Recursion for the Smallest Eigenvalue Density of \(\beta \)-Wishart–Laguerre Ensemble

  • Santosh KumarEmail author


The statistics of the smallest eigenvalue of Wishart–Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of Wishart–Laguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices (\(\beta =1\)) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices (\(\beta =2\)). In the present work we extend this to \(\beta \)-Wishart–Laguerre ensembles for the case when exponent \(\alpha \) in the associated Laguerre weight function, \(\lambda ^\alpha e^{-\beta \lambda /2}\), is a non-negative integer, while \(\beta \) is positive real. This also gives access to the smallest eigenvalue density of fixed trace \(\beta \)-Wishart–Laguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of n (the matrix dimension) and \(\alpha \) also enable us to compare with Tracy–Widom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the Wigner–Smith matrix and are relevant to the problem of quantum chaotic scattering.


\(\beta \)-Wishart–Laguerre ensemble Smallest eigenvalue Recursion relation Tracy–Widom density Large deviations Proper delay times 



The author is grateful to Prof. Katzav for fruitful correspondences. He also thanks the anonymous reviewer whose comments helped improve the manuscript. This work has been supported by the grant EMR/2016/000823 provided by SERB, DST, Government of India.


  1. 1.
    Mehta, M.L.: Random Matrices, 3rd edn. Academic Press, New York (2004)zbMATHGoogle Scholar
  2. 2.
    Forrester, P.J.: Log-Gases and Random Matrices (LMS-34). Princeton University Press, Princeton, NJ (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gnanadesikan, R.: Methods for Statistical Data Analysis of Multivariate Observations, 2nd edn. Wiley, New York (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Park, C.S., Lee, K.B.: Statistical multimode transmit antenna selection for limited feedback MIMO systems. IEEE Trans. Wirel. Commun. 7, 4432 (2008). CrossRefGoogle Scholar
  5. 5.
    Nishigaki, S.M., Damgaard, P.H., Wettig, T.: Smallest Dirac eigenvalue distribution from random matrix theory. Phys. Rev. D 58, 087704 (1998). ADSCrossRefGoogle Scholar
  6. 6.
    Damgaard, P.H., Nishigaki, S.M.: Distribution of the kth smallest Dirac operator eigenvalue. Phys. Rev. D. 63, 045012 (2001). ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Candes, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52, 5406 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Majumdar, S.N., Bohigas, O., Lakshminarayan, A.: Exact minimum eigenvalue distribution of an entangled random pure state. J. Phys. Stat. 131, 33 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Majumdar, S.N.: Extreme eigenvalues of wishart matrices: application to entangled bipartite system. In: Akemann, G. (ed.) Handbook of Random Matrix Theory. Oxford Press, New York (2011)Google Scholar
  10. 10.
    Chen, Y., Liu, D.-Z., Zhou, D.-S.: Smallest eigenvalue distribution of the fixed-trace Laguerre-ensemble. J. Phys. A: Math. Theor. 43, 315303 (2010). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Akemann, G., Vivo, P.: Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without a fixed trace. J. Mech. Stat. 2011, P05020 (2011). MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kumar, S., Sambasivam, B., Anand, S.: Smallest eigenvalue density for regular or fixed-trace complex Wishart-Laguerre ensemble and entanglement in coupled kicked tops. J. Phys. A: Math. Theor. 50, 345201 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Edelman, A., Guionnet, A., Péché, S.: Beyond universality in random matrix theory. Ann. Probab. Appl. 26, 1659 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khatri, C.G.: Distribution of the largest or the smallest characteristic root under null hypothesis concerning complex multivariate normal populations. Ann. Stat. Math. 35, 1807 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Forrester, P.J., Hughes, T.D.: Complex Wishart matrices and conductance in mesoscopic systems: exact results. J. Phys. Math. 35, 6739 (1994). MathSciNetzbMATHGoogle Scholar
  16. 16.
    Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709 (1993). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Forrester, P.J.: Exact results and universal asymptotics in the Laguerre random matrix ensemble. J. Phys. Math. 35, 2539 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nagao, T., Forrester, P.J.: The smallest eigenvalue distribution at the spectrum edge of random matrices. Nucl. Phys. B. 509, 561 (1998). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zanella, A., Chiani, M., Win, M.Z.: On the marginal distribution of the eigenvalues of Wishart matrices. IEEE Trans. Commun. 57, 1050 (2009). CrossRefGoogle Scholar
  20. 20.
