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On the condition number of the critically-scaled Laguerre Unitary Ensemble
1. | Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, United States, United States |
2. | Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, United States |
References:
[1] |
T. H. Baker, P. J. Forrester and P. A. Pearce, Random matrix ensembles with an effective extensive external charge,, J. Phys. A. Math. Gen., 31 (1998), 6087.
doi: 10.1088/0305-4470/31/29/002. |
[2] |
E. Basor, Y. Chen and L. Zhang, PDEs satisfied by extreme eigenvalues distributions of GUE and LUE,, Random Matrices Theory Appl., 1 (2012).
doi: 10.1142/S2010326311500031. |
[3] |
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach,, Amer. Math. Soc., (1999).
|
[4] |
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights,, Internat. Math. Res. Not., 16 (1997), 759.
doi: 10.1155/S1073792897000500. |
[5] |
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights,, Comm. Pure Appl. Math., 52 (1999), 1491.
|
[6] |
P. A. Deift, G. Menon, S. Olver and T. Trogdon, Universality in numerical computations with random data,, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 14973.
doi: 10.1073/pnas.1413446111. |
[7] |
A. Edelman, Eigenvalues and condition numbers of random matrices,, SIAM J. Matrix Anal. Appl., 9 (1988), 543.
doi: 10.1137/0609045. |
[8] |
A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity,, Comm. Math. Phys., 147 (1992), 395.
doi: 10.1007/BF02096594. |
[9] |
P. J. Forrester, The spectrum edge of random matrix ensembles,, Nucl. Phys. B, 402 (1993), 709.
doi: 10.1016/0550-3213(93)90126-A. |
[10] |
H. H. Goldstine and J. von Neumann, Numerical inverting of matrices of high order. II,, Proc. AMS, 2 (1951), 188.
doi: 10.1090/S0002-9939-1951-0041539-X. |
[11] |
A. Greenbaum, Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences,, Linear Algebra Appl., 113 (1989), 7.
doi: 10.1016/0024-3795(89)90285-1. |
[12] |
M. Hestenes and E. Steifel, Method of conjugate gradients for solving linear systems,, J. Res., 20 (1952), 409. Google Scholar |
[13] |
T. Jiang and D. Li, Approximation of rectangular beta-laguerre ensembles and large deviations,, J. Theor. Probab., 28 (2015), 804.
doi: 10.1007/s10959-013-0519-7. |
[14] |
K. Johansson, Shape fluctuations and random matrices,, Commun. Math. Phys., 209 (2000), 437.
doi: 10.1007/s002200050027. |
[15] |
I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis,, Ann. Stat., 29 (2001), 295.
doi: 10.1214/aos/1009210543. |
[16] |
S. Kaniel, Estimates for some computational techniques in linear algebra,, Math. Comput., 20 (1966), 369.
doi: 10.1090/S0025-5718-1966-0234618-4. |
[17] |
P. R. Krishnaiah and T. C. Chang, On the exact distribution of the smallest root of the wishart matrix using zonal polynomials,, Ann. Inst. Stat. Math., 23 (1971), 293.
doi: 10.1007/BF02479230. |
[18] |
A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$,, Adv. Math. (N. Y)., 188 (2004), 337.
doi: 10.1016/j.aim.2003.08.015. |
[19] |
V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices,, Math. USSR-Sbornik, 1 (1967), 457. Google Scholar |
[20] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions,, Cambridge University Press, (2010).
|
[21] |
W.-Y. Qiu and R. Wong, Global asymptotic expansions of the Laguerre polynomials Riemann-Hilbert approach,, Numer. Algorithms, 49 (2008), 331.
doi: 10.1007/s11075-008-9159-x. |
[22] |
B Simon, Trace Ideals and Their Applications, volume 120 of Mathematical Surveys and Monographs,, American Mathematical Society, (2010).
|
[23] |
T. Sugiyama, On the distribution of the largest latent root and the corresponding latent vector for principal component analysis,, Ann. Math. Stat., 37 (1966), 995.
doi: 10.1214/aoms/1177699378. |
[24] |
G. Szegö, Orthogonal Polynomials,, Amer. Math. Soc., (1959). Google Scholar |
[25] |
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel,, Comm. Math. Phys., 159 (1994), 151.
doi: 10.1007/BF02100489. |
[26] |
T. Trogdon, Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions,, PhD thesis, (2013).
|
[27] |
M. Vanlessen, Strong asymptotics of laguerre-type orthogonal polynomials and applications in random matrix theory,, Constr. Approx., 25 (2007), 125.
doi: 10.1007/s00365-005-0611-z. |
[28] |
S.-X. Xu, D. Dai and Y.-Q. Zhao, Critical edge behavior and the bessel to airy transition in the singularly perturbed laguerre unitary ensemble,, Commun. Math. Phys., 332 (2014), 1257.
