The asymptotics of a Bessel-kernel determinant which arises in random matrix theory. (English) Zbl 1241.47059
Summary: In Random Matrix Theory, the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel
\[ B_\alpha(x,y) =\sqrt{x y}\, \frac{J_{\alpha}(x) y J'_\alpha(y)-J_{\alpha}(y)xJ'_\alpha(x)}{x^2-y^2},\qquad x,y>0,\quad \alpha>-1. \]
In particular, the so-called hard edge gap probabilities \(P^{(\alpha)}(R)\) can be expressed as the Fredholm determinants of the corresponding integral operator \(B_\alpha\) restricted to the finite interval \([0,R]\). Using operator theoretic methods, we are going to compute their asymptotics as \(R\to\infty\), i.e., we show that
\[ P^{(\alpha)}(R):=\det(I-B_\alpha)|_{L^2[0,R]}\sim\exp\Big(-\frac{R^2}{4}+\alpha R-\frac{\alpha^2}{2}\log R\Big)\frac{G(1+\alpha)}{(2\pi)^{\alpha/2}}, \]
where \(G\) stands for the Barnes \(G\)-function. In fact, this asymptotic formula will be proved for all complex parameters \(\alpha\) satisfying \(|\operatorname{Re} \alpha|<1\).
\[ B_\alpha(x,y) =\sqrt{x y}\, \frac{J_{\alpha}(x) y J'_\alpha(y)-J_{\alpha}(y)xJ'_\alpha(x)}{x^2-y^2},\qquad x,y>0,\quad \alpha>-1. \]
In particular, the so-called hard edge gap probabilities \(P^{(\alpha)}(R)\) can be expressed as the Fredholm determinants of the corresponding integral operator \(B_\alpha\) restricted to the finite interval \([0,R]\). Using operator theoretic methods, we are going to compute their asymptotics as \(R\to\infty\), i.e., we show that
\[ P^{(\alpha)}(R):=\det(I-B_\alpha)|_{L^2[0,R]}\sim\exp\Big(-\frac{R^2}{4}+\alpha R-\frac{\alpha^2}{2}\log R\Big)\frac{G(1+\alpha)}{(2\pi)^{\alpha/2}}, \]
where \(G\) stands for the Barnes \(G\)-function. In fact, this asymptotic formula will be proved for all complex parameters \(\alpha\) satisfying \(|\operatorname{Re} \alpha|<1\).
MSC:
| 47N30 | Applications of operator theory in probability theory and statistics |
| 60B20 | Random matrices (probabilistic aspects) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 15B52 | Random matrices (algebraic aspects) |
Keywords:
random matrix; gap probability; determinant asymptotics; Wiener-Hopf operator; Toeplitz operator; Hankel operatorReferences:
| [1] | Baik, J.; Buckingham, R.; DiFranco, J., Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function, Comm. Math. Phys., 280, 2, 463-497 (2008) · Zbl 1221.33032 |
| [2] | Baik, J.; Buckingham, R.; DiFranco, J.; Its, A., Total integrals of global solutions to Painlevé II, Nonlinearity, 22, 5, 1021-1061 (2009) · Zbl 1179.33036 |
| [3] | Barnes, E. B., The theory of the \(G\)-function, Quart. J. Pure Appl. Math., XXXI, 264-313 (1990) |
| [4] | Basor, E. L.; Ehrhardt, T., Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices, Math. Nachr., 228, 5-45 (2001) · Zbl 1047.47020 |
| [5] | Basor, E. L.; Ehrhardt, T., Some identities for determinants of structured matrices, Linear Algebra Appl., 343-344, 5-19 (2002) · Zbl 0994.15008 |
| [6] | Basor, E. L.; Ehrhardt, T., Asymptotics of determinants of Bessel operators, Comm. Math. Phys., 234, 491-516 (2003) · Zbl 1022.47015 |
| [7] | Basor, E. L.; Ehrhardt, T., On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants, New York J. Math., 11, 171-203 (2005) · Zbl 1099.47026 |
| [9] | Basor, E. L.; Ehrhardt, T.; Widom, H., On the determinant of a certain Wiener-Hopf + Hankel operator, Integral Equations Operator Theory, 47, 3, 275-288 (2003) · Zbl 1042.47019 |
| [10] | Böttcher, A.; Silbermann, B., Analysis of Toeplitz Operators (2006), Springer: Springer Berlin · Zbl 1098.47002 |
| [11] | Deift, P.; Its, A.; Krasovksy, I., Asymptotics of the Airy-kernel determinant, Comm. Math. Phys., 278, 3, 643-678 (2008) · Zbl 1167.15005 |
| [12] | Deift, P.; Its, A.; Krasovksy, I.; Zhou, X., The Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2), 146, 1, 149-235 (1997) · Zbl 0936.47028 |
| [13] | Deift, P.; Its, A.; Krasovksy, I.; Zhou, X., The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math., 202, 1, 26-47 (2007) · Zbl 1116.15019 |
| [14] | Dyson, F. J., Fredholm determinants and inverse scattering problems, Comm. Math. Phys., 47, 171-183 (1976) · Zbl 0323.33008 |
| [15] | Ehrhardt, T., Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Comm. Math. Phys., 262, 317-341 (2006) · Zbl 1113.82030 |
