Random and complex encounters:physics meets math

Ilya A. Gruzberg

University of Chicago

Brief history of encounters

• Physics and math was done by the same people up to 19th century

• Even in 19th century mathematicians knew physics well (F. Klein)

• By the turn of 20th century – clear separation of physics and math

Galilei Newton

Brief history of encounters

• 20th century. Mostly one-way: physics borrows methods from math

• Examples: general relativity, quantum mechanics

Einstein Levi-Civita Grossmann

Brief history of encounters

• Modern era. In the 80’s string theory and CFT, mathematical physics.

Mostly algebraic

• 90’s. Cardy’s formula for crossing probability in percolation. Laglands et al.

• Lawler and Werner study intersection exponents of conformaly-invariant

2D Brownian motions, known from physics

• 1999: Oded Schramm’s work established

Conformal stochastic geometry

• One missed and two real Fields medals

(2002 Schramm, 2006 Werner, 2010 Smirnov)

Stochastic conformal geometry:applications in physics

Ilya A. Gruzberg

University of Chicago

Stochastic conformal geometry

• What is it? Try Wikipedia. Does not help (yet), though leads to Oded Schramm and Schramm-Loewner evolution. Try to simplify the search.

• Not “stochastic geometry”. From Wikipedia:

• Not “conformal geometry”. From Wikipedia:

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, which extend to the more abstract setting of random measures.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces.

Stochastic conformal geometry

• Goal: precise description of complicated (fractal) shapes

• Geometric characterization: lengths, areas, volumes, (fractal) dimensions,

(multifractal) harmonic measure

• Examples:

– Critical and off-critical clusters and cluster boundaries in statistical mechanics– Disordered systems: spin glasses, Anderson localization (especially quantum

Hall transitions) – Growth processes: diffusion-limited aggregation, Hele-Shaw flows, dielectric

breakdown, molecular beam epitaxy, Kardar-Parisi-Zhang growth,…– Non-equilibrium (driven) patterns: turbulence, coarsening, non-linear waves – Stringy geometry, 2D quantum gravity, random matrices

Stochastic conformal geometry

• Random shapes drawn from a distribution

– Allows to ask probabilistic questions

• Time dependence: stochastic processes

– Actual time dynamics of a growth process– A parameter playing the role of time

Stochastic conformal geometry

• Conformal invariance at continuous phase transitions (critical points)

• Especially powerful in two dimensions (2D):

– Conformal transformations “=” analytic functions

• Modern tools: Schramm-Loewner evolution (SLE) and conformal restriction

• More generally, shapes in 2D can be described by conformal maps:

– Riemann theorem: any simply-connected domain can be mapped

conformally onto the unit disk or the upper half plane– Can use even when there is no conformal invariance

• Loewner chains (Loewner-Kufarev equations)

IPAM meetings

Long programs:

• 2001: Conformal Field Theory and Applications

• 2003: Symplectic Geometry and Physics

• 2004: Multiscale Geometry and Analysis in High Dimensions

• 2007: Random Shapes

Workshop:

• 2009: Laplacian Eigenvalues and Eigenfunctions

Random shapes in nature: growth patterns

• Hele-Shaw flow

• Real patterns in nature:

• Mineral dendrites

• Electrodeposition

Random shapes in nature: turbulence

• Vorticity clusters • Zero vorticity lines

• Inverse cascade in 2D Navier-Stokes turbulence

Critical clusters

• Percolation clusters (site percolation)

Every site is either ON (black) with probability

or OFF (white) with probability

Critical clusters

• Larger critical percolation clusters

• Crossing probability

Critical clusters

• Ising spin clusters

DLA applet

Growth patterns

• Diffusion-limited aggregation T. Witten, L. Sander, 1981

• Many different variants

Viscous fingering vs. DLA

O. Praud and H. L. Swinney, 2005

Random shapes: stochastic geometry

• Fractal properties and interesting subsets

• Multifractal spectrum of harmonic measure

• Crossing probability and other connectivity properties

• Left vs. right passage probability

• Many more …

Stochastic geometry: multifractal exponents

• Lumpy charge distribution on a cluster boundary

• Non-linear is the hallmark of a multifractal

• Cover the curve by small discs of radius

• Charges (probabilities) inside discs

• Moments

Harmonic measure on DLA cluster

Harmonic measure: electric field

Multifractal measures: electric field of a charged cluster

2D critical phenomena

• Scale invariance A. Patashinski, V. Pokrovsky, 1964

L. Kadanoff, 1966

Critical fluctuations (clusters) are self-similar at all scalesThe basis for renormalization group approach

• Conformal invariance

• In 2D conformal maps = analytic functions

A. Polyakov, 1970

2D critical stat mech models

• Traditional approach:

(Algebraic) conformal field theory: correlations of local degrees of freedom Important parameter: central charge

• New focus:

Stochastic geometry: non-local structures – cluster boundaries, their geometric (global and local) and probabilistic properties

Finite geometries: conformal invariance made precise

• Building a dictionary between field theories and stochastic geometry is an important ongoing direction of research

A. Belavin, A. Polyakov, A. Zamolodchikov, 1984

Crossing probability

Is there a left to right crossing of white hexagons?

