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Critical measures for vector energy: global structure of trajectories of quadratic differentials. (English) Zbl 1354.31002
Summary: Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector-valued critical measures, a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures $$\overrightarrow{\mu} = (\mu_1, \mu_2, \mu_3)$$ when the mutual interaction comprises both attracting and repelling forces.
For arbitrary vector polynomial external fields we establish general structural results about critical measures, such as their characterization in terms of an algebraic equation solved by an appropriate combination of their Cauchy transforms, and the symmetry properties (or the $$S$$-properties) exhibited by such measures. In consequence, we conclude that vector-valued critical measures are supported on a finite number of analytic arcs, that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential is the key to the construction of the critical measures.
We illustrate these connections studying in depth a one-parameter family of critical measures under the action of a cubic external field. This choice is motivated by the asymptotic analysis of a family of (non-hermitian) multiple orthogonal polynomials, that is subject of a forthcoming paper. Here we compute explicitly the Riemann surface and the corresponding quadratic differential, and analyze the dynamics of its critical graph as a function of the parameter, giving a detailed description of the occurring phase transitions. When projected back to the complex plane, this construction gives us the complete family of vector-valued critical measures, that in this context turn out to be vector-valued equilibrium measures.

##### MSC:
 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 30F10 Compact Riemann surfaces and uniformization 30F30 Differentials on Riemann surfaces
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##### References:
 [1] Álvarez, G.; Martínez-Alonso, L.; Medina, E., Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials, J. Stat. Mech. Theory Exp., 6, (2013), P06006, 28 [2] Álvarez, G.; Martínez-Alonso, L.; Medina, E., Partition functions and the continuum limit in Penner matrix models, J. Phys. A, 47, 31, (2014), 315205, 29 · Zbl 1296.15017 [3] Aptekarev, A. I., Asymptotics of Hermite-Padé approximants for a pair of functions with branch points, Dokl. Akad. Nauk, 422, 4, 443-445, (2008) [4] Aptekarev, A. I.; Buslaev, V. I.; Martines-Finkelshteĭn, A.; Suetin, S. P., Padé approximants, continued fractions, and orthogonal polynomials, Uspekhi Mat. Nauk, Math. Surveys, 66, 6, 1049-1131, (2011), translation in Russian · Zbl 1242.41014 [5] Aptekarev, A. I.; Koĭèlaars, A. B.È., Hermite-Padé approximations and ensembles of multiple orthogonal polynomials, Uspekhi Mat. Nauk, 66, 6(402), 123-190, (2011) [6] Aptekarev, A. I.; Kuijlaars, A. B.J.; Van Assche, W., Asymptotics of Hermite-Padé rational approximants for two analytic functions with separated pairs of branch points (case of genus 0), Int. Math. Res. Pap., 128, (2008), Art. ID rpm007 · Zbl 1156.41004 [7] Aptekarev, A. I.; Toulyakov, D. N.; Van Assche, W., Hyperelliptic uniformization of algebraic curves of the third order, J. Comput. Appl. Math., 284, 38-49, (2015) · Zbl 1312.33022 [8] Aptekarev, A. I.; Van Assche, W.; Yatsselev, M. L., Hermite-Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports, (2015), preprint [9] Atia, M. J.; Martínez-Finkelshtein, A.; Martínez-González, P.; Thabet, F., Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters, J. Math. Anal. Appl., 416, 1, 52-80, (2014) · Zbl 1295.30015 [10] Baik, J.; Deift, P.; McLaughlin, K. T.