×

On the scaling limits of determinantal point processes with kernels induced by Sturm-Liouville operators. (English) Zbl 1347.15041

Summary: By applying an idea of A. Borodin and G. Olshanski [J. Algebra 313, No. 1, 40–60 (2007; Zbl 1117.60051)], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE).

MSC:

15B52 Random matrices (algebraic aspects)
34B24 Sturm-Liouville theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
60B20 Random matrices (probabilistic aspects)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 1117.60051
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, Milton and Stegun, Irene A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, xiv+1046, (1964), U.S. Government Printing Office, Washington, D.C. · Zbl 0171.38503
[2] Anderson, Greg W. and Guionnet, Alice and Zeitouni, Ofer, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, xiv+492, (2010), Cambridge University Press, Cambridge · Zbl 1184.15023 · doi:10.1017/CBO9780511801334
[3] Andrews, George E. and Askey, Richard and Roy, Ranjan, Special functions, Encyclopedia of Mathematics and its Applications, 71, xvi+664, (1999), Cambridge University Press, Cambridge · Zbl 0920.33001 · doi:10.1017/CBO9781107325937
[4] Borodin, Alexei and Olshanski, Grigori, Asymptotics of {P}lancherel-type random partitions, Journal of Algebra, 313, 1, 40-60, (2007) · Zbl 1117.60051 · doi:10.1016/j.jalgebra.2006.10.039
[5] Collins, Beno{\^{\i}}t, Product of random projections, {J}acobi ensembles and universality problems arising from free probability, Probability Theory and Related Fields, 133, 3, 315-344, (2005) · Zbl 1100.46036 · doi:10.1007/s00440-005-0428-5
[6] Deift, P. A., Orthogonal polynomials and random matrices: a {R}iemann–{H}ilbert approach, Courant Lecture Notes in Mathematics, 3, viii+273, (1999), New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI
[7] Erd{\'e}lyi, Arthur and Magnus, Wilhelm and Oberhettinger, Fritz and Tricomi, Francesco G., Higher transcendental functions. {V}ol. {II}, xvii+396, (1953), New York · Zbl 0052.29502
[8] Forrester, P. J., Log-gases and random matrices, London Mathematical Society Monographs Series, 34, xiv+791, (2010), Princeton University Press, Princeton, NJ · Zbl 1217.82003 · doi:10.1515/9781400835416
[9] Gohberg, Israel and Goldberg, Seymour and Krupnik, Nahum, Traces and determinants of linear operators, Operator Theory: Advances and Applications, 116, x+258, (2000), Birkh\"auser Verlag, Basel · Zbl 0946.47013 · doi:10.1007/978-3-0348-8401-3
[10] Hutson, Vivian and Pym, John S. and Cloud, Michael J., Applications of functional analysis and operator theory, Mathematics in Science and Engineering, 200, xiv+426, (2005), Elsevier B.V., Amsterdam · Zbl 1066.47001
[11] Johansson, Kurt, Shape fluctuations and random matrices, Communications in Mathematical Physics, 209, 2, 437-476, (2000) · Zbl 0969.15008 · doi:10.1007/s002200050027
[12] Johnstone, Iain M., On the distribution of the largest eigenvalue in principal components analysis, The Annals of Statistics, 29, 2, 295-327, (2001) · Zbl 1016.62078 · doi:10.1214/aos/1009210544
[13] Johnstone, Iain M., Multivariate analysis and {J}acobi ensembles: largest eigenvalue, {T}racy–{W}idom limits and rates of convergence, The Annals of Statistics, 36, 6, 2638-2716, (2008) · Zbl 1284.62320 · doi:10.1214/08-AOS605
[14] Kuijlaars, A. B. J., Universality, The {O}xford Handbook of Random Matrix Theory, 103-134, (2011), Oxford University Press, Oxford · Zbl 1236.15071
[15] Lax, Peter D., Functional analysis, Pure and Applied Mathematics, xx+580, (2002), Wiley-Interscience, New York · Zbl 1009.47001
[16] Lubinsky, Doron S., A new approach to universality limits involving orthogonal polynomials, Annals of Mathematics. Second Series, 170, 2, 915-939, (2009) · Zbl 1176.42022 · doi:10.4007/annals.2009.170.915
[17] Reed, Michael and Simon, Barry, Methods of modern mathematical physics. {I}. {F}unctional analysis, xvii+325, (1972), Academic Press, New York – London · Zbl 0242.46001
[18] Simon, Barry, Trace ideals and their applications, Mathematical Surveys and Monographs, 120, viii+150, (2005), Amer. Math. Soc., Providence, RI · Zbl 1074.47001
[19] Stolz, G{\"u}nter and Weidmann, Joachim, Approximation of isolated eigenvalues of ordinary differential operators, Journal f\"ur die Reine und Angewandte Mathematik, 445, 31-44, (1993) · Zbl 0781.34052 · doi:10.1515/crll.1993.445.31
[20] Tao, Terence, Topics in random matrix theory, Graduate Studies in Mathematics, 132, x+282, (2012), Amer. Math. Soc., Providence, RI · Zbl 1256.15020 · doi:10.1090/gsm/132
[21] Tracy, Craig A. and Widom, Harold, Level-spacing distributions and the {A}iry kernel, Communications in Mathematical Physics, 159, 1, 151-174, (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489
[22] Tracy, Craig A. and Widom, Harold, Level spacing distributions and the {B}essel kernel, Communications in Mathematical Physics, 161, 2, 289-309, (1994) · Zbl 0808.35145 · doi:10.1007/BF02099779
[23] Wachter, Kenneth W., The limiting empirical measure of multiple discriminant ratios, The Annals of Statistics, 8, 5, 937-957, (1980) · Zbl 0473.62050 · doi:10.1214/aos/1176345134
[24] Weidmann, Joachim, Linear operators in {H}ilbert spaces, Graduate Texts in Mathematics, 68, xiii+402, (1980), Springer-Verlag, New York – Berlin · Zbl 0434.47001 · doi:10.1007/978-1-4612-6027-1
[25] Weidmann, Joachim, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, 1258, vi+303, (1987), Springer-Verlag, Berlin · Zbl 0647.47052 · doi:10.1007/BFb0077960
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.