Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 016, 23 pages      arXiv:1703.04931
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II

Percy Deift
Department of Mathematics, New York University, 251 Mercer Str., New York, NY 10012, USA

Received October 11, 2016, in final form March 10, 2017; Published online March 14, 2017

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathématiques, Montréal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60$^{\rm th}$ birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849]).

Key words: integrable systems; numerical algorithms; random matrices; random particle systems; Riemann-Hilbert problems.

pdf (480 kb)   tex (47 kb)


  1. Ajanki O., Erdős L., Krüger T., Stability of the matrix Dyson equation and random matrices with correlations, arXiv:1604.08188.
  2. Amir G., Corwin I., Quastel J., Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions, Comm. Pure Appl. Math. 64 (2011), 466-537, arXiv:1003.0443.
  3. Baik J., Deift P., Johansson K., On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119-1178, math.CO/9810105.
  4. Bakhtin Yu., Correll J., A neural computation model for decision-making times, J. Math. Psych. 56 (2012), 333-340.
  5. Bertini L., Cancrini N., The stochastic heat equation: Feynman-Kac formula and intermittence, J. Stat. Phys. 78 (1995), 1377-1401.
  6. Bikbaev R.F., Tarasov V.O., Initial-boundary value problem for the nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 24 (1991), 2507-2516.
  7. Binder I., Damanik D., Goldstein M., Lukic M., Almost periodicity in time of solutions of the KdV equation, arXiv:1509.07373.
  8. Binder I., Damanik D., Lukic M., VandenBoom T., Almost periodicity in time of solutions of the Toda lattice, arXiv:1603.04905.
  9. Bleher P., Its A., Double scaling limit in the random matrix model: the Riemann-Hilbert approach, Comm. Pure Appl. Math. 56 (2003), 433-516, math-ph/0201003.
  10. Bohigas O., Pato M.P., Randomly incomplete spectra and intermediate statistics, Phys. Rev. E 74 (2006), 036212, 6 pages, cond-mat/0608165.
  11. Bornemann F., On the numerical evaluation of Fredholm determinants, Math. Comp. 79 (2010), 871-915, arXiv:0804.2543.
  12. Borodin A., Corwin I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225-400, arXiv:1111.4408.
  13. Borodin A., Okounkov A., Olshanski G., Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481-515, math.CO/9905032.
  14. Borodin A., Olshanski G., The ASEP and determinantal point processes, arXiv:1608.01564.
  15. Borodin A., Petrov L., Integrable probability: from representation theory to Macdonald processes, Probab. Surv. 11 (2014), 1-58, arXiv:1310.8007.
  16. Borot G., Eynard B., Majumdar S.N., Nadal C., Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. Theory Exp. 2011 (2011), P11024, 56 pages, arXiv:1009.1945.
  17. Bothner T., From gap probabilities in random matrix theory to eigenvalue expansions, J. Phys. A: Math. Theor. 49 (2016), 075204, 77 pages, arXiv:1509.07159.
  18. Bothner T., Deift P., Its A., Krasovsky I., On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I, Comm. Math. Phys. 337 (2015), 1397-1463, arXiv:1407.2910.
  19. Bourgade P., Erdös L., Yau H.-T., Edge universality of beta ensembles, Comm. Math. Phys. 332 (2014), 261-353, arXiv:1306.5728.
  20. Bourgade P., Erdös L., Yau H.-T., Yin J., Universality for a class of random band matrices, arXiv:1602.02312.
  21. Bourgade P., Yau H.-T., The eigenvector moment flow and local quantum unique ergodicity, Comm. Math. Phys. 350 (2017), 231-278, arXiv:1312.1301.
  22. Calabrese P., Le Doussal P., Rosso A., Free-energy distribution of the directed polymer at high temperature, Europhys. Lett. 90 (2010), 20002, 6 pages, arXiv:1002.4560.
  23. Charlier C., Claeys T., Thinning and conditioning of the circular unitary ensemble, arXiv:1604.08399.
  24. Claeys T., Fahs B., Random matrices with merging singularities and the Painlevé V equation, SIGMA 12 (2016), 031, 44 pages, arXiv:1508.06734.
