Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. (English) Zbl 1113.82030
Summary: We prove that the asymptotics of the Fredholm determinant of \(I-K_\alpha\), where \(K_\alpha\) is the integral operator with the sine kernel \(\frac{\sin(x-y)}{\pi(x-y)}\) on the interval \([0,\alpha]\), are given by
\[
\log\det(I-K_{2\alpha}) = -\frac{\alpha^2}{2} -\frac{\log\alpha}{4} +\frac{\log 2}{12} + 3\zeta'(-1) + o(1), \qquad \alpha \to \infty.
\]
This formula was conjectured by Dyson. The proof for the first and second order asymptotics was given by Widom, and higher order asymptotics have also been determined. In this paper we identify the constant (or third order) term, which has been an outstanding problem for a long time.
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82B05 | Classical equilibrium statistical mechanics (general) |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 15B52 | Random matrices (algebraic aspects) |
References:
| [1] | Barnes, E. B., The theory of the G-function, Quart. J. Pure Appl. Math., XXXI, 264-313 (1900) · JFM 30.0389.02 |
| [2] | Basor, E. L.; Ehrhardt, T., Some identities for determinants of structured matrices, Linear Algebra Appl., 343/344, 5-19 (2002) · Zbl 0994.15008 |
| [3] | Basor, New York J. Math., 11, 171 (2005) · Zbl 1099.47026 |
| [4] | Basor, Phys. Rev. Lett., 69, 5 (1) · Zbl 0968.82501 · doi:10.1103/PhysRevLett.69.5 |
| [5] | Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators. Berlin: Springer, 1990 · Zbl 0732.47029 |
| [6] | Deift, Ann. Math., 146, 149 (1997) · Zbl 0936.47028 · doi:10.2307/2951834 |
| [7] | Dyson, Commun. Math. Phys., 47, 171 (1976) · Zbl 0323.33008 · doi:10.1007/BF01608375 |
| [8] | Ehrhardt, Operator Theory: Adv. Appl., Vol., 110, 103 (1999) · Zbl 0938.45011 |
| [9] | Gohberg, I., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space. Trans. Math. Monographs 18, Providence, R.I.: Amer. Math. Soc., 1969 · Zbl 0181.13503 |
| [10] | Metha, M.L.: Random Matrices. 6^th ed., San Diego: Academic Press, 1994 |
| [11] | Jimbo, Phys. D, 1, 80 (1) · Zbl 1194.82007 · doi:10.1016/0167-2789(80)90006-8 |
| [12] | 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. (25), 1249-1272 (2004) · Zbl 1077.60079 |
| [13] | Power, Michigan Math. J., 25, 117 (1) · Zbl 0365.47016 |
| [14] | Power, J. Funct. Anal., 31, 52 (31) · Zbl 0401.46036 · doi:10.1016/0022-1236(79)90097-1 |
| [15] | Tracy, C.A., Widom, H.: Introduction to random matrices. In: Lecture Notes in Physics, Vol. 424, Berlin: Springer, 1993, pp. 103-130 · Zbl 0791.15017 |
| [16] | Widom, H., The strong Szegö limit theorem for circular arcs, Indiana Univ. Math. J., 21, 277-283 (1971) · Zbl 0213.34903 |
| [17] | Widom, J. Appr. Theory, 77, 51 (1994) · Zbl 0801.42017 · doi:10.1006/jath.1994.1033 |
| [18] | Widom, Commun. Math. Phys., 171, 159 (1995) · Zbl 0839.47032 · doi:10.1007/BF02103774 |
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.
