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

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. Stochastic Geometry: statistical ensemble of fractal shapes Черноголвка 2007

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

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

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

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

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

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) 8 Черноголвка 2007

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

Fingering Instability 10 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

Finite time singularities Gradient Catastrophe ➠ ➠

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): The main family of singularities - cusps are classified by two integers (p,q):

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

A catastrophe: no physical solution beyond the cusp

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

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

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

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

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

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

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

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

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

Eigenvalues distribution of Norman Random Matrix ensemble Equilibrium measure Черноголвка 2007

Bi-Orthogonal polinomials and Random Matrices

Bi-Orthogonal Polynomials and planar domains Bounded domain measure

Gaussian ensemble Non-Gaussian ensemble

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

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

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

Schwarz function and Boutroux -Krichever curve: Harmonic moments ➩ conserve d Черноголвка 2007

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

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

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