Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T21:24:32.542Z Has data issue: false hasContentIssue false

Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits

Published online by Cambridge University Press:  22 January 2016

P. J. Forrester
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia, [email protected]
N. S. Witte
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Okamoto has obtained a sequence of τ-functions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Adler, M. and Moerbeke, P. van, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math., 54 (2001), no. 2, 153205. math.CO/9912143.3.0.CO;2-5>CrossRefGoogle Scholar
[2] Adler, M. and Moerbeke, P. van, Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice, Comm. Math. Phys., 237 (2003), no. 3, 397440.Google Scholar
[3] Aomoto, K., Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal., 18 (1987), no. 2, 545549.Google Scholar
[4] Baik, J., Riemann-Hilbert problems for last passage percolation, Recent developments in integrable systems and Riemann-Hilbert problems (Birmingham, AL, 2000), Contemp. Math., 326, Amer. Math. Soc., Providence, RI (2003), pp. 121.Google Scholar
[5] Baik, J., Painlevé expressions for LOE, LSE, and interpolating ensembles, Int. Math. Res. Not., 33 (2002), 17391789.Google Scholar
[6] Baik, J. and Rains, E. M., Algebraic aspects of increasing subsequences, Duke Math. J., 109 (2001), no. 1, 165.Google Scholar
[7] Baik, J. and Rains, E. M., Symmetrized random permutations, Random matrix models and their applications (Bleher, P. M. and Its, A. R., eds.), Cambridge Univ. Press, Cambridge (2001), pp. 119.Google Scholar
[8] Baker, T. H. and Forrester, P. J., Random walks and random fixed-point free involutions, J. Phys. A, 34 (2001), no. 28, L381L390.CrossRefGoogle Scholar
[9] Baxter, R. J., Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.Google Scholar
[10] Borodin, A., Discrete gap probabilities and discrete Painlevé equations, Duke Math. J., 117 (2003), no. 3, 489542.Google Scholar
[11] Borodin, A. and Boyarchenko, D., Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys., 234 (2003), no. 2, 287338.Google Scholar
[12] Borodin, A. and Deift, P., Fredholm determinants, Jimbo-Miwa-Ueno τ-functions, and representation theory, Comm. Pure Appl. Math., 55 (2002), no. 9, 11601230.Google Scholar
[13] Borodin, A., Okounkov, A. and Olshanski, G., Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc., 13 (2000), no. 3, 481515, (electronic).Google Scholar
[14] Borodin, A. and Olshanski, G., Infinite random matrices and ergodic measures, Comm. Math. Phys., 223 (2001), 87123.Google Scholar
[15] Borodin, A. and Olshanski, G., z-measures on partitions, Robinson-Schensted-Knuth correspondence, and β = 2 random matrix ensembles, Random matrix models and their applications (P. M. Bleher and Its A. R., eds.), Cambridge Univ. Press, Cambridge (2001), pp. 7194.Google Scholar
[16] Cosgrove, C. M. and Scoufis, G., Painlevé classification of a class of differential equations of the second order and second degree, Stud. Appl. Math., 88 (1993), 2587.CrossRefGoogle Scholar
[17] Deift, P. A., Its, A. R. and Zhou, X., A 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 (1997), no. 1, 149235.Google Scholar
[18] Forrester, P. J., Log Gases and Random Matrices. http://www.ms.unimelb.edu.au/˜matpjf/matpjf.html.Google Scholar
[19] Forrester, P. J., Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles. solvint/0005064.Google Scholar
[20] Forrester, P. J., Exact integral formulas and asymptotics for the correlations in the 1/r2 quantum many-body system., Phys. Lett. A, 179 (1993), no. 2, 127130.CrossRefGoogle Scholar
[21] Forrester, P. J., The spectrum edge of random matrix ensembles, Nucl. Phys. B, 402 (1993), 709728.Google Scholar
[22] Forrester, P. J., Addendum to: “Selberg correlation integrals and the 1/r2 quantum many-body system”, Nucl. Phys. B, 416 (1994), no. 1, 377385.Google Scholar
[23] Forrester, P. J., Exact results and universal asymptotics in the Laguerre random matrix ensemble, J. Math. Phys., 35 (1994), no. 5, 25392551.Google Scholar
