Modulus of continuity of controlled Loewner-Kufarev equations and random matrices. (English) Zbl 1441.30015
Summary: First we introduce the two tau-functions which appeared either as the \(\tau\)-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large \(N\)-matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner-Kufarev equations, which enables us to give a continuity estimate for the solution of such an equation when embedded into the Segal-Wilson Grassmannian.
MSC:
| 30C35 | General theory of conformal mappings |
Keywords:
Loewner-Kufarev equation; Grassmannian; Witt algebra; Faber polynomial; Grunsky coefficientReferences:
| [1] | Amaba, T., Friedrich, R.: Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian (2018). arXiv:1809.00534 · Zbl 1459.35363 |
| [2] | Amaba, T.; Friedrich, R.; Murayama, T., Univalence and Holomorphic Extension of the Solution to \(\omega \)-Controlled Loewner-Kufarev Equations, J. Diff. Eqn., 269, 3, 2697-2704 (2020) · Zbl 1441.93118 · doi:10.1016/j.jde.2020.02.011 |
| [3] | Collins, B.; Mingo, J.; Śniady, P.; Speicher, R., Second order freeness and fluctuations of random matrices III. Higher order freeness and free cumulants, Documenta Math., 12, 1-70 (2007) · Zbl 1123.46047 |
| [4] | Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves: second edition. Math. Surv. Monogr. 88, 400 pp (2004) · Zbl 1106.17035 |
| [5] | Gustafsson, B.; Teodorescu, R.; Vasil’ev, A., Classical and Stochastic Laplacian Growth. Advances and Applications in Fluid Mechanics (2014), Birkhäuser: Springer, Birkhäuser · Zbl 1311.76026 |
| [6] | Juriev, D., On the univalency of regular functions, Ann. di Matematica pura ed applicata (IV), CLXIV, 37-50 (1993) · Zbl 0786.30013 · doi:10.1007/BF01759313 |
| [7] | Kawamoto, N.; Namikawa, Y.; Tsuchiya, A.; Yamada, Y., Geometric realization of conformal field theory on Riemann surfaces, Commun. Math. Phys., 116, 247-308 (1988) · Zbl 0648.35080 · doi:10.1007/BF01225258 |
| [8] | Kostov, I., Krichever, I., Mineev-Weinstein, M., Wiegmann, P., Zabrodin, A.: \( \tau \)-Function for Analytic Curves (2000). arXiv:hep-th/000529 · Zbl 0991.37050 |
| [9] | Kirillov, A.; Yuriev, D., Representations of the Virasoro algebra by the orbit method, JGP, 5, 3, 351-363 (1988) · Zbl 0698.17015 |
| [10] | Krichever, I.; Marshakov, A.; Zabrodin, A., Integrable structure of the Dirichlet boundary problem in multiply-connected domains, Commun. Math. Phys., 259, 1-44 (2005) · Zbl 1091.37019 · doi:10.1007/s00220-005-1387-5 |
| [11] | Krushkal, S., Teichmüller Spaces, Grunsky Operator and Fredholm Eigenvalues. Complex Analysis and Its Applications, 65-88 (2007), Osaka: Osaka Studies, Osaka Municipal University Press, Osaka · Zbl 1151.30032 |
| [12] | Lyons, T., Qian, Z.: System Control and Rough Paths. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford. x+216 pp. ISBN: 0-19-850648-1 (2002) · Zbl 1029.93001 |
| [13] | Malliavin, P., The canonic diffusion above the diffeomorphism group of the circle, C. R. Acad. Sci. Paris Sér. I Math., 329, 4, 325-329 (1999) · Zbl 1006.60073 · doi:10.1016/S0764-4442(00)88575-4 |
| [14] | Marshakov, A.; Wiegmann, P.; Zabrodin, A., Integrable structure of the Dirichlet boundary problem in two dimensions, Commun. Math. Phys., 227, 131-153 (2002) · Zbl 1065.37048 · doi:10.1007/s002200200629 |
| [15] | Mineev-Weinstein, M.; Wiegmann, P.; Zabrodin, A., Integrable structure of interface dynamics, Phys. Rev. Lett., 84, 22, 5106-9 (2000) · doi:10.1103/PhysRevLett.84.5106 |
| [16] | Pommerenke, C.: Univalent Functions (With a Chapter on Quadratic Differentials by Gerd Jensen), Studia Mathematica/Mathematische Lehrbücher, vol. XXV, p. 376. Vandenhoeck & Ruprecht, Göttingen (1975) · Zbl 0298.30014 |
| [17] | Segal, G.; Wilson, G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. No., 61, 5-65 (1985) · Zbl 0592.35112 · doi:10.1007/BF02698802 |
| [18] | Takebe, T.; Teo, L-P; Zabrodin, A., Löwner equations and dispersionless hierarchies, J. Phys. A Math. Gen., 39, 11479-11501 (2006) · Zbl 1104.37046 · doi:10.1088/0305-4470/39/37/010 |
| [19] | Takhtajan, L., Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current correlation functions, Lett. Math. Phys., 56, 181-228 (2001) · Zbl 1072.14521 · doi:10.1023/A:1017999407650 |
| [20] | Takhtajan, LA; Teo, LP, Weil-Petersson metric on the universal Teichmüller space, Mem. Am. Math. Soc., 183, 861, 119 (2006) · Zbl 1243.32010 |
| [21] | Teo, LP, Analytic functions and integrable hierarchiescharacterization of tau functions, Lett. Math. Phys., 64, 1, 75-92 (2003) · Zbl 1021.37040 · doi:10.1023/A:1024969729259 |
| [22] | Teo, L.P.: Conformal Weldings and Dispersionless Toda Hierarchy (2008). arXiv:0808.0727 |
| [23] | Wiegmann, P.; Zabrodin, A., Conformal maps and integrable hierarchies, Commun. Math. Phys., 213, 523-538 (2000) · Zbl 0973.37042 · doi:10.1007/s002200000249 |
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