Duits, Maurice; Geudens, Dries A critical phenomenon in the two-matrix model in the quartic/quadratic case. (English) Zbl 1286.60006 Duke Math. J. 162, No. 8, 1383-1462 (2013). The authors consider a model of two random matrices \(M_{1}\) and \(M_{2}\) related to each other, then they study the eigenvalues of \(M_{1}\) after taking the average over \(M_{2}\). More precisely, they consider the so-called kernel that characterizes the eigenvalues of \(M_{1}\), when the dimension goes to infinity. Such kernel has two parameters \(\alpha\) and \(\tau\), and it is possible to say what are the values of those parameters where there is a critical point, namely \(\alpha=-1\) and \(\tau=1\). Then, the authors perform a scaling of the kernel around that critical point, and the asymptotic analysis of the scaling gives rise to a kernel that is written in terms of the solution of a Riemann-Hilbert problem (RH). The RH problem is to find a \(4\times 4\) matrix-valued function with certain properties, and an important ingredient of the problem is a Painlevé II equation, as well as its so-called Hastings-McLeod solution.The two-matrix model is given (after normalizing) by the following probability measure over the set of Hermitian matrices \[ \exp\{ -nTr(V(M_{1})+W(M_{2})-\tau M_{1}M_{2}) \}, \] where \(V(x):=x^{2}/2\) and \(W(x):=y^{4}/2+\alpha x^{2}/2\). Important tools used by the authors are the biorthogonal polynomials, properties of some meromorphic functions, and the steepest descent analysis of the RH problem. Reviewer: Carlos Gabriel Pacheco (México D. F.) Cited in 9 Documents MSC: 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 33E17 Painlevé-type functions Keywords:two-matrix model; determinantal point process kernel; Riemann-Hilbert problem × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] M. Adler, P. Ferrari, and P. van Moerbeke, Non-intersecting random walks in the neighborhood of a symmetric tacnode , to appear in Ann. Probab., preprint, [math-ph]. 1007.1163v2 · Zbl 1200.60069 · doi:10.1214/09-AOP493 [2] M. Bertola and B. Eynard, The PDEs of biorthogonal polynomials arising in the two-matrix model , Math. Phys. Anal. Geom. 9 (2006), 23-52. · Zbl 1107.33011 · doi:10.1007/s11040-005-9000-x [3] M. Bertola, B. Eynard, and J. Harnad, Duality, biorthogonal polynomials and multi-matrix models , Comm. Math. Phys. 229 (2002), 73-120. · Zbl 1033.15015 · doi:10.1007/s002200200663 [4] M. Bertola, B. Eynard and J. Harnad, Differential systems for biorthogonal polynomials appearing in \(2\)-matrix models and the associated Riemann-Hilbert problem , Comm. Math. Phys. 243 (2003), 193-240. · Zbl 1046.34098 · doi:10.1007/s00220-003-0934-1 [5] P. M. Bleher and A. Its, Double scaling limit in the random matrix model: The Riemann-Hilbert approach , Comm. Pure Appl. Math. 56 (2003), 433-516. · Zbl 1032.82014 · doi:10.1002/cpa.10065 [6] P. M. Bleher and A. B. J. Kuijlaars, Large \(n\) limit of Gaussian random matrices with external source, III: Double scaling limit , Comm. Math. Phys. 270 (2007), 481-517. · Zbl 1126.82010 · doi:10.1007/s00220-006-0159-1 [7] A. Borodin, Biorthogonal ensembles , Nuclear Phys. B 536 (1999), 704-732. · Zbl 0948.82018 · doi:10.1016/S0550-3213(98)00642-7 [8] A. Borodin, “Determinantal point processes” in Oxford Handbook of Random Matrix Theory , Oxford Univ. Press, Oxford, 2011, 231-249. · Zbl 1238.60055 [9] E. Brézin and S. Hikami, Level spacing of random matrices in an external