-R.; Miller, P.; Zhou, X., Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys., 5, 6, 1207-1250, (2001) · Zbl 1016.15022 [11] Balogh, F.; Bertola, M., Regularity of a vector potential problem and its spectral curve, J. Approx. Theory, 161, 1, 353-370, (2009) · Zbl 1190.42009 [12] Balogh, F.; Bertola, M.; Bothner, T., Hankel determinant approach to generalized vorob’ev-yablonski polynomials and their roots, (2015), preprint · Zbl 1376.34073 [13] Beckermann, B.; Kalyagin, V.; Matos, A. C.; Wielonsky, F., Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses, Constr. Approx., 37, 1, 101-134, (2013) · Zbl 1261.31001 [14] Bergkvist, T.; Rullgård, H., On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett., 9, 2-3, 153-171, (2002) · Zbl 1016.34083 [15] Bertola, M., Boutroux curves with external field: equilibrium measures withouta variational problem, Anal. Math. Phys., 1, 2-3, 167-211, (2011) · Zbl 1259.33021 [16] Bertola, M.; Gekhtman, M.; Szmigielski, J., Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model, J. Math. Phys., 54, 4, (2013), 043517, 25 · Zbl 1292.33009 [17] Bertola, M.; Bothner, T., Zeros of large degree vorob’ev-yablonski polynomials via a Hankel determinant identity, (2014), preprint [18] Bertola, M.; Tovbis, A., Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painlevé equation, Constr. Approx., 41, 3, 529-587, (2015) · Zbl 1320.33029 [19] Björk, J.-E.; Borcea, J.; Bøgvad, R., Subharmonic configurations and algebraic Cauchy transforms of probabilitymeasures, (Notions of Positivity and the Geometry of Polynomials, Trends Math., (2011), Birkhäuser/Springer Basel AG Basel), 39-62 · Zbl 1253.31001 [20] Bleher, P. M.; Deaño, A., Painlevé I double scaling limit in the cubic matrix model, (2013), preprint [21] Bleher, P. M.; Delvaux, S.; Kuijlaars, A. B.J., Random matrix model with external source and a constrained vector equilibrium problem, Comm. Pure Appl. Math., 64, 1, 116-160, (2011) · Zbl 1206.60007 [22] Bleher, P. M.; Kuijlaars, A. B.J., Large n limit of Gaussian random matrices with external source. I, Comm. Math. Phys., 252, 1-3, 43-76, (2004) · Zbl 1124.82309 [23] Bleher, P. M.; Kuijlaars, A. B.J., Random matrices with external source and multiple orthogonal polynomials, Int. Math. Res. Not. IMRN, 3, 109-129, (2004) · Zbl 1082.15035 [24] Bleher, P. M.; Liechty, K., Random matrices and the six-vertex model, CRM Monogr. Ser., vol. 32, (2014), American Mathematical Society Providence, RI · Zbl 1279.82001 [25] Bleher, P. M.; Deaño, A., Topological expansion in the cubic random matrix model, Int. Math. Res. Not. IMRN, 12, 2699-2755, (2013) · Zbl 1314.60022 [26] Bleher, P. M.; Kuijlaars, A. B.J., Orthogonal polynomials in the normal matrix model with a cubic potential, Adv. Math., 230, 3, 1272-1321, (2012) · Zbl 1250.42079 [27] Deaño, A.; Huybrechs, D.; Kuijlaars, A. B.J., Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature, J. Approx. Theory, 162, 12, 2202-2224, (2010) · Zbl 1223.41017 [28] Deift, P., Orthogonal polynomials and random matrices: A Riemann-Hilbert approach, Courant Lect. Notes, vol. 3, (2000), American Mathematical Society · Zbl 0997.47033 [29] 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, 1335-1425, (1999) · Zbl 0944.42013 [30] Deift, P.; Its, A.; Krasovsky, I., Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math. (2), 174, 2, 1243-1299, (2011) · Zbl 1232.15006 [31] Deift, P.; Its, A.; Krasovsky, I., Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results, Comm. Pure Appl. Math., 66, 9, 1360-1438, (2013) · Zbl 1292.47016 [32] Duits, M.; Geudens, D.; Kuijlaars, A. B.J., A vector equilibrium problem for the two-matrix model in the quartic/quadratic case, Nonlinearity, 24, 3, 951-993, (2011) · Zbl 1211.31006 [33] Duits, M.; Kuijlaars, A. B.J., Universality in the two-matrix model: a Riemann-Hilbert steepest-descent analysis, Comm. Pure Appl. Math., 62, 8, 1076-1153, (2009) · Zbl 1221.15052 [34] Duits, M.; Kuijlaars, A. B.J.; Mo, M. Y., The Hermitian two matrix model with an even quartic potential, Mem. Amer. Math. Soc., 217, 1022, (2012), v+105 · Zbl 1247.15032 [35] Filipuk, G.; Van Assche, W.; Zhang, L., Multiple orthogonal polynomials associated with an exponential cubic weight, J. Approx. Theory, 190, 1-37, (2015) · Zbl 1309.42037 [36] Fokas, A. S.; Its, A.; Kitaev, A. V., The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys., 147, 395-430, (1992) · Zbl 0760.35051 [37] Geronimo, J. S.; Kuijlaars, A. B.J.; Van Assche, W., Riemann-Hilbert problems for multiple orthogonal polynomials, (Special Functions 2000: Current Perspective and Future Directions, Tempe, AZ, NATO Sci. Ser. II Math. Phys. Chem., vol. 30, (2001), Kluwer Acad. Publ. Dordrecht), 23-59 · Zbl 0997.42012 [38] Gonchar, A. A.; Rakhmanov, E. A., On the convergence of simultaneous Padé approximants for systems of functions of Markov type, Tr. Mat. Inst. Steklova, 157, 31-48, (1981), 234, Number theory, mathematical analysis and their applications · Zbl 0492.41027 [39] Gonchar, A. A.; Rakhmanov, E. A., The equilibrium problem for vector potentials, Uspekhi Mat. Nauk, 40, 4(244), 155-156, (1985) · Zbl 0594.31010 [40] Gonchar, A. A.; Rakhmanov, E. A., Equilibrium distributions and the rate of rational approximation of analytic functions, Mat. Sb., 134(176), 3, 306-352, (1987), 447 · Zbl 0645.30026 [41] Hardy, A.; Kuijlaars, A. B.J., Weakly admissible vector equilibrium problems, J. Approx. Theory, 164, 6, 854-868, (2012) · Zbl 1241.49008 [42] Holst, T.; Shapiro, B., On higher heine-Stieltjes polynomials, Israel J. Math., 183, 321-345, (2011) · Zbl 1227.34087 [43] Jenkins, J. A., Univalent functions and conformal mapping, (Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft18, Moderne Funktionentheorie, (1958), Springer-Verlag Berlin-Göttingen-Heidelberg) · Zbl 0083.29606 [44] Kalyagin, V. A., A class of polynomials determined by two orthogonality relations, Mat. Sb., 110(152), 4, 609-627, (1979) [45] Kamvissis, S., Comment on “existence and regularity for an energy maximization problem in two dimensions”, J. Math. Phys., 50, (2009), 104101, 3 · Zbl 1283.81072 [46] Kamvissis, S.; Rakhmanov, E. A., Existence and regularity for an energy maximization problem in two dimensions, J. Math. Phys., 46, 8, (2005), 083505, 24 · Zbl 1110.81083 [47] Kuijlaars, A. B.J., Multiple orthogonal polynomials in random matrix theory, (Proceedings of the International Congress of Mathematicians. Volume III, New Delhi, (2010), Hindustan Book Agency), 1417-1432 · Zbl 1230.42034 [48] Kuijlaars, A. B.J.; Martínez-Finkelshtein, A.; Wielonsky, F., Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights, Comm. Math. Phys., 286, 1, 217-275, (2009) · Zbl 1188.60018 [49] Kuijlaars, A. B.J.; McLaughlin, K. T.-R., Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter, Comput. Methods Funct. Theory, 1, 1, 205-233, (2001) · Zbl 1035.30027 [50] Kuijlaars, A. B.J.; McLaughlin, K. T-R., Asymptotic zero behavior of Laguerre polynomials with negative parameter, Constr. Approx., 20, 4, 497-523, (2004) · Zbl 1069.33008 [51] Kuijlaars, A. B.J.; Van Assche, W.; Wielonsky, F., Quadratic Hermite-Padé approximation to the exponential function: a Riemann-Hilbert approach, Constr. Approx., 21, 3, 351-412, (2005) · Zbl 1084.41007 [52] Kuijlaars, A. B.J.; López-García, A., The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star, Nonlinearity, 28, 2, 347-406, (2015) · Zbl 1314.31005 [53] Kuijlaars, A. B.J.; Silva, G. L.F., S-curves in polynomial external fields, J. Approx. Theory, 191, 1-37, (2015) · Zbl 1314.31006 [54] Kuijlaars, A. B.J.; Tovbis, A., The supercritical regime in the normal matrix model with cubic potential, Adv. Math., 283, 530-587, (2015) · Zbl 1330.82024 [55] Kuz’mina, G. V., Moduli of families of curves and quadratic differentials, Proc. Steklov Inst. Math., 139, 1-231, (1982) · Zbl 0491.30013 [56] Lehto, O., Univalent functions and Teichmüller spaces, Grad. Texts in Math., vol. 109, (1987), Springer-Verlag New York · Zbl 0606.30001 [57] Martínez-Finkelshtein, A.; Martínez-González, P.; Thabet, F., Trajectories of quadratic differentials for Jacobi polynomials with complex parameters, (2015), preprint [58] Martínez-Finkelshtein, A.; Rakhmanov, E. A., Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials, Comm. Math. Phys., 302, 1, 53-111, (2011) · Zbl 1226.30005 [59] A. Martínez-Finkelshtein, G.L.F. Silva, Critical measures for vector energy: breaking the symmetry for multiple orthogonal polynomials, 2016, in