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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), 6087-6101.
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), 1150003, 21pp.
doi: 10.1142/S2010326311500031. |
[3] |
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Amer. Math. Soc., Providence, RI, 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), 759-782.
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), 1491-1552. |
[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), 14973-14978.
doi: 10.1073/pnas.1413446111. |
[7] |
A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., 9 (1988), 543-560.
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), 395-430.
doi: 10.1007/BF02096594. |
[9] |
P. J. Forrester, The spectrum edge of random matrix ensembles, Nucl. Phys. B, 402 (1993), 709-728.
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), 188-202.
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), 7-63.
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), 409-436. |
[13] |
T. Jiang and D. Li, Approximation of rectangular beta-laguerre ensembles and large deviations, J. Theor. Probab., 28 (2015), 804-847.
doi: 10.1007/s10959-013-0519-7. |
[14] |
K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys., 209 (2000), 437-476.
doi: 10.1007/s002200050027. |
[15] |
I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Stat., 29 (2001), 295-327.
doi: 10.1214/aos/1009210543. |
[16] |
S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comput., 20 (1966), 369-378.
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), 293-295.
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), 337-398.
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), 457-483. |
[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), 331-372.
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, Providence, Rhode Island, mar 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), 995-1001.
doi: 10.1214/aoms/1177699378. |
[24] |
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1959. |
[25] |
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159 (1994), 151-174.
doi: 10.1007/BF02100489. |
[26] |
T. Trogdon, Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions, PhD thesis, University of Washington, nov 2013. |
[27] |
M. Vanlessen, Strong asymptotics of laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx., 25 (2007), 125-175.
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), 1257-1296.
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), 6087-6101.
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), 1150003, 21pp.
doi: 10.1142/S2010326311500031. |
[3] |
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Amer. Math. Soc., Providence, RI, 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), 759-782.
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), 1491-1552. |
[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), 14973-14978.
doi: 10.1073/pnas.1413446111. |
[7] |
A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., 9 (1988), 543-560.
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), 395-430.
doi: 10.1007/BF02096594. |
[9] |
P. J. Forrester, The spectrum edge of random matrix ensembles, Nucl. Phys. B, 402 (1993), 709-728.
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), 188-202.
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), 7-63.
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), 409-436. |
[13] |
T. Jiang and D. Li, Approximation of rectangular beta-laguerre ensembles and large deviations, J. Theor. Probab., 28 (2015), 804-847.
doi: 10.1007/s10959-013-0519-7. |
[14] |
K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys., 209 (2000), 437-476.
doi: 10.1007/s002200050027. |
[15] |
I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Stat., 29 (2001), 295-327.
doi: 10.1214/aos/1009210543. |
[16] |
S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comput., 20 (1966), 369-378.
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), 293-295.
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), 337-398.
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), 457-483. |
[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), 331-372.
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, Providence, Rhode Island, mar 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), 995-1001.
doi: 10.1214/aoms/1177699378. |
[24] |
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1959. |
[25] |
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159 (1994), 151-174.
doi: 10.1007/BF02100489. |
[26] |
T. Trogdon, Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions, PhD thesis, University of Washington, nov 2013. |
[27] |
M. Vanlessen, Strong asymptotics of laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx., 25 (2007), 125-175.
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), 1257-1296.
doi: 10.1007/s00220-014-2131-9. |
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