The realizability problem as a special case of the infinite-dimensional truncated moment problem
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- by Raúl E. Curto and Maria Infusino;
- Proc. Amer. Math. Soc. 152 (2024), 2145-2155
- DOI: https://doi.org/10.1090/proc/16710
- Published electronically: March 25, 2024
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Abstract:
The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated $K-$moment problem where $K$ is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited by Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253–1281], where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained by Curto, Ghasemi, Infusino, and Kuhlmann [J. Operator Theory 90 (2023), pp. 223–261] for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results of Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253–1281], allowing to finally embed them in the above-mentioned unified framework for the infinite-dimensional truncated moment problem.References
- Daniel Alpay, Palle E. T. Jorgensen, and David P. Kimsey, Moment problems in an infinite number of variables, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015), no. 4, 1550024, 14. MR 3447225, DOI 10.1142/S0219025715500241
- Yu. M. Berezanskiĭ and Yu. G. Kondrat′ev, Spektral′nye metody v beskonechnomernom analize, “Naukova Dumka”, Kiev, 1988 (Russian). MR 978630
- Yuri M. Berezansky, Yuri G. Kondratiev, Tobias Kuna, and Eugene Lytvynov, On a spectral representation for correlation measures in configuration space analysis, Methods Funct. Anal. Topology 5 (1999), no. 4, 87–100. MR 1773905
- Ju. M. Berezans′kiĭ and S. N. Šifrin, A generalized symmetric power moment problem, Ukrain. Mat. Ž. 23 (1971), 291–306. (errata insert) (Russian). MR 300130
- H. J. Borchers and J. Yngvason, Integral representations for Schwinger functionals and the moment problem over nuclear spaces, Comm. Math. Phys. 43 (1975), no. 3, 255–271. MR 383099
- E. Caglioti, T. Kuna, J. L. Lebowitz, and E. R. Speer, Point processes with specified low order correlations, Markov Process. Related Fields 12 (2006), no. 2, 257–272. MR 2249631
- Emanuele Caglioti, Maria Infusino, and Tobias Kuna, Translation invariant realizability problem on the $d$-dimensional lattice: an explicit construction, Electron. Commun. Probab. 21 (2016), Paper No. 45, 9. MR 3510253, DOI 10.1214/16-ECP4620
- Raúl E. Curto, Mehdi Ghasemi, Maria Infusino, and Salma Kuhlmann, The truncated moment problem for unital commutative $\Bbb R$-algebras, J. Operator Theory 90 (2023), no. 1, 223–261. MR 4615733
- D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2003. Elementary theory and methods. MR 1950431
- R. M. Erdahl, Representability, Int. J. Quantum Chem. 13 (1978), 697–718.
- Claude Garrod and Jerome K. Percus, Reduction of the $N$-particle variational problem, J. Mathematical Phys. 5 (1964), 1756–1776. MR 170658, DOI 10.1063/1.1704098
- Mehdi Ghasemi, Maria Infusino, Salma Kuhlmann, and Murray Marshall, Moment problem for symmetric algebras of locally convex spaces, Integral Equations Operator Theory 90 (2018), no. 3, Paper No. 29, 19. MR 3796386, DOI 10.1007/s00020-018-2453-7
- Mehdi Ghasemi, Salma Kuhlmann, and Murray Marshall, Application of Jacobi’s representation theorem to locally multiplicatively convex topological $\Bbb {R}$-algebras, J. Funct. Anal. 266 (2014), no. 2, 1041–1049. MR 3132736, DOI 10.1016/j.jfa.2013.09.001
- Mehdi Ghasemi, Salma Kuhlmann, and Murray Marshall, Moment problem in infinitely many variables, Israel J. Math. 212 (2016), no. 2, 989–1012. MR 3505409, DOI 10.1007/s11856-016-1318-5
- J. P. Hansen and I. R. McDonald, Theory of simple liquids, 2nd ed., Academic Press, New York, 1987.
- Gerhard C. Hegerfeldt, Extremal decomposition of Wightman functions and of states on nuclear *-algebras by Choquet theory, Comm. Math. Phys. 45 (1975), no. 2, 133–135. MR 454672
- Maria Infusino, Tobias Kuna, and Aldo Rota, The full infinite dimensional moment problem on semi-algebraic sets of generalized functions, J. Funct. Anal. 267 (2014), no. 5, 1382–1418. MR 3229795, DOI 10.1016/j.jfa.2014.06.012
- Maria Infusino and Tobias Kuna, The full moment problem on subsets of probabilities and point configurations, J. Math. Anal. Appl. 483 (2020), no. 1, 123551, 29. MR 4019810, DOI 10.1016/j.jmaa.2019.123551
- Maria Infusino, Salma Kuhlmann, Tobias Kuna, and Patrick Michalski, Projective limit techniques for the infinite dimensional moment problem, Integral Equations Operator Theory 94 (2022), no. 2, Paper No. 12, 44. MR 4403194, DOI 10.1007/s00020-022-02692-6
- O. Kallenberg, Lectures on random measures, Institute of Statistics Mimeo Series No. 963, (1974).
