Translation invariant realizability problem on the $d$-dimensional lattice: an explicit construction
| Autor: | Tobias Kuna, Emanuele Caglioti, Maria Infusino |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
Pure mathematics 82B20 FOS: Physical sciences 44A60 60G55 82B20 Fixed point 01 natural sciences Upper and lower bounds translation invariant Point process Infinite dimensional moment problem Point processes Realizability Translation invariant Truncated moment problem Lattice (order) 0103 physical sciences FOS: Mathematics statistics probability and uncertainty truncated moment problem 0101 mathematics 010306 general physics Mathematical Physics point processes Mathematics Probability (math.PR) 010102 general mathematics infinite dimensional moment problem realizability statistics and probability Mathematical Physics (math-ph) Radial distribution 16. Peace & justice 44A60 60G55 Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Electron. Commun. Probab. |
| ISSN: | 1083-589X |
| DOI: | 10.1214/16-ecp4620 |
| Popis: | 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. Comment: 9 pages, 4 figures |
| Databáze: | OpenAIRE |
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