Possible sets of autocorrelations and the Simplex algorithm

Keren, Shahar; Kfir, Haggai; Kanter, Ido

doi:10.1088/0305-4470/39/16/004

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…As the notation indicates, we are still considering translation invariant correlations; we may in addition require that the two-point correlations ρ 2 (x, y) of the realizing measure all be translation invariant and approach ρ 2 as |x − y| → ∞. This is the problem discussed in [15] for r ∈ Z, or just on a ring. It turns out that, at least for the case r ∈ Z and specified (ρ, g (1), .…”

Section: Discussionmentioning

confidence: 99%

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Realizability of Point Processes

2007

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There are various situations in which it is natural to ask whether a given collection of k functions, ρ 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 ρ j 's for this to be true. Our primary examples are X = R d , X = Z d , and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ρ 1 (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 ρ and translation invariant ρ 2 are specified on Z; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established.

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Section: Discussionmentioning

confidence: 99%

“…, k, are specified only on part of the domain X; for example, on the lattice we might only specify the nearest neighbor correlations. See [15] for a similar problem in error correcting codes. We will not consider this case further here, except for some comments at the end of section 6.…”

Section: Introductionmentioning

confidence: 99%

Realizability of Point Processes

2007

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“…In this and other situations it is natural to consider a closely related problem in which the CONDITIONS FOR REALIZABILITY 5 correlation functions are specified only on part of the domain X; for example, if X is a lattice then we might only specify the nearest neighbor correlations. See [17] for a similar problem in error correcting codes. This will not be considered here; see, however, [20], Section 7.…”

Section: Introductionmentioning

confidence: 99%

Necessary and sufficient conditions for realizability of point processes

Kuna¹,

Lebowitz²,

Speer³

2011

Ann. Appl. Probab.

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We give necessary and sufficient conditions for a pair of (generalized) functions $\rho_1(\mathbf{r}_1)$ and $\rho_2(\mathbf{r}_1,\mathbf{r}_2)$, $\mathbf{r}_i\in X$, to be the density and pair correlations of some point process in a topological space $X$, for example, $\mathbb {R}^d$, $\mathbb {Z}^d$ or a subset of these. This is an infinite-dimensional version of the classical "truncated moment" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of $X$. Without this restriction we obtain, for compact $X$, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further requirement---the existence of a finite third order moment. We generalize the latter conditions in two distinct ways when $X$ is not compact.Comment: Published in at http://dx.doi.org/10.1214/10-AAP703 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Possible sets of autocorrelations and the Simplex algorithm

Cited by 2 publications

References 13 publications

Realizability of Point Processes

Realizability of Point Processes

Necessary and sufficient conditions for realizability of point processes

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