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Determinantal identity for multilevel ensembles and finite determinantal point processes

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Abstract

We give a simple algebraic derivation of a useful determinantal identity for multilevel ensembles such as random matrix chains and finite determinantal point processes, with applications to the calculation of point correlators, gap probabilities and Janossy densities.

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References

  1. Andréief C.: Note sur une relation pour les intégrales définies des produits des fonctions. Mém. Soc. Sci. Bordeaux 2, 1–14 (1883)

    Google Scholar 

  2. Borodin, A.: Determinantal point processes. In: Akemann, G., Baik, J., Di Francesco P. (eds.) Oxford Handbook of Random Matrix Theory, Chapt. 11. Oxford University Press (2011)

  3. Eynard B., Mehta M.L.: Matrices coupled in a chain: eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Harnad J.: Janossy densities, multimatrix spacing distributions and Fredholm resolvents. Int. Math. Res. Not 48, 2599–2609 (2004)

    Article  MathSciNet  Google Scholar 

  5. Harnad J., Orlov A.Yu.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A 39, 8783–8809 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Itzykson C., Zuber J.B.: The planar approximation (II). J. Math. Phys 21, 411–421 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Johansson K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Johansson, K.: Random matrices and determinantal point processes. In: Bovier, A., Dunlop, F., Van Enter, A., Den Hollander, F., Dalibard, J. (eds.) Mathematical Statistical Physics, pp. 1–55. Proceedings of the 2005 Les Houches Summer School. Elsevier B.V., Amsterdam (2006)

  9. Prähofer M., Spohn H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)

    Article  MATH  Google Scholar 

  10. Prats-Ferrer A., Eynard B., di Francesco P., Zuber J.-B.: Correlation functions of Harish–Chandra integrals over the orthogonal and symplectic groups. J. Stat. Phys. 129, 885–935 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Soshnikov A.: Janossy densities of coupled random matrices. Commun. Math. Phys. 251, 447–471 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Tracy C.A., Widom H.: Correlation functions, cluster functions and spacing distributions for random matrices. J. Stat. Phys. 92, 809–835 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tracy C.A., Widom H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre de recherches mathématiques, Université de Montréal, C. P. 6128, Succ. Centre Ville, Montreal, QC, H3C 3J7, Canada

    J. Harnad

  2. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, QC, H3G 1M8, Canada

    J. Harnad

  3. Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect, Moscow, 117851, Russia

    A. Yu. Orlov

Authors
  1. J. Harnad
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  2. A. Yu. Orlov
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Correspondence to J. Harnad.

Additional information

Work of J. Harnad supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec. Work of A. Yu. Orlov supported by joint RFBR Consortium EINSTEIN grants Nos. 06-01-92054; 05-01-00498 and RAS Program “Fundamental Methods in Nonlinear Physics”.

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Harnad, J., Orlov, A.Y. Determinantal identity for multilevel ensembles and finite determinantal point processes. Anal.Math.Phys. 2, 105–121 (2012). https://doi.org/10.1007/s13324-012-0029-2

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  • DOI: https://doi.org/10.1007/s13324-012-0029-2

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