Duke Mathematical Journal

Discrete gap probabilities and discrete Painlevé equations

Alexei Borodin

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Abstract

We prove that Fredholm determinants of the form $\det(1-K\sb s)$, where $K\sb s$ is the restriction of either the discrete Bessel kernel or the discrete $\sb 2F\sb 1$-kernel to $\{s, s + 1,\ldots\}$, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations.

These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a $z$-measure, or as normalized Toeplitz determinants with symbols $e\sp {\eta(\zeta+\zeta\sp {-1})}$ and $(1 +\sqrt {\xi}\zeta)\sp z(1 +\sqrt {\xi}/\zeta)\sp {z\sp \prime}$.

The proofs are based on a general formalism involving discrete integrable operators and discrete Riemann-Hilbert problems. A continuous version of the formalism has been worked out in [BD].

Article information

Source
Duke Math. J., Volume 117, Number 3 (2003), 489-542.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598403

Digital Object Identifier
doi:10.1215/S0012-7094-03-11734-2

Mathematical Reviews number (MathSciNet)
MR1979052

Zentralblatt MATH identifier
1034.39013

Subjects
Primary: 39Axx: Difference equations {For dynamical systems, see 37-XX; for dynamic equations on time scales, see 34N05}
Secondary: 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20] 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

Citation

Borodin, Alexei. Discrete gap probabilities and discrete Painlevé equations. Duke Math. J. 117 (2003), no. 3, 489--542. doi:10.1215/S0012-7094-03-11734-2. https://projecteuclid.org/euclid.dmj/1085598403


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References

  • M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54 (2001), 153--205. \CMP1 794 352
  • --------, Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice.
  • D. Aldous and P. Diaconis, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 413--432.
  • J. Baik, Riemann-Hilbert problems for last passage percolation, preprint.
  • J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119--1178.
  • --. --. --. --., On the distribution of the length of the second row of a Young diagram under Plancherel measure, Geom. Funct. Anal. 10 (2000), 702--731.
  • J. Baik, P. Deift and E. Rains, A Fredholm determinant identity and the convergence of moments for random Young tableaux, Comm. Math. Phys. 223 (2001), 627--672. \CMP1 866 169
  • E. Basor and H. Widom, On a Toeplitz determinant identity of Borodin and Okounkov, Integral Equations Operator Theory 37 (2000), 397--401.
  • A. M. Borodin, Riemann-Hilbert problem and the discrete Bessel kernel, Internat. Math. Res. Notices 2000, 467--494.
  • --. --. --. --., Harmonic analysis on the infinite symmetric group and the Whittaker kernel, St. Petersburg Math. J. 12 (2001), 733--759. MR 2002a:05253
  • A. Borodin and D. Boyarchenko, Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys. 234 (2003), 287--338.
  • A. Borodin and P. Deift, Fredholm determinants, Jimbo-Miwa-Ueno $\tau$-functions, and representation theory, Comm. Pure Appl. Math. 55 (2002), 1160--1230. \CMP1 908 746
  • A. Borodin and A. Okounkov, A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory 37 (2000), 386--396.
  • A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481--515.
  • A. Borodin and G. Olshanski, Point processes and the infinite symmetric group, Math. Res. Lett. 5 (1998), 799--816.
  • --. --. --. --., Distributions on partitions, point processes, and the hypergeometric kernel, Comm. Math. Phys. 211 (2000), 335--358.
  • --. --. --. --. ``Z-measures on partitions, Robinson-Schensted-Knuth correspondence, and $\beta=2$ random matrix ensembles'' in Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ. 40, Cambridge Univ. Press, Cambridge, 2001, 71--94.
  • --------, Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, to appear in Ann. of. Math. (2).
  • A. Böttcher, Featured review of 2001g:47042a, b, Math. Reviews, 2001.
  • E. Brézin and V. A. Kazakov, Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990), 144--150.
  • P. Deift, ``Integrable operators'' in Differential Operators and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2 189, Adv. Math. Sci. 41, Amer. Math. Soc., Providence, 1999, 69--84.
  • --. --. --. --., Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631--640.
  • 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 (1997), 149--235.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, McGraw-Hill, New York, 1953.
  • A. S. Fokas, A. R. Its, and A. V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313--344.
  • P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: P$_V$, P$_III$, the LUE, JUE and CUE, Comm. Pure Appl. Math. 55 (2002), 679--727. \CMP1 885 665
  • I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), 257--285.
  • B. Grammaticos, F. W. Nijhoff, and A. Ramani, ``Discrete Painlevé equations'' in The Painlevé Property, CRM Ser. Math. Phys., Springer, New York, 1999, 413--516.
  • 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 (2001), 1085--1132.
  • J. Harnad and A. R. Its, Integrable Fredholm operators and dual isomonodromic deformations, Comm. Math. Phys. 226 (2002), 497--530.
  • M. Hisakado, Unitary matrix models and Painlevé, III, Modern Phys. Lett. A 11 (1996), 3001--3010.
  • M. E. H. Ismail and N. S. Witte, Discriminants and functional equations for polynomials orthogonal on the unit circle, J. Approx. Theory 110 (2001), 200--228.
  • A. R. Its, A. G. Izergin, V. E. Korepin, and N. A. Slavnov, ``Differential equations for quantum correlation functions'' in Proceedings of the Conference on Yang-Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory (Canberra, Australia, 1989), Internat. J. Modern Phys. B 4, World Sci., Singapore, 1990, 1003--1037.
  • M. Jimbo and T. Miwa, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients, II, Phys. D 2 (1981), 407--448.
  • M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), 80--158.
  • M. Jimbo and H. Sakai, A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145--154.
  • K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437--476.
  • --. --. --. --., Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259--296.
  • A. A. Kapaev and E. Hubert, A note on the Lax pairs for Painlevé equations, J. Phys. A 32 (1999), 8145--8156.
  • S. Kerov, G. Olshanski, and A. Vershik, Harmonic analysis on the infinite symmetric group: A deformation of the regular representation, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 773--778.
  • M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281--295.
  • Y. Ohta, A. Ramani, B. Grammaticos, and K. M. Tamizhmani, From discrete to continuous Painlevé equations: A bilinear approach, Phys. Lett. A 216 (1996), 255--261.
  • A. Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 2000, 1043--1095.
  • V. Periwal and D. Shevitz, Unitary-matrix models as eactly solvable string theories, Phys. Rev. Lett. 64 (1990), 1326--1329.
  • J. Palmer, Deformation analysis of matrix models, Phys. D 78 (1994), 166--185.
  • H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165--229. \CMP1 882 403
  • A. Soshnikov, Determinantal random point fields, Russian Math. Surveys 55 (2000), 923--975.
  • C. A. Tracy, Whittaker kernel and the fifth Painlevé transcendent, letter to A. Borodin and G. Olshanski, 1998.
  • C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), 33--72.
  • --. --. --. --., Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151--174.
  • --. --. --. --., Random unitary matrices, permutations and Painlevé, Comm. Math. Phys. 207 (1999), 665--685.
  • H. Widom, On convergence of moments for random Young tableaux and a random growth model, Internat. Math. Res. Notices 2002, 455--464.