doi: 10.1007/s00220-014-2131-9. |
show all references
References:
[1] |
T. H. Baker, P. J. Forrester and P. A. Pearce, Random matrix ensembles with an effective extensive external charge,, J. Phys. A. Math. Gen., 31 (1998), 6087.
doi: 10.1088/0305-4470/31/29/002. |
[2] |
E. Basor, Y. Chen and L. Zhang, PDEs satisfied by extreme eigenvalues distributions of GUE and LUE,, Random Matrices Theory Appl., 1 (2012).
doi: 10.1142/S2010326311500031. |
[3] |
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach,, Amer. Math. Soc., (1999).
|
[4] |
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights,, Internat. Math. Res. Not., 16 (1997), 759.
doi: 10.1155/S1073792897000500. |
[5] |
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights,, Comm. Pure Appl. Math., 52 (1999), 1491.
|
[6] |
P. A. Deift, G. Menon, S. Olver and T. Trogdon, Universality in numerical computations with random data,, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 14973.
doi: 10.1073/pnas.1413446111. |
[7] |
A. Edelman, Eigenvalues and condition numbers of random matrices,, SIAM J. Matrix Anal. Appl., 9 (1988), 543.
doi: 10.1137/0609045. |
[8] |
A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity,, Comm. Math. Phys., 147 (1992), 395.
doi: 10.1007/BF02096594. |
[9] |
P. J. Forrester, The spectrum edge of random matrix ensembles,, Nucl. Phys. B, 402 (1993), 709.
doi: 10.1016/0550-3213(93)90126-A. |
[10] |
H. H. Goldstine and J. von Neumann, Numerical inverting of matrices of high order. II,, Proc. AMS, 2 (1951), 188.
doi: 10.1090/S0002-9939-1951-0041539-X. |
[11] |
A. Greenbaum, Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences,, Linear Algebra Appl., 113 (1989), 7.
doi: 10.1016/0024-3795(89)90285-1. |
[12] |
M. Hestenes and E. Steifel, Method of conjugate gradients for solving linear systems,, J. Res., 20 (1952), 409. Google Scholar |
[13] |
T. Jiang and D. Li, Approximation of rectangular beta-laguerre ensembles and large deviations,, J. Theor. Probab., 28 (2015), 804.
doi: 10.1007/s10959-013-0519-7. |
[14] |
K. Johansson, Shape fluctuations and random matrices,, Commun. Math. Phys., 209 (2000), 437.
doi: 10.1007/s002200050027. |
[15] |
I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis,, Ann. Stat., 29 (2001), 295.
doi: 10.1214/aos/1009210543. |
[16] |
S. Kaniel, Estimates for some computational techniques in linear algebra,, Math. Comput., 20 (1966), 369.
doi: 10.1090/S0025-5718-1966-0234618-4. |
[17] |
P. R. Krishnaiah and T. C. Chang, On the exact distribution of the smallest root of the wishart matrix using zonal polynomials,, Ann. Inst. Stat. Math., 23 (1971), 293.
doi: 10.1007/BF02479230. |
[18] |
A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$,, Adv. Math. (N. Y)., 188 (2004), 337.
doi: 10.1016/j.aim.2003.08.015. |
[19] |
V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices,, Math. USSR-Sbornik, 1 (1967), 457. Google Scholar |
[20] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions,, Cambridge University Press, (2010).
|
[21] |
W.-Y. Qiu and R. Wong, Global asymptotic expansions of the Laguerre polynomials Riemann-Hilbert approach,, Numer. Algorithms, 49 (2008), 331.
doi: 10.1007/s11075-008-9159-x. |
[22] |
B Simon, Trace Ideals and Their Applications, volume 120 of Mathematical Surveys and Monographs,, American Mathematical Society, (2010).
|
[23] |
T. Sugiyama, On the distribution of the largest latent root and the corresponding latent vector for principal component analysis,, Ann. Math. Stat., 37 (1966), 995.
doi: 10.1214/aoms/1177699378. |
[24] |
G. Szegö, Orthogonal Polynomials,, Amer. Math. Soc., (1959). Google Scholar |
[25] |
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel,, Comm. Math. Phys., 159 (1994), 151.
doi: 10.1007/BF02100489. |
[26] |
T. Trogdon, Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions,, PhD thesis, (2013).
|
[27] |
M. Vanlessen, Strong asymptotics of laguerre-type orthogonal polynomials and applications in random matrix theory,, Constr. Approx., 25 (2007), 125.
doi: 10.1007/s00365-005-0611-z. |
[28] |
S.-X. Xu, D. Dai and Y.-Q. Zhao, Critical edge behavior and the bessel to airy transition in the singularly perturbed laguerre unitary ensemble,, Commun. Math. Phys., 332 (2014), 1257.
doi: 10.1007/s00220-014-2131-9. |
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