Nagoya Mathematical Journal

Application of the $\tau$-function theory of Painlevé equations to random matrices: $\rm {P}_{\rm VI}$, the JUE, CyUE, cJUE and scaled limits

P. J. Forrester and N. S. Witte

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Okamoto has obtained a sequence of $\tau$-functions for the $P_{VI}$ 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 multi-dimensional 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 $P_{VI}$ theory. We show that the Hamiltonian also satisfies an equation related to the discrete $P_{V}$ 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 $P_{V}$ e transcendent in $\sigma$ 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 $\tau$-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.

Article information

Nagoya Math. J., Volume 174 (2004), 29-114.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33E17: Painlevé-type functions
Secondary: 15A52 34A05: Explicit solutions and reductions 34M55: Painlevé and other special equations; classification, hierarchies; 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37N20: Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]


Forrester, P. J.; Witte, N. S. Application of the $\tau$-function theory of Painlevé equations to random matrices: $\rm {P}_{\rm VI}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174 (2004), 29--114.

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  • M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices , Comm. Pure Appl. Math., 54 , no. 2, 153–205 (2001). math.CO/9912143.
  • M. Adler and P. van Moerbeke, Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice , Comm. Math. Phys., 237 (2003, no. 3), 397–440.
  • K. Aomoto, Jacobi polynomials associated with Selberg integrals , SIAM J. Math. Anal., 18 , no. 2, 545–549 (1987).
  • J. Baik, Riemann-Hilbert problems for last passage percolation , Recent developments in integrable systems and Riemann-Hilbert problems (Birmingham, AL, 2000), Contemp. Math., 326, 1–21, Amer. Math. Soc., Providence, RI (2003).
  • J. Baik, Painlevé expressions for LOE, LSE, and interpolating ensembles , Int. Math. Res. Not., 33 , 1739–1789 (2002).
  • J. Baik and E. M. Rains, Algebraic aspects of increasing subsequences , Duke Math. J., 109 , no. 1, 1–65 (2001).
  • J. Baik and E. M. Rains, Symmetrized random permutations (P. M. Bleher and A. R. Its, Random matrix models and their applications, eds.), 1–19, Cambridge Univ. Press, Cambridge (2001).
  • T. H. Baker and P. J. Forrester, Random walks and random fixed-point free involutions , J. Phys. A, 34 , no. 28, L381–L390 (2001).
  • R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).
  • A. Borodin, Discrete gap probabilities and discrete Painlevé equations , Duke Math. J., 117 (2003, no. 3), 489–542.
  • A. Borodin and D. Boyarchenko, Distribution of the first particle in discrete orthogonal polynomial ensembles , Comm. Math. Phys., 234 (2003, no. 2), 287–338.
  • A. Borodin and P. Deift, Fredholm determinants, Jimbo-Miwa-Ueno $\tau$-functions, and representation theory , Comm. Pure Appl. Math., 55 (2002, no. 9), 1160–1230.
  • A. Borodin, A. Okounkov and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups , J. Amer. Math. Soc., 13 , no. 3, 481–515, (electronic (2000)).
  • A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures , Comm. Math. Phys., 223 , 87–123 (2001).
  • A. Borodin and G. Olshanski, $z$-measures on partitions, Robinson-Schensted-Knuth correspondence, and $\beta=2$ random matrix ensembles (P. M. Bleher and A. R. Its, Random matrix models and their applications, eds.), 71–94, Cambridge Univ. Press, Cambridge (2001).
  • C. M. Cosgrove and G. Scoufis, Painlevé classification of a class of differential equations of the second order and second degree , Stud. Appl. Math., 88 , 25–87 (1993).
  • P. A. Deift, A. R. Its and X. Zhou, 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 , no. 1, 149–235 (1997).
  • P. J. Forrester, Log Gases and Random Matrices . $\tilde\phantomx$matpjf/matpjf.html.