  25. Claeys T., Kuijlaars A.B.J., Universality of the double scaling limit in random matrix models, Comm. Pure Appl. Math. 59 (2006), 1573-1603, math-ph/0501074.
  26. Corwin I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76 pages, arXiv:1106.1596.
  27. Corwin I., Kardar-Parisi-Zhang universality, Notices Amer. Math. Soc. 63 (2016), 230-239, arXiv:1606.06602.
  28. Corwin I., Hammond A., KPZ line ensemble, Probab. Theory Related Fields 166 (2016), 67-185, arXiv:1312.2600.
  29. Corwin I., Liu Z., Wang D., Fluctuations of TASEP and LPP with general initial data, Ann. Appl. Probab. 26 (2016), 2030-2082, arXiv:1412.5087.
  30. Corwin I., Quastel J., Remenik D., Renormalization fixed point of the KPZ universality class, J. Stat. Phys. 160 (2015), 815-834, arXiv:1103.3422.
  31. Damanik D., Goldstein M., On the inverse spectral problem for the quasi-periodic Schrödinger equation, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 217-401, arXiv:1209.4331.
  32. Damanik D., Goldstein M., On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data, J. Amer. Math. Soc. 29 (2016), 825-856, arXiv:1212.2674.
  33. Davies E.B., Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, Vol. 106, Cambridge University Press, Cambridge, 2007.
  34. Deift P., Some open problems in random matrix theory and the theory of integrable systems, in Integrable Systems and Random Matrices, Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849.
  35. Deift P., Its A., Krasovsky I., Zhou X., The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math. 202 (2007), 26-47, math.FA/0601535.
  36. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., 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.
  37. Deift P., Kriecherbauer T., Venakides S., Forced lattice vibrations. Part I, Comm. Pure Appl. Math. 48 (1995), 1187-1249.
  38. Deift P., Li L.C., Nanda T., Tomei C., The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39 (1986), 183-232.
  39. Deift P., Li L.C., Tomei C., Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 42 (1989), 443-521.
  40. Deift P., Menon G., Olver S., Trogdon T., Universality in numerical computations with random data, Proc. Natl. Acad. Sci. USA 111 (2014), 14973-14978, arXiv:1407.3829.
  41. Deift P., Park J., Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res. Not. 2011 (2011), 5505-5624, arXiv:1006.4909.
  42. Deift P., Trogdon T., Universality for the Toda algorithm to compute the eigenvalues of a random matrix, arXiv:1604.07384.
  43. Deift P., Trogdon T., Universality for eigenvalue algorithms on sample covariance matrices, arXiv:1701.01896.
  44. Deift P., Trogdon T., Menon G., On the condition number of the critically-scaled Laguerre unitary ensemble, Discrete Contin. Dyn. Syst. 36 (2016), 4287-4347, arXiv:1507.00750.
  45. Deift P., Venakides S., Zhou X., New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Not. 1997 (1997), 286-299.
  46. Deift P., Zhou X., Perturbation theory for infinite-dimensional integrable systems on the line. A case study, Acta Math. 188 (2002), 163-262,.
  47. Deift P., Zhou X., Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math. 56 (2003), 1029-1077, math.AP/0206222.
  48. Dotsenko V., Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers, Europhys. Lett. 90 (2010), 20003, 5 pages.
  49. Dyson F.J., A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962), 1191-1198.
  50. Dyson F.J., Statistical theory of the energy levels of complex systems. II, J. Math. Phys. 3 (1962), 157-165.
  51. Dyson F.J., Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), 171-183.
  52. Erdős L., Universality for random matrices and log-gases, in Current Developments in Mathematics 2012, Int. Press, Somerville, MA, 2013, 59-132, arXiv:1212.0839.
  53. Erdős L., Knowles A., Yau H.-T., Yin J., Delocalization and diffusion profile for random band matrices, Comm. Math. Phys. 323 (2013), 367-416, arXiv:1205.5669.
