RECORD
Realizability of point processes
Mathematical Physics · Submitted 23 Dec 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here we describe some necessary and sufficient conditions on the $\rho_j$'s for this to be true.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
There are various situations in which it is natural to ask whether a given collection of $k$ functions, $\rho_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. Here we describe some necessary and sufficient conditions on the $\rho_j$'s for this to be true. Our primary examples are $X=\mathbb{R}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities $\rho_1(\mathbf{r})$. Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density $\rho$ and translation invariant $\rho_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $\rho$ and the largest $\rho$ for which realizability can be established.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There are various situations in which it is natural to ask whether a given collection of $k$ functions, $\rho_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. | background 1.00 | 0.08 | 0.04 | asserted 0.70 |
| s1 | Here we describe some necessary and sufficient conditions on the $\rho_j$'s for this to be true. | contribution 0.91 | 0.92 | 0.43 | asserted 0.98 |
| s2 | Our primary examples are $X=\mathbb{R}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. | contribution 0.54 | 0.59 | 0.04 | asserted 0.96 |
| s3 | In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities $\rho_1(\mathbf{r})$. | contribution 0.69 | 0.77 | 0.09 | asserted 1.00 |
| s4 | Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. | result 0.91 | 0.94 | 0.17 | asserted 0.96 |
| s5 | We also investigate in detail a simple example in which a uniform density $\rho$ and translation invariant $\rho_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $\rho$ and the largest $\rho$ for which realizability can be established. | limitation 0.43 | 0.94 | 0.06 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
There are various situations in which it is natural to ask whether a given collection of $k$ functions, $\rho_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. Here we describe some necessary and sufficient conditions on the $\rho_j$'s for this to be true. Our primary examples are $X=\mathbb{R}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities $\rho_1(\mathbf{r})$. Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density $\rho$ and translation invariant $\rho_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $\rho$ and the largest $\rho$ for which realizability can be established.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.