Growth phenomena
Free Access
Math. Model. Nat. Phenom.
Volume 15, 2020
Growth phenomena
Article Number 9
Number of page(s) 14
Published online 17 February 2020
  1. O. Alekseev and M. Mineev-Weinstein, Stochastic Laplacian growth. Phys. Rev. E 94 (2016) 060103. [Google Scholar]
  2. O. Alekseev and M. Mineev-Weinstein, Theory of stochastic Laplacian growth. J. Stat. Phys. 168 (2017) 68–91. [Google Scholar]
  3. O. Alekseev and M. Mineev-Weinstein, Statistical mechanics of stochastic growth phenomena. Phys. Rev. E 96 (2017), 010103. [Google Scholar]
  4. F. Balogh, M. Bertola, S.Y. Lee and K.D.T.-R. McLaughlin, Strong asymptotics of the orthogonal polynomial with respect to a measure supported on the plane. Preprint arXiv.math-ph.:1209.6366 (2012). [Google Scholar]
  5. M. Bauer and D. Bernard, 2D growth processes: SLE and Loewner chains. Phys. Rep. 432 (2006) 115. [Google Scholar]
  6. E. Ben-Jacob, From snowflake formation to the growth of bacterial colonies. Part 2: Cooperative formation of complex colonial patterns. Contempt. Phys. 38 (1997) 205–241. [CrossRef] [Google Scholar]
  7. D. Bensimon, L. Kadanoff, S. Liang, B. Shraiman and C. Tang, Viscous flows in two dimensions. Rev. Mod. Phys. 58 (1986) 977. [Google Scholar]
  8. E. Bettelheim, Classical and Quantum Integrability in Laplacian Growth. Preprint arXiv:1506.01463 [nlin.PS] (2015). [Google Scholar]
  9. P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150 (1999) 185–266. [Google Scholar]
  10. P. Bleher and A.B.J. Kuijlaars, Large n limit of Gaussian random matrices with external source, part I. Commun. Math. Phys. 252 (2004) 43–76. [CrossRef] [Google Scholar]
  11. P. Bleher and A. Kuijlaars, Orthogonal polynomials in the normal matrix model with a cubic potential. Adv. Math. 230 (2012) 1272–1321. [CrossRef] [Google Scholar]
  12. P. Bleher, A. Kuijllars and S. Delvaux, Random matrix model with external source and a constrained vector equilibrium problem. Commun. Pure Appl. Math. 64 (2011) 116160. [Google Scholar]
  13. L. Carleson and N. Makarov, Aggregation in the plane and Loewner’s equation. Commun. Math. Phys. 48 (2001) 538–607. [Google Scholar]
  14. E. DiBenedetto and A. Friedman, Bubble growth in porous media. Indiana Univ. Math. J. 35 (1986) 573–606. [CrossRef] [Google Scholar]
  15. M. Duits and A.B.J. Kuijlaars, Painlevé I asymptotics for orthogonal polynomials with respect to a varying weight. Nonlinearity 19 (2006) 2211–2245. [Google Scholar]
  16. P. Ebenfelt, B. Gustafsson, D. Khavinson and M. Putinar eds., Quadrature Domains and Their Applications, The Harold S. Shapiro Anniversary Volume, Birkhäuser, Basel (2005). [CrossRef] [Google Scholar]
  17. P. Elbau and G. Felder, Density of eigenvalues of random normal matrices. Commun. Math. Phys. 259 (2005) 433–450. [CrossRef] [Google Scholar]
  18. A.S. Fokas, A.R. Its and A.V. Kitaev, The isomonodromy approach to matrix problems in 2D quantum gravity. Commun. Math. Phys. 147 (1992) 395–430. [CrossRef] [Google Scholar]
  19. L.A. Galin, Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47 (1945) 250–253 (In Russian); English trasl., (Dokl.)Acad. Sci. URSS 47 (1945) 246–249. [Google Scholar]
  20. S. Garoufalidis, A. Its, A. Kapaev and M. Mari-o, Asymptotics of the Instantons of Painlev I. Int. Math. Res. Notices 2012 (2012) 561–606. [CrossRef] [Google Scholar]
  21. C. Gomez, M. Ruiz-Altaba and G. Sierra, Quantum groups in two-dimensional physics. Cambridge Univ. Press (1996). [CrossRef] [Google Scholar]
  22. A.A. Gonchar and E.A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials. Mat. Sbornik. 125 (1984) 117–127. Translation from Mat. Sb., Nov. Ser. 134 (1987) 306–352. [Google Scholar]