[24] Forrester, P. J., Normalization of the wavefunction for the Calogero-Sutherland model with internal degrees of freedom, Internat. J. Modern Phys. B, 9 (1995), no. 10, 12431261.Google Scholar
[25] Forrester, P. J., Inter-relationships between gap probabilities in random matrix theory, preprint (1999).Google Scholar
[26] Forrester, P. J., Nagao, T. and Honner, G., Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nucl. Phys. B, 553 (1999), no. 3, 601643.Google Scholar
[27] Forrester, P. J. and Rains, E. M., Interrelationships between orthogonal, unitary and symplectic matrix ensembles, Random matrix models and their applications (P. M. Bleher and Its A. R., eds.), Cambridge Univ. Press, Cambridge (2001), pp. 171207. solv-int/9907008.Google Scholar
[28] Forrester, P. J. and Witte, N. S., Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles, Lett. Math. Phys., 53 (2000), 195200.Google Scholar
[29] Forrester, P. J. and Witte, N. S., Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Commun. Math. Phys., 219 (2001), 357398.Google Scholar
[30] Forrester, P. J. and Witte, N. S., Application of the τ-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE and CUE, Commun. Pure Appl. Math., 55 (2002), 679727.Google Scholar
[31] Forrester, P. J. and Witte, N. S., τ-Function evaluation of gap probabilities in orthogonal and symplectic matrix ensembles, Nonl., 15 (2002), 937954. solv-int/0203049.Google Scholar
[32] Forrester, P. J. and Rains, E. M., Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter, in preparation (2002).Google Scholar
[33] Grammaticos, B., Ohta, Y., Ramani, A. and Sakai, H., Degeneration through coalescence of the q-Painlevé VI equation, J. Phys. A, 31 (1998), no. 15, 35453558.Google Scholar
[34] Gravner, J., Tracy, C. A. and Widom, H., Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Statist. Phys., 102 (2001), no. 56, 10851132.Google Scholar
[35] Haine, L. and Semengue, J.-P., The Jacobi polynomial ensemble and the Painlevé VI equation, J. Math. Phys., 40 (1999), 21172134.Google Scholar
[36] Hochstadt, H., The Functions of Mathematical Physics, Wiley-Interscience, New York, 1971.Google Scholar
[37] Jimbo, M. and Miwa, T., Studies on holonomic quantum fields. XVII, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), no. 9, 405410.CrossRefGoogle Scholar
[38] Jimbo, M. and Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D, 2 (1981), no. 3, 407448.Google Scholar
[39] Jimbo, M., Miwa, T., Môri, Y. and Sato, M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D, 1 (1980), no. 1, 80158.Google Scholar
[40] Johansson, K., Shape fluctuations and random matrices, Comm. Math. Phys., 209 (2000), no. 2, 437476.CrossRefGoogle Scholar
[41] Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y., Determinant formulas for the Toda and discrete Toda equations, Funkcialaj Ekvacioj, 44 (2001), 291307. solv-int/9908007.Google Scholar
[42] Kaneko, J., Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal., 24 (1993), no. 4, 10861110.Google Scholar
[43] Malmquist, J., Sur les équations différentielles du second ordre dont l’intégrale générale a ses points critiques fixes, Arkiv Mat., Astron. Fys., 18 (1922), no. 8, 189.Google Scholar
[44] McCoy, B. and Wu, T. T., The Two-Dimensional Ising Model, Harvard University Press, Harvard, 1973.Google Scholar
[45] Mu˘gan, U. and Sakka, A., Schlesinger transformations for Painlevé VI equation, J. Math. Phys., 36 (1995), no. 3, 12841298.CrossRefGoogle Scholar
[46] Muir, T., The Theory of Determinants in the Historical Order of Development, Dover, New York, 1960.Google Scholar
[47] Nickel, B., On the singularity structure of the 2d Ising model susceptibility, J. Phys. A, 32 (1999), 38893906.Google Scholar
[48] Nickel, B., Addendum to ‘On the singularity structure of the 2d Ising model susceptibility’, J. Phys. A, 33 (2000), 16931711.Google Scholar