| [16] | Ehrhardt, T., Dyson’s constants in the asymptotics of the determinants of Wiener-Hopf-Hankel operators with the sine kernel, Comm. Math. Phys., 272, 683-698 (2007) · Zbl 1135.47023 |
| [17] | Ehrhardt, T.; Silbermann, B., Approximate identities and stability of discrete convolution operators with flip, (Oper. Theory Adv. Appl., vol. 110 (1999), Birkhäuser: Birkhäuser Basel), 103-132 · Zbl 0938.45011 |
| [18] | Fokas, A.; Its, A.; Kapaev, A.; Novokshenov, V., Painlevé transcendents, the Riemann-Hilbert approach, (Math. Surveys Monogr., vol. 128 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 1111.34001 |
| [19] | Forrester, P. J., The spectrum edge of random matrix ensembles, Nuclear Phys. B, 402, 3, 709-728 (1993) · Zbl 1043.82538 |
| [20] | Gohberg, I.; Krein, M. G., Introduction to the theory of linear nonselfadjoint operators in Hilbert space, (Transl. Math. Monogr., vol. 18 (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0181.13504 |
| [21] | Hagen, R.; Roch, S.; Silbermann, B., Spectral theory of approximation methods for convolution equations, (Oper. Theory Adv. Appl., vol. 74 (1995), Birkhäuser: Birkhäuser Basel) · Zbl 0804.65141 |
| [22] | Jimbo, M., Monodromy problem and the boundary problem for some Painlevé equations, Publ. Res. Inst. Math. Sci., 18, 1137-1161 (1982) · Zbl 0535.34042 |
| [23] | Krasovsky, I. V., Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle, Int. Math. Res. Not. IMRN, 25, 1249-1272 (2004) · Zbl 1077.60079 |
| [24] | Kuijlaars, A. B.J.; Vanlessen, M., Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. IMRN, 30, 1575-1600 (2002) · Zbl 1122.30303 |
| [25] | Mehta, M. L., Random Matrices (1994), Academic Press: Academic Press San Diego · Zbl 0594.60067 |
| [26] | Nagao, T.; Forrester, P. J., Asymptotic correlations at the spectrum edge of random matrices, Nuclear Phys. B, 435, 3, 401-420 (1995) · Zbl 1020.82588 |
| [27] | Nagao, T.; Wadati, M., Eigenvalue distribution of random matrices at the spectrum edge, J. Phys. Soc. Japan, 62, 11, 3845-3856 (1993) · Zbl 0972.82543 |
| [28] | Okamoto, K., Studies on the Painlevé equations IV. Third Painlevé equation \(P_{III}\), Funkcial. Ekvac., 30, 305-332 (1987) · Zbl 0639.58013 |
| [29] | Olver, F. W.J., Asymptotics and Special Functions, AKP Classics (1997), A.K. Peters, Ltd.: A.K. Peters, Ltd. Wellesley, MA · Zbl 0982.41018 |
| [30] | Power, S., The essential spectrum of a Hankel operator with piecewise continuous symbol, Michigan Math. J., 25, 1, 117-121 (1978) · Zbl 0365.47016 |
| [31] | Power, S., \(C^\ast \)-algebras generated by Hankel operators and Toeplitz operators, J. Funct. Anal., 31, 52-68 (1979) · Zbl 0401.46036 |
| [33] | Roch, S.; Santos, P. A.; Silbermann, B., Finite section method in some algebras of multiplication and convolution operators and a flip, Z. Anal. Anwend., 16, 3, 575-606 (1997) · Zbl 0887.65141 |
| [35] | Simon, B., Trace Ideals and Their Applications, Math. Surveys Monogr., vol. 120 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1074.47001 |
| [36] | Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (1937), Clarendon Press: Clarendon Press Oxford · Zbl 0017.40404 |
| [37] | Tracy, C. A.; Widom, H., Introduction to random matrices, (Lecture Notes in Phys., vol. 424 (1993), Springer: Springer Berlin), 103-130 · Zbl 0791.15017 |
| [38] | Tracy, C. A.; Widom, H., Level spacing distributions and the Airy kernel, Comm. Math. Phys., 159, 151-174 (1994) · Zbl 0789.35152 |
| [39] | Tracy, C. A.; Widom, H., Level spacing distributions and the Bessel kernel, Comm. Math. Phys., 161, 289-309 (1994) · Zbl 0808.35145 |
| [40] | Tracy, C. A.; Widom, H., Fredholm determinants, differential equations and matrix models, Comm. Math. Phys., 163, 33-72 (1994) · Zbl 0813.35110 |
| [41] | Vanlessen, M., Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx., 25, 2, 125-175 (2007) · Zbl 1117.15025 |
| [42] | Widom, H., The strong Szegö limit theorem for circular arcs, Indiana Univ. Math. J., 21, 277-283 (1971/1972) · Zbl 0213.34903 |
| [43] | Widom, H., The asymptotics of a continuous analogue of orthogonal polynomials, J. Approx. Theory, 77, 51-64 (1994) · Zbl 0801.42017 |
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.