    Forrester, P.J.: Eigenvalue distributions for some correlated complex sample covariance matrices. J. Phys. A: Math. Theor. 40, 11093 (2007). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wirtz, T., Guhr, T.: Distribution of the smallest eigenvalue in the correlated Wishart model. Phys. Lett. Rev. 111, 094101 (2013). ADSCrossRefGoogle Scholar
  22. 22.
    Edelman, A.: Eigenvalues and condition numbers of random matrices. Ph.D. thesis, MIT. (1989)
  23. 23.
    Edelman, A.: The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type. Linear Appl. Alg. 159, 55 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Phys. Lett. B 305, 115 (1993). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Phys. Math. 159, 151 (1994). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tracy, C.A., Widom, H.: Level spacing distributions and the Bessel kernel. Commun. Phys. Math. 161, 289 (1994). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Feldheim, O.N., Sodin, S.: A universality result for the smallest eigenvalues of certain sample covariance matrices. Geom. Funct. Anal. 20, 88 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Katzav, E., Castillo, I.P.: Large deviations of the smallest eigenvalue of the Wishart-Laguerre ensemble. Phys. Rev. E 82(R), 040104 (2010). ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Haake, F., Kuś, M., Scharf, R.: Classical and quantum chaos for a kicked top. Z. Phys. B Condens. Matter 65, 381 (1987). ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Haake, F.: Quantum Signatures of Chaos, 3rd edn. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  31. 31.
    Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43, 5830 (2002). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Eisenbud, L.: The formal properties of nuclear collisions. PhD Thesis. Princeton University, Princeton (1948)Google Scholar
  33. 33.
    Wigner, E.P.: Lower limit for the energy derivative of the scattering phase shift. Phys. Rev. 98, 145 (1995). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Smith, F.T.: Lifetime matrix in collision theory. Phys. Rev. 118, 349 (1960). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Brouwer, P.W., Frahm, K.M., Beenakker, C.W.J.: Quantum mechanical time-delay matrix in chaotic scattering. Phys. Rev. Lett. 78, 4737 (1997). ADSCrossRefGoogle Scholar
  36. 36.
    Brouwer, P.W., Frahm, K.M., Beenakker, C.W.J.: Distribution of the quantum mechanical time-delay matrix for a chaotic cavity. Waves Random Media 9, 91 (1999). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sommers, H.-J., Savin, D.V., Sokolov, V.V.: Distribution of proper delay times in quantum chaotic scattering: a crossover from ideal to weak coupling. Phys. Rev. Lett. 87, 094101 (2001). ADSCrossRefGoogle Scholar
  38. 38.
    Texier, C.: Wigner time delay and related concepts: application to transport in coherent conductors. Phys. E Low Dimens. Syst. Nanostruct. 82, 16 (2016). ADSCrossRefGoogle Scholar
  39. 39.
    Fyodorov, Y.V., Sommers, H.-J.: Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. 38, 1918 (1997). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ramírez, J., Rider, B., Virág, B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Math. Soc. Am. 24, 919 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Majumdar, S.N., Schehr, G.: Top eigenvalue of a random matrix: large deviations and third order phase transition. J. Stat. Mech. 2014, P01012 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Borot, G., Eynard, B., Majumdar, S.N., Nadal, C.: Large deviations of the maximal eigenvalue of random matrices. J. Stat. Mech. Theory Exp. 2011, P11024 (2011). MathSciNetCrossRefGoogle Scholar
  43. 43.
    Borot, G., Nadal, C.: Right tail expansion of Tracy-Widom beta laws. Random Matrices: Theory Appl. 01, 1250006 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Dumaz, L., Virág, B.: The right tail exponent of the Tracy-Widom-distribution. Ann. Inst. H. Poincaré Probab. Stat. 49, 915 (2013). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Forrester, P.J., Rahman, A.A., Witte, N.S.: Large N expansions for the Laguerre and Jacobi-ensembles from the loop equations. J. Math. Phys. 58, 113303 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Killip, R., Nenciu, I.: Matrix models for circular ensembles. Int. Math. Res. Not. 2004, 2664 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Forrester, P.J.: Beta Random Matrix Ensembles. Lecture Notes Series, IMS, NUS, vol. 18. World Scientific, Singapore (2009)Google Scholar
  48. 48.