  54. Erdős L., Péché S., Ramírez J.A., Schlein B., Yau H.-T., Bulk universality for Wigner matrices, Comm. Pure Appl. Math. 63 (2010), 895-925, arXiv:0905.4176.
  55. Erdős L., Yau H.-T., Yin J., Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math. 229 (2012), 1435-1515, arXiv:1007.4652.
  56. Fokas A.S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), 1411-1443.
  57. Fokas A.S., Lenells J., The unified method: I. Nonlinearizable problems on the half-line, J. Phys. A: Math. Theor. 45 (2012), 195201, 38 pages, arXiv:1109.4935.
  58. Fröhlich J., Spencer T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151-184.
  59. Fyodorov Y.V., Mirlin A.D., Scaling properties of localization in random band matrices: a $\sigma$-model approach, Phys. Rev. Lett. 67 (1991), 2405-2409.
  60. Grava T., Its A., Kapaev A., Mezzadri F., On the Tracy-Widom$_\beta$ distribution for $\beta=6$, SIGMA 12 (2016), 105, 26 pages, arXiv:1607.01351.
  61. Hairer M., Solving the KPZ equation, Ann. of Math. 178 (2013), 559-664, arXiv:1109.6811.
  62. Its A., Shepelsky D., Initial boundary value problem for the focusing nonlinear Schrödinger equation with Robin boundary condition: half-line approach, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), 20120199, 14 pages, arXiv:1201.5948.
  63. Johansson K., Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476, math.CO/9903134.
  64. Johansson K., Transversal fluctuations for increasing subsequences on the plane, Probab. Theory Related Fields 116 (2000), 445-456, math.PR/9910146.
  65. Johansson K., Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Comm. Math. Phys. 215 (2001), 683-705, math-ph/0006020.
  66. Johansson K., Discrete polynuclear growth and determinantal processes, Comm. Math. Phys. 242 (2003), 277-329, math.PR/0206208.
  67. Johansson K., Two time distribution in Brownian directed percolation, Comm. Math. Phys. 351 (2017), 441-492, arXiv:1502.00941.
  68. Kamvissis S., McLaughlin K.D.T.-R., Miller P.D., Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation, Annals of Mathematics Studies, Vol. 154, Princeton University Press, Princeton, NJ, 2003.
  69. Kardar M., Parisi G., Zhang Y.-C., Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
  70. Kriecherbauer T., Shcherbina M., Fluctuations of eigenvalues of matrix models and their applications, arXiv:1003.6121.
  71. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), 253-290.
  72. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math. 36 (1983), 571-593.
  73. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math. 36 (1983), 809-829.
  74. Li X.H., Menon G., Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles, J. Stat. Phys. 153 (2013), 801-812, arXiv:1306.1179.
  75. Manakov S.V., Nonlinear Fraunhofer diffraction, Sov. Phys. JETP 38 (1974), 693-696.
  76. McKean H.P., Trubowitz E., Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143-226.
  77. Mehta M.L., Random matrices, Pure and Applied Mathematics (Amsterdam), Vol. 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004.
  78. Miller P.D., Xu Z., On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data, Comm. Pure Appl. Math. 64 (2011), 205-270, arXiv:1002.3178.
  79. Minami N., Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), 709-725.
  80. Morse P.M., Rubenstein P.J., The diffraction of waves by ribbons and by slits, Phys. Rev. 54 (1938), 895-898.
  81. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  82. Olver S., Numerical solution of Riemann-Hilbert problems: Painlevé II, Found. Comput. Math. 11 (2011), 153-179.
  83. Olver S., Nadakuditi R.R., Trogdon T., Sampling unitary ensembles, Random Matrices Theory Appl. 4 (2015), 1550002, 22 pages, arXiv:1404.0071.
  84. Osipov A., Rokhlin V., Xiao H., Prolate spheroidal wave functions of order zero. Mathematical tools for bandlimited approximation, Applied Mathematical Sciences, Vol. 187, Springer, New York, 2013.