source , Phys. Rev. E (3) 58 (1998), 7176-7185. · doi:10.1103/PhysRevE.58.7176 [10] E. Brézin and S. Hikami, Universal singularity at the closure of a gap in a random matrix theory , Phys. Rev. E (3) 57 (1998), 4140-4149. · doi:10.1103/PhysRevE.57.4140 [11] T. Claeys and A. B. J. Kuijlaars, Universality of the double scaling limit in random matrix models , Comm. Pure Appl. Math. 59 (2006), 1573-1603. · Zbl 1111.35031 · doi:10.1002/cpa.20113 [12] T. Claeys, A. B. J. Kuijlaars, and M. Vanlessen, Multi-critical unitary random matrix ensembles and the general Painlevé II equation , Ann. of Math. (2) 168 (2008), 601-642. · Zbl 1179.15037 · doi:10.4007/annals.2008.168.601 [13] J.-M. Daul, V. A. Kazakov, and I. K. Kostov, Rational theories of 2D gravity from the two-matrix model , Nuclear Phys. B 409 (1993), 311-338. · Zbl 1043.81684 · doi:10.1016/0550-3213(93)90582-A [14] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , Courant Lect. Notes Math. 3 , Amer. Math. Soc., Providence, 1999. · Zbl 0997.47033 [15] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to varying exponential weights , Comm. Pure Appl. Math. 52 (1999), 1491-1552. · Zbl 1026.42024 · doi:10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-# [16] P. 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. · Zbl 0944.42013 · doi:10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1 [17] P. 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. · Zbl 0771.35042 · doi:10.2307/2946540 [18] S. Delvaux and A. B. J. Kuijlaars, A phase transition for nonintersecting Brownian motions, and the Painlevé II equation , Int. Math. Res. Not. IMRN 2009 , no. 19, 3639-3725. · Zbl 1182.60023 · doi:10.1093/imrn/rnp069 [19] S. Delvaux, A. B. J. Kuijlaars, and L. Zhang, Critical behavior of nonintersecting Brownian motions at a tacnode , Comm. Pure and Appl. Math 64 (2011), 1305-1383. · Zbl 1231.60085 · doi:10.1002/cpa.20373 [20] K. Deschout and A. B. J. Kuijlaars, “Double scaling limit for modified Jacobi-Angelesco polynomials” in Notions of Positivity and the Geometry of Polynomials , Trends Math., Springer, Basel, 2011, 115-161. · Zbl 1254.30046 [21] M. R. Douglas, “The two-matrix model” in Random Surfaces and Quantum Gravity (Cargèse, 1990) , NATO Adv. Sci. Inst. Ser. B Phys. 262 , Plenum, New York, 1991, 77-83. · Zbl 0772.58074 · doi:10.1007/978-1-4615-3772-4_6 [22] M. Duits, D. Geudens, and A. B. J. Kuijlaars, A vector equilibrium problem for the two-matrix model in the quartic/quadratic case , Nonlinearity 24 (2011), 951-993. · Zbl 1211.31006 · doi:10.1088/0951-7715/24/3/012 [23] M. Duits and A. B. J. Kuijlaars, Universality in the two-matrix model: A Riemann-Hilbert steepest-descent analysis , Comm. Pure Appl. Math. 62 (2009), 1076-1153. · Zbl 1221.15052 · doi:10.1002/cpa.20269 [24] M. Duits, A. B. J. Kuijlaars, and M. Y. Mo, The Hermitian two-matrix model with an even quartic potential , Mem. Amer. Math. Soc. 217 (2012). · Zbl 1247.15032 · doi:10.1090/S0065-9266-2011-00639-8 [25] N. M. Ercolani and K. T.-R. McLaughlin, Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model , Phys. D 152/153 (2001), 232-268. · Zbl 1038.82042 · doi:10.1016/S0167-2789(01)00173-7 [26] B. Eynard, Large-\(N\) expansion of the \(2\)-matrix model , J. High Energy Phys. 2003 , no. 1, 051. [27] B. Eynard and M. L. Mehta, Matrices coupled in a chain, I: Eigenvalue correlations , J. Phys. A 31 (1998), 4449-4456. · Zbl 0938.15012 · doi:10.1088/0305-4470/31/19/010 [28] H. Flaschka and A. C. Newell, Monodromy- and spectrum-preserving deformations, I , Comm. Math. Phys. 76 (1980), 65-116. · Zbl 0439.34005 · doi:10.1007/BF01197110 [29] A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach , Math. Surveys Monogr. 