    Desrosiers, P., Liu, D.-Z.: Asymptotics for products of characteristic polynomials in classical \(\beta \)-ensembles. Constr. Approx. 39, 273 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Desrosiers, P., Forrester, P.J.: Hermite and Laguerre \(\beta \)-ensembles: asymptotic corrections to the eigenvalue density. Nucl. Phys. B 743, 307 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Caër G, L., Male, C., Delannay, R.: Nearest-neighbour spacing distributions of the \(\beta \)-Hermite ensemble of random matrices. Physica A 383, 190 (2007). ADSCrossRefGoogle Scholar
  51. 51.
    Dumitriu, I., Edelman, A.: Global spectrum fluctuations for the \(\beta \)-Hermite and \(\beta \)-Laguerre ensembles via matrix models. J. Math. Phys. 47, 063302 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Papenbrock, T., Pluhar, Z., Weidenmüller, H.A.: Level repulsion in constrained Gaussian random-matrix ensembles. J. Phys. A: Math. Gen. 39, 9709 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Shukla, P., Sadhukhan, S.: Random matrix ensembles with column/row constraints: I. J. Phys. A: Math. Theor. 48, 415002 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Shukla, P., Sadhukhan, S.: Random matrix ensembles with column/row constraints: II. J. Phys. A: Math. Theor. 48, 415003 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Rosenzweig, N.: In: Uhlenbeck, G. et al. (eds.) Statistical Physics. Benjamin, New York (1963)Google Scholar
  56. 56.
    Bronk, B.V.: Topics in the Theory of Random Matrices. Ph. D. thesis. Princeton University, Princeton (1964)Google Scholar
  57. 57.
    Akemann, G., Cicuta, G.M., Molinari, L., Vernizzi, G.: Compact support probability distributions in random matrix theory. Phys. Rev. E 59, 1489 (1999). ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Lloyd, S., Pagels, H.: Complexity as thermodynamic depthe. Ann. Phys. 188, 186 (1988). ADSCrossRefGoogle Scholar
  59. 59.
    Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A: Math. Gen. 34, 7111 (2001). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Kumar, S., Pandey, A.: Entanglement in random pure states: spectral density and average von Neumann entropy. J. Phys. A: Math. Theor. 44, 445301 (2011). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Vivo, P., Pato, M.P., Oshanin, G.: Random pure states: quantifying bipartite entanglement beyond the linear statistics. Phys. Rev. E 93, 052106 (2016). ADSCrossRefGoogle Scholar
  63. 63.
    Wei, L.: Proof of Vivo-Pato-Oshanin’s conjecture on the fluctuation of von Neumann entropy. Phys. Rev. E 96, 022106 (2017). ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    Forrester, P.J.: Recurrence equations for the computation of correlations in the \(1/r^2\) quantum many-body system. J. Stat. Phys. 72, 39 (1993). ADSCrossRefzbMATHGoogle Scholar
  65. 65.
    Forrester, P.J., Rains, E.M.: A Fuchsian matrix differential equation for Selberg correlation integrals. Commun. Math. Phys. 309, 771 (2012). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Forrester, P.J., Ito, M.: Difference system for Selberg correlation integrals. J. Phys. A: Math. Theor. 43, 175202 (2010). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Savin, D.V., Sommers, H.-J., Wieczorek, W.: Nonlinear statistics of quantum transport in chaotic cavities. Phys. Rev. B 77, 125332 (2008). ADSCrossRefGoogle Scholar
  68. 68.
    Akemann, G., Guhr, T., Kieburg, M., Wegner, R., Wirtz, T.: Completing the picture for the smallest eigenvalue of real Wishart matrices. Phys. Rev. Lett. 113, 250201 (2014). ADSCrossRefzbMATHGoogle Scholar
  69. 69.
    Wirtz, T., Akemann, G., Guhr, T., Kieburg, M., Wegner, R.: The smallest eigenvalue distribution in the real Wishart-Laguerre ensemble with even topology. J. Phys. A: Math. Theor. 48, 245202 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Fyodorov, Y.V., Nock, A.: On random matrix averages involving half-integer powers of GOE characteristic polynomials. J. Stat. Phys. 159, 731 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Berbenni-Bitsch, M.E., Meyer, S., Wettig, T.: Microscopic universality with dynamical fermions. Phys. Rev. D 58(R), 71502 (1998). ADSCrossRefGoogle Scholar
  72. 72.
    Wolfram Research Inc. Mathematica Version 11.0. Wolfram Research Inc, Champaign, IL (2016)Google Scholar
  73. 73.
