Duke Mathematical Journal

The asymptotic behavior of the discrete holomorphic map Za via the Riemann–Hilbert method

Alexander I. Bobenko and Alexander Its

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We study the asymptotic behavior of the discrete analogue of the holomorphic map za. The analysis is based on the use of the Riemann–Hilbert approach. Specifically, using the Deift–Zhou nonlinear steepest descent method we prove the asymptotic formulas which were conjectured in 2000.

Article information

Duke Math. J., Volume 165, Number 14 (2016), 2607-2682.

Received: 10 September 2014
Revised: 30 October 2015
First available in Project Euclid: 13 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 39A12: Discrete version of topics in analysis 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20]

discrete differential geometry discrete integrable systems Riemann–Hilbert problem discrete complex analysis


Bobenko, Alexander I.; Its, Alexander. The asymptotic behavior of the discrete holomorphic map $Z^{a}$ via the Riemann–Hilbert method. Duke Math. J. 165 (2016), no. 14, 2607--2682. doi:10.1215/00127094-3620012. https://projecteuclid.org/euclid.dmj/1468414744

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  • [1] S. I. Agafonov, Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations, Discrete Comput. Geom. 29 (2003), 305–319.
  • [2] S. I. Agafonov, Asymptotic behavior of discrete holomorphic maps $z^{c}$ and $\log(z)$, J. Nonlinear Math. Phys. 12 (2005), 1–14.
  • [3] S. I. Agafonov and A. I. Bobenko, Discrete $Z^{{\gamma}}$ and Painlevé equations, Internat. Math. Res. Not. IMRN (2000), no. 4, 165–193.
  • [4] H. Ando, M. Hay, K. Kajiwara, and T. Masuda, An explicit formula for the discrete power function associated with circle patterns of Schramm type, Funkcial. Ekvac. 57 (2014), 1–41.
  • [5] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vols. I, II, McGraw-Hill, New York, 1953.
  • [6] A. I. Bobenko, “Discrete conformal maps and surfaces” in Symmetries and Integrability of Differential Equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser. 255, Cambridge Univ. Press, Cambridge, 1999, 97–108.
  • [7] A. I. Bobenko, C. Mercat, and Y. B. Suris, Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function, J. Reine Angew. Math. 583 (2005), 117–161.
  • [8] A. I. Bobenko and U. Pinkall, Discrete isothermic surfaces, J. Reine Angew. Math. 475 (1996), 187–208.
  • [9] A. I. Bobenko and Y. B. Suris, Discrete Differential Geometry: Integrable Structure, Grad. Stud. Math. 98, Amer. Math. Soc., Providence, 2008.
  • [10] U. Bücking, Rigidity of quasicrystallic and $Z^{{\gamma}}$-circle patterns, Discrete Comput. Geom. 46 (2011), 223–251.
  • [11] P. A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Lect. Notes Math. 3, Amer. Math. Soc., Providence, 1999.
  • [12] P. A. Deift and A. R. Its, eds., Painlevé Equations, Part I, Constr. Approx. 39, Springer, New York, 2014.
  • [13] 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.
  • [14] P. A. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425.
  • [15] P. A. Deift and X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), 295–368.
  • [16] A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Novokshenov, Painlevé Transcendents: The Riemann–Hilbert Approach, Math. Surveys Monogr. 128, Amer. Math. Soc., Providence, 2006.
  • [17] A. S. Fokas, A. R. Its, and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395–430.
  • [18] Z.-X. He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), 1–33.
  • [19] A. R. Its, “Large $N$ asymptotics in random matrices: The Riemann-Hilbert approach” in Random Matrices, Random Processes and Integrable Systems, CRM Ser. Math. Phys., Springer, New York, 2011, 351–413.
  • [20] M. Jimbo, T. Miwa, and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I, Phys. D 2 (1981), 306–352.
  • [21] R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2002), 409–439.
  • [22] L. Kharevych, B. Springborn, and P. Schröder, Discrete conformal maps via circle patterns, ACM Trans. Graphics 25 (2006), 412–438.
  • [23] F. Nijhoff, “On some ‘Schwarzian’ equations and their discrete analogues” in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl. 26, Birkhäuser, Boston, 1997, 237–260.
  • [24] F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math. 39 (1995), 133–158.
  • [25] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010.
  • [26] A. Schiftner, M. Höbinger, J. Wallner, and H. Pottmann, Packing circles and spheres on surfaces, ACM Trans. Graphics 28 (2009), no. 139.
  • [27] O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J. 86 (1997), 347–389.
  • [28] K. Stephenson, Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge Univ. Press, Cambridge, 2005.
  • [29] W. P. Thurston, The finite Riemann mapping theorem, invited talk at the “Symposium on the Occasion of the Proof of the Bieberbach Conjecture,” Purdue Univ., West Lafayette, Indiana, 1985.
  • [30] M. Vanlessen, Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, J. Approx. Theory 125 (2003), 198–237.