- Yuri G. Kondratiev and Tobias Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 2, 201–233. MR 1914839, DOI 10.1142/S0219025702000833
- Yuri G. Kondratiev, Tobias Kuna, and Maria João Oliveira, Holomorphic Bogoliubov functionals for interacting particle systems in continuum, J. Funct. Anal. 238 (2006), no. 2, 375–404. MR 2253724, DOI 10.1016/j.jfa.2006.06.001
- Leonid Koralov, Existence of pair potential corresponding to specified density and pair correlation, Lett. Math. Phys. 71 (2005), no. 2, 135–148. MR 2134693, DOI 10.1007/s11005-005-0343-9
- Klaus Krickeberg, Moments of point processes, Probability and information theory, II, Lecture Notes in Math., Vol 296, Springer, Berlin-New York, 1973, pp. 70–101. MR 378095
- Hans Kummer, $n$-representability problem for reduced density matrices, J. Mathematical Phys. 8 (1967), 2063–2081. MR 239827, DOI 10.1063/1.1705122
- T. Kuna, J. L. Lebowitz, and E. R. Speer, Realizability of point processes, J. Stat. Phys. 129 (2007), no. 3, 417–439. MR 2351408, DOI 10.1007/s10955-007-9393-y
- Tobias Kuna, Joel L. Lebowitz, and Eugene R. Speer, Necessary and sufficient conditions for realizability of point processes, Ann. Appl. Probab. 21 (2011), no. 4, 1253–1281. MR 2857448, DOI 10.1214/10-AAP703
- A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal. 59 (1975), no. 3, 219–239. MR 391830, DOI 10.1007/BF00251601
- D. A. Mazziotti, Structure of fermionic density matrices: complete N-representability conditions, Phys. Rev. Lett. 108 (2012), 263002.
- David A. Mazziotti, Quantum many-body theory from a solution of the $N$-representability problem, Phys. Rev. Lett. 130 (2023), no. 15, Paper No. 153001, 7. MR 4582676, DOI 10.1103/physrevlett.130.153001
- Raphael Lachieze-Rey and Ilya Molchanov, Regularity conditions in the realisability problem with applications to point processes and random closed sets, Ann. Appl. Probab. 25 (2015), no. 1, 116–149. MR 3297768, DOI 10.1214/13-AAP990
- J. K. Percus, The pair distribution function in classical statistical mechanics, The Equilibrium Theory of Classical Fluids, Benjamin, New York, 1964. Edited by H. L. Frisch and J. L. Lebowitz.
- Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR 1056697, DOI 10.1007/978-3-0348-7469-4
- Konrad Schmüdgen, On the infinite-dimensional moment problem, Ark. Mat. 56 (2018), no. 2, 441–459. MR 3893784, DOI 10.4310/ARKIV.2018.v56.n2.a14
- Salvatore Torquato, Random heterogeneous materials, Interdisciplinary Applied Mathematics, vol. 16, Springer-Verlag, New York, 2002. Microstructure and macroscopic properties. MR 1862782, DOI 10.1007/978-1-4757-6355-3
- S. Torquato and F. H. Stillinger, New conjectural lower bounds on the optimal density of sphere packings, Experiment. Math. 15 (2006), no. 3, 307–331. MR 2264469
- Ge Zhang and Salvatore Torquato, Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions, Phys. Rev. E 101 (2020), no. 3, 032124, 23. MR 4083991
- G. F. Us, A truncated symmetric generalized power moment problem, Ukrain. Mat. Ž. 26 (1974), 348–358, 429 (Russian). MR 348550
Bibliographic Information
- Raúl E. Curto
- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52246
- MR Author ID: 53500
- ORCID: 0000-0002-1776-5080
- Email: raul-curto@uiowa.edu
- Maria Infusino
- Affiliation: Dipartimento di Matematica e Informatica, Universitá degli Studi di Cagliari, Palazzo delle Scienze, Via Ospedale 72, 09124 Cagliari, Italy
- MR Author ID: 886339
- ORCID: 0000-0003-3438-5503
- Email: maria.infusino@unica.it
- Received by editor(s): May 17, 2023
- Received by editor(s) in revised form: October 19, 2023
- Published electronically: March 25, 2024
- Additional Notes: The authors wish to thank the Autonomous Region of Sardinia for funding the visit of the first author to University of Cagliari within the programme "Visiting Professor/Scientist 2022" (LR 7/2007). The second author received partial support by the INdAM-GNAMPA Project E53C23001670001 and also by the University of Cagliari. In addition, the first author was partially supported by U.S. NSF grant DMS-2247167.
- Communicated by: Javad Mashreghi
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 2145-2155
- MSC (2020): Primary 44A60, 47A57, 60G55, 28C05; Secondary 46J05, 28E99
- DOI: https://doi.org/10.1090/proc/16710
- MathSciNet review: 4728479