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P. Wiegmann University of Chicago

Laplacian Growth: singularities of growing patterns Random Matrices and Asymptotes of Orthogonal polynomials,. P. Wiegmann University of Chicago. 1. Stochastic Geometry : statistical ensemble of fractal shapes. Diffusion-Limited Aggregation, or DLA,

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P. Wiegmann University of Chicago

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  1. Laplacian Growth: singularities of growing patternsRandom Matrices and Asymptotes of Orthogonal polynomials, P. Wiegmann University of Chicago 1

  2. Stochastic Geometry: statistical ensemble of fractal shapes Diffusion-Limited Aggregation, or DLA, is an extraordinarily simple computer simulation of the formation of clusters by particles diffusing through a medium that jostles the particles as they move. Черноголвка 2007

  3. Hypothesis: The pattern is related to asymptotes of distribution of zerosof Bi-Orthogonal Polynomials. Черноголвка 2007

  4. Continuous problem ( a hydrodynamic limit): a size of particles tends to zero

  5. Laplacian growth - velocity of moving planar interface is a gradient of a harmonic field A probability of a Brownian mover to arrive and join the aggregate at a point z is a harmonic measure of the domain z Черноголвка 2007

  6. Hele-Shaw cell (1894) water oil Oil (exterior)-incompressible liquid with high viscosity Water (interior) - incompressible liquid with low viscosity Черноголвка 2007 6

  7. Laplacian growth - velocity of a moving planar interface = = a gradient of a harmonic field Черноголвка 2007 7

  8. Random matrix theory; ✓ • Topological Field Theory; • Quantum Gravity; • Non-linear waves and soliton theory; • Whitham universal hierarchies; • Integrable hierarchies and Painleve transcendants • Isomonodromic deformation theory; • Asymptotes of orthogonal polynomials ✓ • Non-Abelian Riemann Hilbert problem; • Stochastic Loewner Evolution (anticipated) Черноголвка 2007 8

  9. Integrability of continuum problem (fluid mechanics) A. Zabrodin and P.W. (2001)

  10. Fingering Instability Any almost all fronts are unstable - an arbitrary small deviation from a plane front causes a complex set of fingers growing out of control Linear analysis is due to Saffman&Taylor 1956 10

  11. ➠ Finite time singularities ➠ Gradient Catastrophe

  12. Finite time singularities:any but plain algebraic domain lead to cusp like singularities which occur at a finite time (the area of the domain) Hypotrocoid: a map of the unit circle -- Universal character of of singularities: The main family of singularities - cusps are classified by two integers (p,q): 12

  13. Self-similar (universal) shapes of the singularities Chebyshev-polynomials Generic singularity (2,3) is related to solutions of KdV equation 13

  14. A catastrophe: no physical solution beyond the cusp

  15. Richardson’s theorem: Cauchy transform of the exterior (oil) It follows that harmonic moments - are conserved 15

  16. Hydrodynamic problem is ill defined Problem of regularization hydrodynamic singularities Черноголвка 2007

  17. Text Singular limit of non-linear waves Riemann Equation Weak solutions: discontinuities - shocks Черноголвка 2007

  18. ➠ Hamiltonian Regularization vs Diffusion regularization Non-vanishing size of particles

  19. S.-Y. Lee, R. Teodoerescu, P. W. E. Bettelheim, I. Krichever, A. Zabrodin, O. Agam

  20. Weak solution of hydrodynamics: preserving the algebraic structure of the curve (i.e. integrable structure) Pressure is harmonic everywhere except moving lines of discontinuities - shocks Shocks are uniquely determined by integrability

  21. Orthogonal polynomials Szegotheorem: If V(x) real (real orthogonal polynomials 1) zeros of are distributed along a real axis ➠ Asymptotes: 2) Zeros form dense segments of the real axis, 3) Asymptotes at the edges is of Airy type ➠

  22. ➠ Eigenvalues distribution of Hermitian Random Matrices ➠ Equilibrium measure of real orthogonal polynomials

  23. Bi-Orthogonal polynomials ➠ Asymptotes: ➠ Zeros are distributed along a branching graph ➠ Asymptotes at the edges are Painleve transcendants

  24. Eigenvalues distribution of Norman Random Matrix ensemble Equilibrium measure Черноголвка 2007

  25. Bi-Orthogonal polinomials and Random Matrices

  26. Bi-Orthogonal Polynomials and planar domains Bounded domain measure

  27. Gaussian ensemble Non-Gaussian ensemble

  28. Semiclassical limit of Matrix Growth: N→ N+1 is equivalent to the Hele-Shaw flow. Proved by Haakan Hedenmalm and Nikolai Makarov

  29. Bi-Orthogonal Polynomials Semiclassical Limit: back to hydrodynamics Asymptotes of Orthogonal Polynomials solve Hele-Shaw flow

  30. Classical limit: does not exists at the anti-Stokes lines, where polynomials accumulate zeros: anti-Stokes lines - lines of discontinuities pressure and velocity - - shock fronts of the flow

  31. Schwarz function and Boutroux -Krichever curve: Harmonic moments conserved ➩ Черноголвка 2007

  32. Asymptotes of Bi-Orthogonal polynomials A graph of zeros (or anti-Stokes lines, or shocks) is determined by two conditions

  33. A planar domain ➠ measure of bi-orthogonal polynomials➠ evolving Boutroux-Krichever curve ➠ evolving anti-Stokes graph ➠ branching tree

  34. Elliptic curve Boutroux self-similar curve - an elementary branch Черноголвка 2007

  35. Черноголвка 2007

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