  23. B. Gustafsson, R. Teodorescu and A. Vasil’ev, Classical and stochastic Laplacian growth. Springer International Publishing (2014). [Google Scholar]
  24. A. Hassel and S. Zelditch, Determinants of Laplacians in exterior domains. IMRN (1999) 971–1004. [CrossRef] [Google Scholar]
  25. M.B. Hastings and L.S. Levitov, Laplacian growth as one-dimensional turbulence. Physica D 116 (1998) 244. [Google Scholar]
  26. H. Hedenmalm and S. Shimorin, Hele-Shaw flow on hyperbolic surfaces. J. Math. Pures Appl. 81 (2002) 187–222. [Google Scholar]
  27. S.D. Howison, Fingering in Hele-Shaw cells. J. Fluid Mech. 167 (1986) 439–453. [Google Scholar]
  28. S.D. Howison, Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Math. 3 (1992) 209–224. [Google Scholar]
  29. S. Howison, I. Loutsenko and J. Ockendon, A class of exactly solvable free-boundary inhomogeneous porous medium flows. Appl. Math. Lett. 20 (2007) 93–97. [Google Scholar]
  30. A.R. Its and L.A. Takhtajan, Normal matrix models, ∂̅-problem, and orthogonal polynomials on the complex plane. Preprint arXiv.math.:0708.3867 (2007). [Google Scholar]
  31. J. Jenkins, Univalent functions and conformal mapping. Springer-Verlag (1958). [Google Scholar]
  32. F. Johansson Viklund, A. Sola and A. Turner, Scaling limits of anisotropic Hastings-Levitov clusters. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 235–357. [CrossRef] [Google Scholar]
  33. F. Johansson Viklund, A. Sola and A. Turner, Small-particle limits in a regularized Laplacian growth model. Commun. Math. Phys. 334 (2015) 331–366. [CrossRef] [Google Scholar]
  34. Q. Kang, D. Zhang and S. Chen, Immiscible displacement in a channel: simulations of fingering in two dimensions. Adv. Wat. Res. 27 (2004) 13–22. [CrossRef] [Google Scholar]
  35. L. Karp, Construction of quadrature domains in Rn from quadrature domains in R2. Complex Var. Theory Appl. 17 (1992) 179–188. [Google Scholar]
  36. D. Khavinson, M. Mineev-Weinstein and M. Putinar, Planar eliptic growth. Complex Anal. Oper. Theory 3 (2009) 425–451. [CrossRef] [Google Scholar]
  37. D. Khavinson, M. Mineev-Weinstein, M. Putinar and R. Teodorescu, Lemniscates are destroyed by eliptic growth. Math. Res. Lett. 17 (2010) 337. [CrossRef] [Google Scholar]
  38. I. Kostov, I. Krichever, M. Mineev-Weinstein, P. Wiegmann and A. Zabrodin, τ-function for analytic curves. Vol. 40 of Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ. Cambridge University Press (2001) 285–299. [Google Scholar]
  39. I. Krichever, A. Marshakov and A. Zabrodin. Integrable structure of the Dirichlet boundary problem in multiply-connected domains. Commun. Math. Phys. 259 (2005) 1–44. [CrossRef] [Google Scholar]
  40. I. Krichever, M. Mineev-Weinstein, P. Wiegmann and A. Zabrodin, Laplacian growth and Whitham equations of soliton theory. Physica D 198 (2004) 1–28. [Google Scholar]
  41. P.P. Kufarev, A solution of the boundary problem of an oil well in a circle. Dokl. Acad. Nauk SSSR. 60 (1948) 1333–1334. [Google Scholar]
  42. J.S. Langer, Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52 (1980) 1–28. [Google Scholar]
  43. S.-Y. Lee and N. Makarov, Topology of quadrature domains. arXiv:1307.0487 [math.CV] (2015). [Google Scholar]
  44. S.-Y. Lee, R. Teodorescu and P. Wiegmann, Shocks and finite-time singularities in Hele-Shaw flow. Physica D 238 (2009) 1113–1128. [Google Scholar]
  45. S.-Y. Lee, R. Teodorescu and P. Wiegmann, Weak solution of the Hele-Shaw problem: shocks and viscous fingering. JETP Lett. 92 (2010) 9196. [Google Scholar]
  46. S.-Y. Lee, R. Teodorescu and P. Wiegmann, Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé I transcendent. Physica D 240 (2011) 1080–1091. [Google Scholar]
  47. D.S. Lubinsky, H.N. Mhaskar and E.B. Saff, A proof of Freud’s conjecture for exponential weights. Constr. Approx. 4 (1988) 65–83. [Google Scholar]
