Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T19:32:03.109Z Has data issue: false hasContentIssue false
  • English
  • Français

τ-function of discrete isomonodromy transformations and probability

Part of: Difference equations Difference and functional equations Curves

Published online by Cambridge University Press:  01 May 2009

D. Arinkin  and
A. Borodin
D. Arinkin
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA (email: [email protected])
A. Borodin
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA, USA (email: [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.

We introduce the τ-function of a difference rational connection (d-connection) and its isomonodromy transformations. We show that in a continuous limit ourτ-function agrees with the Jimbo–Miwa–Ueno τ-function. We compute the τ-function for the isomonodromy transformations leading to difference Painlevé V and difference Painlevé VI equations. We prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the τ-function for an associated d-connection.

Keywords

discrete Painlevé equationsτ-functions

MSC classification

Secondary: 39A10: Difference equations, additive 14H60: Vector bundles on curves and their moduli
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 747 - 772
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Adler, M. and van Moerbeke, P., Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. of Math. (2) 153 (2001), 149–189.CrossRefGoogle Scholar
[2]Adler, M. and van Moerbeke, P., Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice, Comm. Math. Phys. 237 (2003), 397–440.CrossRefGoogle Scholar
[3]Arinkin, D. and Borodin, A., Moduli spaces of d-connections and difference Painlevé equations, Duke Math. J. 134 (2006), 515–556.CrossRefGoogle Scholar
[4]Baik, J., Riemann–Hilbert problems for last passage percolation, in Recent developments in integrable systems and Riemann–Hilbert problems, Birmingham, AL, 2000, Contemporary Mathematics, vol. 326 (American Mathematical Society, Providence, RI, 2003), 1–21.Google Scholar
[5]Borodin, A. and Boyarchenko, D., Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys. 234 (2003), 287–338.CrossRefGoogle Scholar
[6]Borodin, A. and Deift, P., Fredholm determinants, Jimbo–Miwa–Ueno τ-functions, and representation theory, Comm. Pure Appl. Math. 55 (2002), 1160–1230.CrossRefGoogle Scholar
[7]Borodin, A. and Olshanski, G., Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. (2) 161 (2005), 1319–1422.CrossRefGoogle Scholar
[8]Borodin, A., Discrete gap probabilities and discrete Painlevé equations, Duke Math. J. 117 (2003), 489–542.CrossRefGoogle Scholar
[9]Borodin, A., Isomonodromy transformations of linear systems of difference equations, Ann. of Math. (2) 160 (2004), 1141–1182.CrossRefGoogle Scholar
[10]Daems, E. and Kuijlaars, A. B. J., Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, J. Approx. Theory 146 (2007), 91–114.CrossRefGoogle Scholar
[11]Dubrovin, B., Geometry of 2D topological field theories, in Integrable systems and quantum groups, Montecatini Terme, 1993, Lecture Notes in Mathematics, vol. 1620 (Springer, Berlin, 1996), 120–348.CrossRefGoogle Scholar
[12]Forrester, P. J. and Witte, N. S., Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys. 219 (2001), 357–398.CrossRefGoogle Scholar
[13]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, Comm. Pure Appl. Math. 55 (2002), 679–727.CrossRefGoogle Scholar
[14]Forrester, P. J. and Witte, N. S., Discrete Painlevé equations and random matrix averages, Nonlinearity 16 (2003), 1919–1944.CrossRefGoogle Scholar
[15]Forrester, P. J. and Witte, N. S., Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J. 174 (2004), 29–114.CrossRefGoogle Scholar
[16]Forrester, P. J. and Witte, N. S., Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N-recurrences for averages over U(N)— PIII′ and PV τ-functions, Int. Math. Res. Not. (2004), 160–183.Google Scholar
[17]Grammaticos, B., Ramani, A. and Ohta, Y., A unified description of the asymmetric q-P V and d-P IV equations and their Schlesinger transformations, J. Nonlinear Math. Phys. 10 (2003), 215–228.CrossRefGoogle Scholar
[18]Harnad, J. and Its, A. R., Integrable Fredholm operators and dual isomonodromic deformations, Comm. Math. Phys. 226 (2002), 497–530.CrossRefGoogle Scholar
[19]Jimbo, M. and Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Physica D 2 (1981), 407–448.CrossRefGoogle Scholar
[20]Jimbo, M. and Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Physica D 4 (1981/82), 26–46.CrossRefGoogle Scholar
[21]Jimbo, M., Miwa, T., Môri, Y. and Sato, M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D 1 (1980), 80–158.CrossRefGoogle Scholar
[22]Jimbo, M., Miwa, T. and Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Physica D 2 (1981), 306–352.CrossRefGoogle Scholar
[23]Johansson, K., Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002), 225–280.CrossRefGoogle Scholar
[24]Jimbo, M. and Sakai, H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145–154.CrossRefGoogle Scholar
[25]Knudsen, F. F. and Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’, Math. Scand. 39 (1976), 19–55.CrossRefGoogle Scholar
[26]Krichever, I. M., Analytic theory of difference equations with rational and elliptic coefficients and the Riemann–Hilbert problem, Uspekhi Mat. Nauk 59 (2004), 111–150.Google Scholar
[27]Laszlo, Y. and Sorger, C., The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), 499–525.CrossRefGoogle Scholar
[28]Mehta, M. L., A nonlinear differential equation and a Fredholm determinant, J. Physique I 2 (1992), 1721–1729.CrossRefGoogle Scholar
[29]Palmer, J., Deformation analysis of matrix models, Physica D 78 (1994), 166–185.CrossRefGoogle Scholar
[30]Sakai, H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165–229.CrossRefGoogle Scholar
[31]Sakai, H., Lax form of the q-Painlevé equation associated with the A 2(1) surface, J. Phys. A: Math. Gen. 39 (2006), 12203–12210.Google Scholar
[32]Tracy, C. A. and Widom, H., Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), 33–72.CrossRefGoogle Scholar