128 , Amer. Math. Soc., Providence, 2006. · Zbl 1111.34001 [30] 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. · Zbl 0760.35051 · doi:10.1007/BF02096594 [31] A. Hardy and A. B. J. Kuijlaars, Weakly admissible vector equilibrium problems , J. Approx. Theory 164 (2012), 854-868. · Zbl 1241.49008 · doi:10.1016/j.jat.2012.03.009 [32] S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation , Arch. Ration. Mech. Anal. 73 (1980), 31-51. · Zbl 0426.34019 · doi:10.1007/BF00283254 [33] J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Determinantal processes and independence , Probab. Surv. 3 (2006), 206-229. · Zbl 1189.60101 · doi:10.1214/154957806000000078 [34] K. Johansson, “Random matrices and determinantal processes” in Mathematical Statistical Physics , Elsevier, Amsterdam, 2006, 1-55. · Zbl 1411.60144 [35] K. Johansson, Non-colliding Brownian motions and the extended tacnode process , Comm. Math. Phys. 319 (2013), 231-267. · Zbl 1268.60104 · doi:10.1007/s00220-012-1600-2 [36] A. A. Kapaev, Riemann-Hilbert problem for bi-orthogonal polynomials , J. Phys. A 36 (2003), 4629-4640. · Zbl 1078.34072 · doi:10.1088/0305-4470/36/16/312 [37] W. König, Orthogonal polynomial ensembles in probability theory , Probab. Surv. 2 (2005), 385-447. · Zbl 1189.60024 · doi:10.1214/154957805100000177 [38] A. B. J. Kuijlaars, A. Martínez-Finkelshtein, and F. Wielonsky, Non-intersecting squared Bessel paths: Critical time and double scaling limit , Comm. Math. Phys. 308 (2011), 227-279. · Zbl 1245.60044 · doi:10.1007/s00220-011-1322-x [39] A. B. J. Kuijlaars and K. T.-R. McLaughlin, A Riemann-Hilbert problem for biorthogonal polynomials , J. Comput. Appl. Math. 178 (2005), 313-320. · Zbl 1096.42009 · doi:10.1016/j.cam.2004.01.043 [40] R. Lyons, Determinantal probability measures , Publ. Math. Inst. Hautes Études Sci. 98 (2003), 167-212. · Zbl 1055.60003 · doi:10.1007/s10240-003-0016-0 [41] M. L. Mehta and P. Shukla, Two coupled matrices: Eigenvalue correlations and spacing functions , J. Phys. A 27 (1994), 7793-7803. · Zbl 0842.15006 · doi:10.1088/0305-4470/27/23/022 [42] M. Y. Mo, Universality in the two matrix model with a monomial quartic and a general even polynomial potential , Comm. Math. Phys. 291 (2009), 863-894. · Zbl 1179.81136 · doi:10.1007/s00220-009-0893-2 [43] A. Okounkov and N. Reshetikhin, Random skew plane partitions and the Pearcey process , Comm. Math. Phys. 269 (2007), 571-609. · Zbl 1115.60011 · doi:10.1007/s00220-006-0128-8 [44] E. B. Saff and V. Totik, Logarithmic Potentials with External Field , Grundlehren Math. Wiss. 316 , Springer, Berlin, 1997. · Zbl 0881.31001 [45] A. Soshnikov, Determinantal random point fields (in Russian), Uspekhi Mat. Nauk 55 (2000), no. 5 (335), 107-160; English translation in Russian Math. Surveys 55 (2000), 923-975. · Zbl 0991.60038 · doi:10.4213/rm321 [46] C. A. Tracy and H. Widom, The Pearcey process , Comm. Math. Phys 263 (2006), 381-400. · Zbl 1129.82031 · doi:10.1007/s00220-005-1506-3 [47] W. Wasow, “Asymptotic expansions for ordinary differential equations” in Pure and Applied Mathematics , Vol. XIV, John Wiley, New York, 1965. · Zbl 0133.35301 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.