  48. E. Lundberg and D. Khavinson, Gravitational lensing by a collection of objects with radial densities. Anal. Math. Phys. 1 (2011) 139–145. [CrossRef] [Google Scholar]
  49. P. Macklin and J. Lowengrub, An improved geometry-aware curvature discretization for level set methods: application to tumor growth. J. Comput. Phys. 215 (2006) 392–401. [Google Scholar]
  50. O. Marchal and M. Cafasso, Double-scaling limits of random matrices and minimal (2m, 1) models: the merging of two cuts in a degenerate case. J. Stat. Mech. Theory Exp. 2011 (2011) P04013. [CrossRef] [Google Scholar]
  51. A. Marshakov, P. Wiegmann and A. Zabrodin, Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227 (2002) 131–153. [CrossRef] [Google Scholar]
  52. A. Martinez-Finkelshtein, P. Martinez-González and R. Orive, Asymptotics of polynomial solutions of a class of generalized Lamé differential equations. Electr. Trans. Numer. Anal. 19 (2005) 18–28. [Google Scholar]
  53. A. Martínez-Finkelshtein and E.A. Rakhmanov. Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials. Comm. Math. Phys. 302 (2011) 53–111. [CrossRef] [Google Scholar]
  54. A. Martínez-Finkelshtein and E.B. Saff, Asymptotic properties of Heine-Stieltjes and Van Vleck polynomials. J. Approx. Theory, 118 (2002) 131–151. [Google Scholar]
  55. E. Memin and P. Perez, Fluid motion recovery by coupling dense and parametric vector fields. IEEE CVPR (1999) 620–625. [Google Scholar]
  56. H.N. Mhaskar and E.B. Saff, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). Constr. Approx. 1 (1985) 71–91. [Google Scholar]
  57. M.B. Mineev, A finite polynomial solution of the two-dimensional interface dynamics. Physica D 43 (1990) 288–292. [Google Scholar]
  58. M. Mineev-Weinstein, Multidimensional pattern formation has an infinite number of constants of motion. Phys. Rev. E 47 (1993) R2241–R2244. [Google Scholar]
  59. M.B. Mineev-Weinstein, Selection of the Saffman-Taylor finger in the absence of surface tension: an exact result. Phys. Rev. Lett. 80 (1998) 2113–2116. [Google Scholar]
  60. M.B. Mineev-Weinstein and S.P. Dawson, A class of non-singular exact solutions for Laplacian pattern formation. Phys. Rev. E 50 (1994) R24–R27. [Google Scholar]
  61. M. Mineev-Weinstein, P.B. Wiegmann and A. Zabrodin, Integrable structure of interface dynamics. Phys. Rev. Lett. 84 (2000) 5106. [CrossRef] [PubMed] [Google Scholar]
  62. M. Mineev-Weinstein, M. Putinar, L Sander and A. Zabrodin, eds, Physics and mathematics of growing interfaces. Physica D (2007) 235. [Google Scholar]
  63. M. Mineev-Weinstein, M. Putinar and R. Teodorescu, Random matrices in 2D Laplacian growth and operator theory. J. Phys. A: Math. Theor. 41 (2008) 263001. [CrossRef] [Google Scholar]
  64. M.Y. Mo, The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. Int. Math. Res. Not. IMRN 51 (2008) rnn042. [Google Scholar]
  65. J. Norris and A. Turner, Hastings-Levitov aggregation in the small-particle limit. Commun. Math. Phys. 316 (2012) 809–841. [CrossRef] [Google Scholar]
  66. R. Orive and Z. García, On a class of equilibrium problems in the real axis. J. Comput. Appl. Math. 235 (2010) 1065–1076. [Google Scholar]
  67. P. Ya. Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell. Dokl. Akad. Nauk USSR 47 (1945) 254–257 (in Russian); English transl., On the displacement of the oil-bearing contour. C. R. (Dokl.) Acad. Sci. URSS 47 (1945) 250–254. [Google Scholar]
  68. Ch. Pommerenke, Univalent functions, with a chapter on quadratic differentials by G. Jensen. Vandenhoeck & Ruprecht, Göttingen (1975). [Google Scholar]
  69. Ch. Pommerenke, Boundary behaviour of conformal maps. Springer, Berlin (1992). [CrossRef] [Google Scholar]
