Electronic Communications in Probability

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

Emanuele Caglioti, Maria Infusino, and Tobias Kuna

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1464281070

Digital Object Identifier
doi:10.1214/16-ECP4620

Mathematical Reviews number (MathSciNet)
MR3510253

Zentralblatt MATH identifier
1346.44003

Subjects
Primary: 44A60: Moment problems 60G55: Point processes 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
truncated moment problem realizability point processes translation invariant infinite dimensional moment problem

Rights
Creative Commons Attribution 4.0 International License.

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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