[49] Noumi, M., Okada, S., Okamoto, K. and Umemura, H., Special polynomials associated with the Painlevé equations II, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publishing, River Edge, NJ (1998), pp. 349372.Google Scholar
[50] Noumi, M. and Yamada, Y., Affine Weyl group symmetries in Painlevé type equations, Toward the exact WKB analysis of Differential Equations, Linear or Non-Linear (Howls, C. J., Kawai, T. and Takei, Y., eds.), Kyoto University Press (2000), pp. 245259.Google Scholar
[51] Noumi, M. and Yamada, Y., A new Lax pair for the sixth Painlevé equation associated with sbo(8), math-ph/0203029 (2002).Google Scholar
[52] Noumi, M. and Yamada, Y., Affine Weyl groups, discrete dynamical systems and Painlevé equations, Commun. Math. Phys., 199 (1998), 281295.Google Scholar
[53] Okamoto, K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV , Math. Ann., 275 (1986), no. 2, 221255.Google Scholar
[54] Okamoto, K., Studies on the Painlevé equations. I. Sixth Painlevé equation PVI , Ann. Mat. Pura Appl. (4), 146 (1987), 337381.CrossRefGoogle Scholar
[55] Okamoto, K., Studies on the Painlevé equations. II. Fifth Painlevé equation PV , Japan. J. Math. (N.S.), 13 (1987), no. 1, 4776.Google Scholar
[56] Okamoto, K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII , Funkcial. Ekvac., 30 (1987), no. 23, 305332.Google Scholar
[57] Orrick, W. P., Nickel, B., Guttmann, A. J. and Perk, J. H. H., The susceptibility of the square lattice Ising model: New developments, J. Stat. Phys., 102 (2001), 795841.Google Scholar
[58] Orrick, W. P., Nickel, B. G., Guttmann, A. J. and Perk, J. H. H., Critical behaviour of the two-dimensional Ising susceptibility, Phys. Rev. Lett., 86 (2001), 41204123.Google Scholar
[59] Rains, E. M., Increasing subsequences and the classical groups, Electron. J. Combin., 5 (1998), no. 1, Research Paper 12, 9 pp., (electronic).CrossRefGoogle Scholar
[60] Ramani, A., Ohta, Y. and Grammaticos, B., Discrete integrable systems from continuous Painlevé equations through limiting procedures, Nonlinearity, 13 (2000), 10731085.Google Scholar
[61] Sakai, H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys., 220 (2001), no. 1, 165229.Google Scholar
[62] Selberg, A., Remarks on a multiple integral, Norsk Mat. Tidsskr., 26 (1944), 7178.Google Scholar
[63] Szegö, G., Orthogonal Polynomials, Colloquium Publications 23, third edition, American Mathematical Society, Providence, Rhode Island, 1967.Google Scholar
[64] Taneda, M., Representation of Umemura polynomials for the sixth Painlevé equation by the generalized Jacobi polynomials, Physics and combinatorics 1999 (Nagoya), World Sci. Publishing, River Edge, NJ (2001), pp. 366376.Google Scholar
[65] Tracy, C. A. and Widom, H., Fredholm determinants, differential equations and matrix models, Commun. Math. Phys., 163 (1994), no. 1, 3372.Google Scholar
[66] Tracy, C. A. and Widom, H., Level spacing distributions and the Bessel kernel, Commun. Math. Phys., 161 (1994), no. 2, 289309.Google Scholar
[67] Moerbeke, P. van, Integrable lattices: random matrices and random permutations, Random matrix models and their applications (Bleher, P. M. and Its, A. R., eds.), Cambridge Univ. Press, Cambridge (2001), pp. 321406.Google Scholar
[68] Watanabe, H., Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1998), 379425.Google Scholar
[69] Widom, H., The asymptotics of a continuous analogue of orthogonal polynomials, J. Approx. Theory, 77 (1994), no. 1, 5164.Google Scholar
[70] Witte, N. S., Gap probabilities for double intervals in hermitian random matrix ensembles as τ-functions – the Bessel kernel case, in preparation (2001).Google Scholar
[71] Witte, N. S. and Forrester, P. J., Gap probabilities in the finite and scaled Cauchy random matrix ensembles, Nonl., 13 (2000), 19651986.Google Scholar
[72] Witte, N. S., Forrester, P. J. and Cosgrove, C. M., Integrability, random matrices and Painlevé transcendents, Kruskal, 2000 (Adelaide), ANZIAM J., 44 (2002), no. 1, 4150.Google Scholar
[73] Yan, Z. M., A class of generalized hypergeometric functions in several variables, Canad. J. Math., 44 (1992), no. 6, 13171338.Google Scholar