preparation. [60] Martínez-Finkelshtein, A.; Rakhmanov, E. A., On asymptotic behavior of heine-Stieltjes and Van vleck polynomials, (Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemp. Math., vol. 507, (2010), Amer. Math. Soc. Providence, RI), 209-232 · Zbl 1207.30058 [61] Martínez-Finkelshtein, A.; Rakhmanov, E. A.; Suetin, S. P., Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later, (Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Contemp. Math., vol. 578, (2012), Amer. Math. Soc. Providence, RI), 165-193 · Zbl 1318.42033 [62] Monakhov, V. N.; Leĭfman, L. I.A., Boundary-value problems with free boundaries for elliptic systems of equations, Transl. Math. Monogr., (1983), American Mathematical Society [63] Nikishin, E. M.; Sorokin, V. N., Rational approximations and orthogonality, Transl. Math. Monogr., vol. 92, (1991), American Mathematical Society Providence, RI, translated from the Russian by R.P. Boas · Zbl 0733.41001 [64] Nuttall, J., Padé polynomial asymptotics from a singular integral equation, Constr. Approx., 6, 2, 157-166, (1990) · Zbl 0685.41014 [65] Ortega-Cerdà, J.; Pridhnani, B., The Pólya-tchebotaröv problem, harmonic analysis and partial differential equations, Contemp. Math., vol. 505, 153-170, (2010), Amer. Math. Soc. Providence, RI · Zbl 1196.30003 [66] E. Perevoznikova, E.A. Rakhmanov, Variations of the equilibrium energy and S-property of compacta of minimal capacity, 1994, manuscript. [67] Pommerenke, C., Univalent functions - with a chapter on quadratic differentials by G. Jensen, (1975), Vandenhoeck & Ruprecht Gottingen · Zbl 0298.30014 [68] Rakhmanov, E. A., On the asymptotics of Hermite-Padé polynomials for two Markov functions, Mat. Sb., 202, 1, 133-140, (2011) [69] Rakhmanov, E. A., Orthogonal polynomials and S-curves, Contemp. Math., vol. 578, (2012), Amer. Math. Soc. Providence, RI · Zbl 1318.30056 [70] Rakhmanov, E. A., The asymptotics of Hermite-Padé polynomials for two Markov-type functions, Sb. Math., 202, 1, 127, (2011) · Zbl 1218.41007 [71] Schiffer, M., Variation of domain functionals, Bull. Amer. Math. Soc., 60, 303-328, (1954) · Zbl 0056.32702 [72] Schiffer, M., A method of variation within the family of simple functions, Proc. Lond. Math. Soc., S2-44, 6, 432-449, (1938) · JFM 64.0307.01 [73] Solynin, A. Y., Quadratic differentials and weighted graphs on compact surfaces, (Analysis and Mathematical Physics, Trends Math., (2009), Birkhäuser Basel), 473-505 · Zbl 1297.30068 [74] Stahl, H., Orthogonal polynomials with complex-valued weight function. I, II, Constr. Approx., 2, 3, 225-240, (1986), 241-251 · Zbl 0592.42016 [75] Stahl, H., Orthogonal polynomials with respect to complex-valued measures, (Orthogonal Polynomials and Their Applications, Erice, 1990, IMACS Ann. Comput. Appl. Math., vol. 9, (1991), Baltzer Basel), 139-154 · Zbl 0852.42009 [76] Stieltjes, T. J., Sur certains polynômes que vérifient une équation différentielle linéaire du second ordre et sur la teorie des fonctions de Lamé, Acta Math., 6, 321-326, (1885) · JFM 17.0310.01 [77] Strebel, K., Quadratic differentials, Ergeb. Math. Grenzgeb. (3), vol. 5, (1984), Springer-Verlag Berlin, [Results in Mathematics and Related Areas (3)] · Zbl 0547.30001 [78] Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, (1975), American Mathematical Society · JFM 61.0386.03 [79] Teichmüller, O., Extremale quasikonforme abbildungen und quadratische differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat.wiss. Kl., 1939, 22, 197, (1940) · JFM 66.1252.01 [80] Van Assche, W., Padé and Hermite-Padé approximation and orthogonality, Surv. Approx. Theory, 2, 61-91, (2006) · Zbl 1102.41017 [81] Vasiliev, A., Moduli of families of curves for conformal and quasiconformal mappings, Lecture Notes in Math., vol. 1788, (2002), Springer-Verlag Berlin · Zbl 0999.30001 [82] Zdravkovska, S., Conversation with vladimir igorevich arnol’d, Math. Intelligencer, 9, 4, 28-32, (1987), (Arnold interviewed by Smilka Zdravkovska) · Zbl 0627.01013
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