    Koev, P., Edelman, A.: The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comput. 75, 833 (2006).
  74. 74.
    Koev, P.: Hypergeometric Function of a Matrix Argument, Online (2008).
  75. 75.
    Borodin, A., Forrester, P.J.: Increasing subsequences and the hard-to-soft edge transition in matrix ensembles. J. Phys. A: Math. Gen. 36, 2963 (2003). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Ma, Z.: Accuracy of the Tracy-Widom limits for the extreme eigenvalues in white Wishart matrices. Bernoulli 18, 322 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Baik, J., Buckingham, R., DiFranco, J.: Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Commun. Math. Phys. 280, 463 (2008). ADSCrossRefzbMATHGoogle Scholar
  78. 78.
    Brouwer, P.W., van Langen, S.A., Frahm, K.M., Büttiker, M., Beenakker, C.W.J.: Distribution of parametric conductance derivatives of a quantum dot. Phys. Rev. Lett. 79, 913 (1997). ADSCrossRefGoogle Scholar
  79. 79.
    Schomerus, H., van Bemmel, K.J.H., Beenakker, C.W.J.: Localization-induced coherent backscattering effect in wave dynamics. Phys. Rev. E 63, 026605 (2001). ADSCrossRefGoogle Scholar
  80. 80.
    Marciani, M., Brouwer, P.W., Beenakker, C.W.J.: Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard. Phys. Rev. B 90, 045403 (2014). ADSCrossRefGoogle Scholar
  81. 81.
    Schomerus, H., Marciani, M., Beenakker, C.W.J.: Effect of chiral symmetry on chaotic scattering from majorana zero modes. Phys. Rev. Lett. 114, 166803 (2015). ADSCrossRefGoogle Scholar
  82. 82.
    Mezzadri, F., Simm, N.J.: Moments of the transmission eigenvalues, proper delay times, and random matrix theory: I. J. Math. Phys. 52, 103511 (2011). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Mezzadri, F., Simm, N.J.: Moments of the transmission eigenvalues, proper delay times and random matrix theory: II. J. Math. Phys. 53, 053504 (2012). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Mezzadri, F., Simm, N.J.: \(\tau \)-function theory of quantum chaotic transport with \(\beta \) = 1, 2, 4. Commun. Math. Phys. 324, 465 (2013). ADSCrossRefzbMATHGoogle Scholar
  85. 85.
    Texier, C., Majumdar, S.N.: Wigner time-delay distribution in chaotic cavities and freezing transition. Phys. Rev. Lett. 110, 250602 (2013). ADSCrossRefGoogle Scholar
  86. 86.
    Kuipers, J., Savin, D.V., Sieber, M.: Efficient semiclassical approach for time delays. New J. Phys. 16, 123018 (2014). ADSCrossRefGoogle Scholar
  87. 87.
    Cunden, F.D.: Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities. Phys. Rev. E 91(R), 060102 (2015). ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    Cunden, F.D., Mezzadri, F., Simm, N., Vivo, P.: Correlators for the Wigner-Smith time-delay matrix of chaotic cavities. J. Phys. A: Math. Theor. 49, 18LT01 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    Cunden, F.D., Mezzadri, F., Simm, N., Vivo, P.: Large-N expansion for the time-delay matrix of ballistic chaotic cavities. J. Math. Phys. 57, 111901 (2016). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Mahaux, C., Weidenmüller, H.A.: Shell Model Approach to Nuclear Reactions. North Holland, Amsterdam (1969)Google Scholar
  91. 91.
    Verbaarschot, J.J.M., Weidenmüller, H.A., Zirnbauer, M.R.: Grassmann integration in stochastic quantum physics: the case of compound-nucleus scattering. Phys. Rep. 129, 367 (1985). ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    Kumar, S., Nock, A., Sommers, H.-J., Guhr, T., Dietz, B., Miski-Oglu, M., Richter, A., Schäfer, : Distribution of scattering matrix elements in quantum chaotic scattering. Phys. Rev. Lett. 111, 030403 (2013). ADSCrossRefGoogle Scholar
  93. 93.
    Nock, A., Kumar, S., Sommers, H.-J., Guhr, T.: Distributions of off-diagonal scattering matrix elements: exact results. Ann. Phys. 342, 103 (2014). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997). ADSCrossRefGoogle Scholar
  95. 95.
    Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random matrix theory. J. Phys. Math. 37, 4986 (1996). MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Shiv Nadar UniversityGautam Buddha NagarIndia

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