  • P. J. Forrester, Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles . solv-int/0005064.
  • P. J. Forrester, Exact integral formulas and asymptotics for the correlations in the $1/r\sp 2$ quantum many-body system. , Phys. Lett. A, 179 , no. 2, 127–130 (1993).
  • P. J. Forrester, The spectrum edge of random matrix ensembles , Nucl. Phys. B, 402 , 709–728 (1993).
  • P. J. Forrester, Addendum to: “Selberg correlation integrals and the $1/r\sp 2$ quantum many-body system” , Nucl. Phys. B, 416 , no. 1, 377–385 (1994).
  • P. J. Forrester, Exact results and universal asymptotics in the Laguerre random matrix ensemble , J. Math. Phys., 35 , no. 5, 2539–2551 (1994).
  • P. J. Forrester, Normalization of the wavefunction for the Calogero-Sutherland model with internal degrees of freedom , Internat. J. Modern Phys. B, 9 , no. 10, 1243–1261 (1995).
  • P. J. Forrester, Inter-relationships between gap probabilities in random matrix theory , preprint (1999).
  • P. J. Forrester, T. Nagao and G. Honner, Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges , Nucl. Phys. B, 553 , no. 3, 601–643 (1999).
  • P. J. Forrester and E. M. Rains, Interrelationships between orthogonal, unitary and symplectic matrix ensembles (P. M. Bleher and A. R. Its, Random matrix models and their applications, eds.), 171–207, Cambridge Univ. Press, Cambridge (2001). solv-int/9907008.
  • P. J. Forrester and N. S. Witte, Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles , Lett. Math. Phys., 53 , 195–200 (2000).
  • P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE , Commun. Math. Phys., 219 , 357–398 (2001).
  • P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE and CUE , Commun. Pure Appl. Math., 55 , 679–727 (2002).
  • P. J. Forrester and N. S. Witte, $\tau$-Function evaluation of gap probabilities in orthogonal and symplectic matrix ensembles , Nonl., 15 , 937–954 (2002). solv-int/0203049.
  • P. J. Forrester and E. M. Rains, Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter , in preparation (2002).
  • B. Grammaticos, Y. Ohta, A. Ramani and H. Sakai, Degeneration through coalescence of the $q$-Painlevé VI equation , J. Phys. A, 31 , no. 15, 3545–3558 (1998).
  • J. Gravner, C. A. Tracy and H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models , J. Statist. Phys., 102 , no. 5–6, 1085–1132 (2001).
  • L. Haine and J.-P. Semengue, The Jacobi polynomial ensemble and the Painlevé VI equation , J. Math. Phys., 40 , 2117–2134 (1999).
  • H. Hochstadt, The Functions of Mathematical Physics, Wiley-Interscience, New York (1971).
  • M. Jimbo and T. Miwa, Studies on holonomic quantum fields. XVII , Proc. Japan Acad. Ser. A Math. Sci., 56 , no. 9, 405–410 (1980).
  • M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II , Phys. D, 2 , no. 3, 407–448 (1981).
  • M. Jimbo, T. Miwa, Y. Môri and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent , Phys. D, 1 , no. 1, 80–158 (1980).
  • K. Johansson, Shape fluctuations and random matrices , Comm. Math. Phys., 209 , no. 2, 437–476 (2000).
  • K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Determinant formulas for the Toda and discrete Toda equations , Funkcialaj Ekvacioj, 44 , 291–307 (2001). solv-int/9908007.
  • J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials , SIAM J. Math. Anal., 24 , no. 4, 1086–1110 (1993).
  • J. Malmquist, Sur les équations différentielles du second ordre dont l'intégrale générale a ses points critiques fixes , Arkiv Mat., Astron. Fys., 18 , no. 8, 1–89 (1922).
  • B. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Harvard University Press, Harvard (1973).
  • U. Muğan and A. Sakka, Schlesinger transformations for Painlevé VI equation , J. Math. Phys., 36 , no. 3, 1284–1298 (1995).
  • T. Muir, The Theory of Determinants in the Historical Order of Development, Dover, New York (1960).
  • B. Nickel, On the singularity structure of the $2d$ Ising model susceptibility , J. Phys. A, 32 , 3889–3906 (1999).
