Necessary and sufficient conditions for realizability of point processes

Kuna, Tobias; Lebowitz, Joel L.; Speer, Eugène R.

doi:10.1214/10-aap703

Cited by 27 publications

(65 citation statements)

References 30 publications

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“…Some progress in this direction have been recently done e.g. in [28], [29], [30], where a systematic study of the truncated case for point processes and random sets was initiated. However, to the best of our knowledge, at the moment there is no results for the truncated moment problem for general R−algebras.…”

Section: The Moment Problem On Symmetric Algebras Of Lc Real Spacesmentioning

confidence: 99%

“…Recent approaches are based on the interpretation of these problems as infinite dimensional truncated moment problems (see e.g. [23], [25], [28], [29]). This new point of view was exploited not only in the analysis of interacting particle systems but also in the study of the realizability problem for random closed sets (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…This kind of structural investigation has been recently started and brought to new theoretical developments on the infinite dimensional full moment problem, which are essentially based on the interplay between the finite and the infinite dimensional case (see e.g. [1], [14], [18], [19], [25], [28], [29]). In particular, the results in [14] revealed intriguing relations between real algebraic geometry, functional and harmonic analysis, leading us to several open questions which we are going to present in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Infinite dimensional moment problem: open questions and applications

Infusino¹,

Kuhlmann²

2017

Ordered Algebraic Structures and Related Topics

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This paper is dedicated to Murray Marshall, who posed many of the questions here addressed and was still working on them in the very last days of his life. We lost a wonderful collaborator and a dear friend. We sorely miss him.Abstract. Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.

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Section: The Moment Problem On Symmetric Algebras Of Lc Real Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Infinite dimensional moment problem: open questions and applications

Infusino¹,

Kuhlmann²

2017

Ordered Algebraic Structures and Related Topics

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“…V. Conditions for realizability have also been found for special types of many-particle systems [10]. Moreover, necessary and sufficient conditions for the particular class of point configurations with "hard" cores have been identified [15,16], but these conditions are difficult to check in practice. Thus, knowledge of necessary conditions beyond inequalities (1), (2), and (3) that can be applied to determine the realizability of general pair correlation functions are, for the most part, lacking.…”

Section: Introductionmentioning

confidence: 99%

Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions

Zhang

Torquato

2020

Phys. Rev. E

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The capacity to identify realizable many-body configurations associated with targeted functional forms for the pair correlation function g2(r) or its corresponding structure factor S(k) is of great fundamental and practical importance. While there are obvious necessary conditions that a prescribed structure factor at number density ρ must satisfy to be configurationally realizable, sufficient conditions are generally not known due to the infinite degeneracy of configurations with different higher-order correlation functions. A major aim of this paper is to expand our theoretical knowledge of the class of pair correlation functions or structure factors that are realizable by classical disordered ensembles of particle configurations, including exotic "hyperuniform" varieties. We first introduce a theoretical formalism that provides a means to draw classical particle configurations from canonical ensembles with certain pairwise-additive potentials that could correspond to targeted analytical functional forms for the structure factor. This formulation enables us to devise an improved algorithm to construct systematically canonical-ensemble particle configurations with such targeted pair statistics, whenever realizable. As a proof-of-concept, we test the algorithm by targeting several different structure factors across dimensions that are known to be realizable and one hyperuniform target that is known to be nontrivially unrealizable. Our algorithm succeeds for all realizable targets and appropriately fails for the unrealizable target, demonstrating the accuracy and power of the method to numerically investigate the realizability problem. Subsequently, we also target several families of structure-factor functions that meet the known necessary realizability conditions but were heretofore not known to be realizable by disordered hyperuniform point configurations, including d-dimensional Gaussian structure factors, d-dimensional generalizations of the 2D one-component plasma (OCP), the d-dimensional Fourier duals of the previous OCP cases. Moreover, we also explore unusual nonhyperuniform targets, including "hyposurficial" and "antihyperuniform" examples. In all of these instances, the targeted structure factors were achieved with high accuracy, suggesting that they are indeed realizable by equilibrium configurations with pairwise interactions at positive temperatures. Remarkably, we also show that the structure factor of nonequilibrium "perfect glass" specified by two-, three-, and four-body interactions, can also be realized by equilibrium pair interactions at positive temperatures. Our findings lead us to the conjecture that any realizable structure factor corresponding to either a translationally invariant equilibrium or nonequilibrium system can be attained by an equilibrium ensemble involving only effective pair interactions. Our investigation not only broadens our knowledge of analytical functional forms for g2(r) and S(k) associated with disordered point configurations across dimensions but also deepens our under...

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“…Moreover, in [42], it has been shown that this dual is generated by the Bell's inequalities (see [43] for an overview of the role played by these inequalities in quantum mechanics) for n = 3, 4, but not for n ≥ 5. Finally, we mention the paper [44], where the problem of realizability of correlations has been extended to the more general setting of random points in a topological space.…”

Section: Introductionmentioning

confidence: 99%