  85. Pfrang C.W., Deift P., Menon G., How long does it take to compute the eigenvalues of a random symmetric matrix?, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ., Vol. 65, Cambridge University Press, New York, 2014, 411-442, arXiv:1203.4635.
  86. Prähofer M., Spohn H., Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys. 108 (2002), 1071-1106, math.PR/0105240.
  87. Ramírez J.A., Rider B., Virág B., Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc. 24 (2011), 919-944, math.PR/0607331.
  88. Rumanov I., Beta ensembles, quantum Painlevé equations and isomonodromy systems, in Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemp. Math., Vol. 651, Amer. Math. Soc., Providence, RI, 2015, 125-155, arXiv:1408.3847.
  89. Rumanov I., Painlevé representation of Tracy-Widom$_\beta$ distribution for $\beta=6$, Comm. Math. Phys. 342 (2016), 843-868, arXiv:1408.3779.
  90. Sasamoto T., Spohn H., One-dimensional KPZ equation: an exact solution and its universality, Phys. Rev. Lett. 104 (2010), 230602, 4 pages, arXiv:1002.1883.
  91. Schenker J., Eigenvector localization for random band matrices with power law band width, Comm. Math. Phys. 290 (2009), 1065-1097, arXiv:0809.4405.
  92. Shcherbina M., Orthogonal and symplectic matrix models: universality and other properties, Comm. Math. Phys. 307 (2011), 761-790, arXiv:1004.2765.
  93. Shcherbina T., On the second mixed moment of the characteristic polynomials of 1D band matrices, Comm. Math. Phys. 328 (2014), 45-82, arXiv:1209.3385.
  94. Shcherbina T., Universality of the second mixed moment of the characteristic polynomials of the 1D band matrices: real symmetric case, J. Math. Phys. 56 (2015), 063303, 23 pages, arXiv:1410.3084.
  95. Sinai Ya.G., Soshnikov A.B., A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Funct. Anal. Appl. 32 (1998), 114-131.
  96. Slepian D., Some asymptotic expansions for prolate spheroidal wave functions, J. Math. and Phys. 44 (1965), 99-140.
  97. Sodin S., The spectral edge of some random band matrices, Ann. of Math. 172 (2010), 2223-2251, arXiv:0906.4047.
  98. Sommerfeld A., Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), 317-374.
  99. Soshnikov A., Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207 (1999), 697-733, math-ph/9907013.
  100. Symes W.W., The $QR$ algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D 4 (1982), 275-280.
  101. Tao T., Vu V., Random matrices: universality of local eigenvalue statistics, Acta Math. 206 (2011), 127-204, arXiv:0906.0510.
  102. Tracy C.A., Widom H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174, hep-th/9211141.
  103. Tracy C.A., Widom H., A Fredholm determinant representation in ASEP, J. Stat. Phys. 132 (2008), 291-300, arXiv:0804.1379.
  104. Tracy C.A., Widom H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815-844, Erratum, Comm. Math. Phys. 304 (2011), 875-878, arXiv:0704.2633.
  105. Tracy C.A., Widom H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129-154, arXiv:0807.1713.
  106. Trefethen L.N., Embree M., Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, NJ, 2005.
  107. Trogdon T., Olver S., Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
  108. Trogdon T., Olver S., Deconinck B., Numerical inverse scattering for the Korteweg-de Vries and modified Korteweg-de Vries equations, Phys. D 241 (2012), 1003-1025.
  109. Venakides S., The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38 (1985), 125-155.
  110. Venakides S., The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990), 335-361.
  111. Venakides S., Deift P., Oba R., The Toda shock problem, Comm. Pure Appl. Math. 44 (1991), 1171-1242.
  112. Witte N.S., Bornemann F., Forrester P.J., Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles, Nonlinearity 26 (2013), 1799-1822, arXiv:1209.2190.
  113. Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316-374.

Previous article  Next article   Contents of Volume 13 (2017)