EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 15 (2020) Math. Model. Nat. Phenom., 15 (2020) 9 References
Growth phenomena
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Growth phenomena
Article Number 9
Number of page(s) 14
DOI https://doi.org/10.1051/mmnp/2019032
Published online 17 February 2020
  1. O. Alekseev and M. Mineev-Weinstein, Stochastic Laplacian growth. Phys. Rev. E 94 (2016) 060103. [PubMed] [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. [PubMed] [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. [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. [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. [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. [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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.