  70. O. Praud and H.L. Swinney, Fractal dimensions and unscreened angles measured for radial viscous fingering. Phys. Rev. E 72 (2005) 011406. [Google Scholar]
  71. A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1986). [Google Scholar]
  72. G. Prokert, Existence results for Hele-Shaw flow driven by surface tension. Eur. J. Appl. Math. 9 (1998) 195–221. [Google Scholar]
  73. E.A. Rakhmanov, The convergence of diagonal Padé approximants. Mat. Sb. (N.S.) 104 (1977) 271–291, 335. English translation: Math. USSR-Sb. 33 (1977) 243–260. [Google Scholar]
  74. S. Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56 (1972) 609–618. [Google Scholar]
  75. P.G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. London. Ser. A 245 (1958) 312–329. [CrossRef] [MathSciNet] [Google Scholar]
  76. E.B. Saff and V. Totik, Logarithmic Potentials with External Fields. Vol. 316 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (1997). [CrossRef] [Google Scholar]
  77. Y. Sawada, A. Dougherty and J.P. Gollub, Dendritic and fractal patterns in electrolytic metal deposits. Phys. Rev. Lett. 56 (1986) 1260–1263. [CrossRef] [PubMed] [Google Scholar]
  78. B. Shraiman and D. Bensimon, Singularities in nonlocal interface dynamics. Phys. Rev. A 30 (1984) 2840. [Google Scholar]
  79. M.G. Stepanov and L.S. Levitov, Laplacian growth with separately controlled noise and anisotropy. Phys. Rev. E 63 (2001) 061102. [Google Scholar]
  80. K. Takasaki and T. Takebe, Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7 (1995) 743. [CrossRef] [MathSciNet] [Google Scholar]
  81. G. Taylor and P.G. Saffman, A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Math. 12 (1959) 265–279. [Google Scholar]
  82. R. Teodorescu, Generic critical points of normal matrix ensembles. J. Phys. A: Math. Theor. 39 (2006) 8921. [Google Scholar]
  83. R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin and P. Wiegmann, Normal random matrix ensemble as a growth problem. Nucl. Phys. B 704 (2005) 407–444. [Google Scholar]
  84. R. Teodorescu, P. Wiegmann and A. Zabrodin, Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the Kortweg-de Vries hierarchy. Phys. Rev. Lett. 95 (2005) 044502. [CrossRef] [PubMed] [Google Scholar]
  85. Y. Tu, Saffman-Taylor problem in sector geometry: solution and selection. Phys. Rev. A 44 (1991) 1203–1210. [CrossRef] [PubMed] [Google Scholar]
  86. W. Van Assche, J. Geronimo and A.B.J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, in Special Functions 2000 edited by J. Bustoz et al. Kluwer, Dordrecht (2001) 23–59. [CrossRef] [Google Scholar]
  87. A.N. Varchenko and P.I. Etingof, Why the boundary of a round drop becomes a curve of order four. Vol. 3 of University Lecture Series. American Mathematical Society, Providence, RI (1992). [Google Scholar]
  88. Yu.P. Vinogradov and P.P. Kufarev, On some particular solutions of the problem of filtration. Doklady Akad. Nauk SSSR (N.S.) 57 (1947) 335–338. [Google Scholar]
  89. D.V. Voiculescu, Free probability for pairs of faces II: 2-variables bi-free partial R-transform and systems with rank ≤ 1 commutation. Ann. Inst. Henri Poincaré Probab. Statist. 52 (2016) 1–15. [CrossRef] [Google Scholar]
  90. P.B. Wiegmann and A. Zabrodin, Conformal maps and integrable hierarchies. Commun. Math. Phys. 213 (2000) 523–538. [CrossRef] [Google Scholar]
  91. P. Wiegmann and A. Zabrodin, Large scale correlations in normal and general non-Hermitian matrix ensembles. J. Phys. A 36 (2003) 3411–3424. [CrossRef] [Google Scholar]
  92. P. Wiegmann and A. Zabrodin, Large N expansion for normal and complex matrix ensembles. Frontiers Number Theory, Physics, and Geometry I. Springer, Berlin/Heidelberg, Part I (2006) 213–229. [CrossRef] [Google Scholar]

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