2016.


Received: 10 October 2015; Accepted: 11 April 2016; Published: 2016
First available in Project Euclid: 26 May 2016

zbMATH: 1346.44003
MathSciNet: MR3510253
Digital Object Identifier: 10.1214/16-ECP4620

Primary: 44A60 , 60G55 , 82B20

Keywords: infinite dimensional moment problem , Point processes , realizability , translation invariant , truncated moment problem

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