laplacian growth: singularities of growing patterns random matrices and asymptotes of orthogonal...

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.


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


Hele-Shaw cell (1894)

waterwater

Oil (exterior)-incompressible liquid with high viscosity

Oil (exterior)-incompressible liquid with high viscosity

Water (interior) - incompressible liquid with low viscosityWater (interior) - incompressible liquid with low viscosity

6

oiloil


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

7Черноголвка 2007Черноголвка 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)

8Черноголвка 2007Черноголвка 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

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

Linear analysis is due to Saffman&Taylor 1956

Finite time singularitiesFinite time singularities

Gradient CatastropheGradient 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 Hypotrocoid: a map of the unit circle

12

-- Universal character of of singularities:-- 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):

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Self-similar (universal) shapes of the singularitiesSelf-similar (universal) shapes of the singularitiesChebyshev-polynomialsChebyshev-polynomials

Generic singularity (2,3) is related to solutions of KdV equation

Generic singularity (2,3) is related to solutions of KdV equation

A catastrophe: no physical solution beyond the cuspA catastrophe: no physical solution beyond the cusp

Richardson’s theorem: Richardson’s theorem:

Cauchy transform of the exterior (oil)Cauchy transform of the exterior (oil)

It follows that harmonic moments It follows that harmonic moments

- are conserved- are conserved

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Problem of regularization hydrodynamic singularities

Hydrodynamic problem is ill defined


Riemann Equation

Singular limit of non-linear waves

Weak solutions: discontinuities - shocks


➠ 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


Bi-Orthogonal polinomials and Random Matrices

Bi-Orthogonal Polynomials and planar domainsBi-Orthogonal Polynomials and planar domains

Bounded domainmeasure

Bounded domainmeasure

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

➩conserv

ed


Asymptotes of Bi-Orthogonal polynomialsAsymptotes of Bi-Orthogonal polynomials

A graph of zeros (or anti-Stokes lines, or shocks) is determined by two conditionsA 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


QuickTime™ and aMotion JPEG OpenDML decompressor

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laplacian growth: singularities of growing patterns random matrices and asymptotes of orthogonal...

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