  • B. Nickel, Addendum to `On the singularity structure of the $2d$ Ising model susceptibility' , J. Phys. A, 33 , 1693–1711 (2000).
  • M. Noumi, S. Okada, K. Okamoto and H. Umemura, Special polynomials associated with the Painlevé equations II , Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 349–372, World Sci. Publishing, River Edge, NJ (1998).
  • M. Noumi and Y. Yamada, Affine Weyl group symmetries in Painlevé type equations , Toward the exact WKB analysis of Differential Equations, Linear or Non-Linear (C. J. Howls, T. Kawai and Y. Takei, eds.), 245–259, Kyoto University Press (2000).
  • M. Noumi and Y. Yamada, A new Lax pair for the sixth Painlevé equation associated with $\widehatso(8)$ , math-ph/0203029 (2002).
  • M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations , Commun. Math. Phys., 199 , 281–295 (1998).
  • K. Okamoto, Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_\rm II$ and $P_\rm IV$ , Math. Ann., 275 , no. 2, 221–255 (1986).
  • K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation $P_\rm VI$ , Ann. Mat. Pura Appl. (4), 146 , 337–381 (1987).
  • K. Okamoto, Studies on the Painlevé equations. II. Fifth Painlevé equation $P_\rm V$ , Japan. J. Math. (N.S.), 13 , no. 1, 47–76 (1987).
  • K. Okamoto, Studies on the Painlevé equations. IV. Third Painlevé equation $P_\rm III$ , Funkcial. Ekvac., 30 , no. 2–3, 305–332 (1987).
  • W. P. Orrick, B. Nickel, A. J. Guttmann and J. H. H. Perk, The susceptibility of the square lattice Ising model: New developments , J. Stat. Phys., 102 , 795–841 (2001).
  • W. P. Orrick, B. G. Nickel, A. J. Guttmann and J. H. H. Perk, Critical behaviour of the two-dimensional Ising susceptibility , Phys. Rev. Lett., 86 , 4120–4123 (2001).
  • E. M. Rains, Increasing subsequences and the classical groups , Electron. J. Combin., 5 , no. 1, Research Paper 12, 9 pp., (electronic (1998)).
  • A. Ramani, Y. Ohta and B. Grammaticos, Discrete integrable systems from continuous Painlevé equations through limiting procedures , Nonlinearity, 13 , 1073–1085 (2000).
  • H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations , Comm. Math. Phys., 220 , no. 1, 165–229 (2001).
  • A. Selberg, Remarks on a multiple integral , Norsk Mat. Tidsskr., 26 , 71–78 (1944).
  • G. Szegö, Orthogonal Polynomials, Colloquium Publications 23, third edition, American Mathematical Society, Providence, Rhode Island (1967).
  • M. Taneda, Representation of Umemura polynomials for the sixth Painlevé equation by the generalized Jacobi polynomials , Physics and combinatorics 1999 (Nagoya), 366–376, World Sci. Publishing, River Edge, NJ (2001).
  • C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models , Commun. Math. Phys., 163 , no. 1, 33–72 (1994).
  • C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel , Commun. Math. Phys., 161 , no. 2, 289–309 (1994).
  • P. van Moerbeke, Integrable lattices: random matrices and random permutations (P. M. Bleher and A. R. Its, Random matrix models and their applications, eds.), 321–406, Cambridge Univ. Press, Cambridge (2001).
  • H. Watanabe, Birational canonical transformations and classical solutions of the sixth Painlevé equation , Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 , 379–425 (1998).
  • H. Widom, The asymptotics of a continuous analogue of orthogonal polynomials , J. Approx. Theory, 77 , no. 1, 51–64 (1994).
  • N. S. Witte, Gap probabilities for double intervals in hermitian random matrix ensembles as $\tau$-functions – the Bessel kernel case , in preparation (2001).
  • N. S. Witte and P. J. Forrester, Gap probabilities in the finite and scaled Cauchy random matrix ensembles , Nonl., 13 , 1965–1986 (2000).
  • N. S. Witte, P. J. Forrester and C. M. Cosgrove, Integrability, random matrices and Painlevé transcendents , Kruskal, 2000 (Adelaide), ANZIAM J., 44 (2002, no. 1), 41–50.
  • Z. M. Yan, A class of generalized hypergeometric functions in several variables , Canad. J. Math., 44 , no. 6, 1317–1338 (1992).