• Crossing probabilities J. Cardy, 1992

Crossing probability

And now?

Cluster boundaries

• Focus on one domain wall using certain boundary conditions

• Conformal invariance allows to consider systems in simple domains, for example, upper half plane

Exploration process

• Cluster boundary can be “grown” step-by-step

• Each step is determined by local environment

• Description in terms of differential equations in continuum

Conformal maps

• Specify a 2D shape by a function that maps it to a simple shape

• Always possible (for simply-connected domains) by Riemann’s theorem

Loewner equation

• Describes upper half plane with a cut along a curve

C. Loewner, 1923

SLE postulates

• Conformal invariance of the measure on curves

Schramm-Loewner evolution

• Conformal invariance leads uniquely to Loewner equation driven by a Brownian motion:

O. Schramm, 1999

SLE generates conformally-invariant measures on random curvesthat are continuum limits of critical cluster boundaries

• Noise strength is an important parameter:

SLE versus CFT

• SLE describes all critical 2D systems with

CFT correlators = SLE martingales

• Relation between noise strength and central charge:

M. Bauer and D. Bernard, 2002R. Friedrich and W. Werner, 2002

A path in a uniform spanning tree: loop-erased random walk

Self-avoiding walk

Self-avoiding walk (coupling to GFF, courtesy Scott Sheffield)

Ising spin cluster boundary (coupling to GFF, courtesy Scott Sheffield)

Double domino tilings

Level lines of Gaussian free field (courtesy Scott Sheffield)

Boundary of a percolation cluster

Random Peano curve

Calculations with SLE

• Shift

• Simple way of deriving crossing probabilities, various critical exponents and scaling functions

• SLE as a Langevin equation

• Langevin dynamics diffusion equation

• Multifractal spectra for critical clusters

Crossing probability

J. Cardy, 1992

Crossing probability

L. Carleson

S. Smirnov, 2001

“Most difficult theorem about the identity function”

P. Jones

Conformal multifractality

B. Duplantier, 2000

• For critical clusters with central charge

• Can obtain from SLE

• Can now obtain this and more using traditional CFT

• Originally obtained by quantum gravity

E. Bettelheim, I. Rushkin, IAG, P. Wiegmann, 2005A. Belikov, IAG, I. Rushkin, 2008

D. Belyaev, S.Smirnov, 2008

Applications of SLE

• Numerical applications based on effective “zipper” algorithms that test ensembles of curves for conformal invariance

• Each curve is discretized and “unzipped”, giving the driving function

• These functions are tested as Brownian motions (Gaussianity, independence of increments

D. Marshall

T. Kennedy

Applications of SLE

• 2D turbulence D. Bernard, G. Boffetta, A. Celani, G. Falkovich, 2006

• Zero vorticity contours are (percolation?)

• Temperature isolines are (Gaussian free field?)

Applications of SLE

• 2D quantum chaosJ. P. Keating, J. Marklof, and I. G. Williams, 2006

E. Bogomolny, R. Dubertrand, C. Schmit, 2006

• Nodal lines of chaotic wave functions are with (expect percolation ) · = 6· ¼5:3 (6:05;5:92)

Applications of SLE

• 2D Ising spin glass

C. Amoruso, A. K. Hartmann, M. B. Hastings, M. A. Moore, 2006D. Bernard, P. Le Doussal, A. Middleton, 2006

• Domain walls are with

(somewhat disappointing: nothing special about this value)

Applications of SLE

• Morphology of thin ballistically deposited films

• Height isolines are (Ising?)

Applications of SLE

• Morphology of KPZ surfaces

• Height isolines are (self-avoiding random walk?)

Multifractal spectrum is known numerically

What about growth patterns?

Both for DLA and Hele-Shaw patterns

Very few analytical results

Continuous Loewner chains

Growth along the whole boundary of a domain

Discrete Loewner chains

Iterated conformal maps M. Hastings and L. Levitov, 1998

Discrete Loewner chains

Excellent tool for generating and studying DLA-like patterns Variants and generalizations: interpolations between DLA and LG, Laplacian random walks, …

M. Stepanov and L. Levitov, 2001

Anderson transitions in 2D and conformal restriction

• Metal-insulator transitions induced by disorder • For most Anderson transitions no analytical results are available, even in 2D where conformal invariance is expected to help

• CFTs for Anderson transitions in 2D have

• Appropriate stochastic/geometric notion is conformal restriction

G. Lawler, O. Schramm, W. Werner, 2003

E. Bettelheim, IAG, A.W. W. Ludwig, in progress

Current directions and challenges

• Stochastic geometry of off-critical systems (massive SLE)

• Stochastic geometry in disordered systems: spin glasses, Anderson localization and quantum Hall transitions

• Stochastic geometry in non-equilibrium (driven) systems: turbulence, etc.

• Analysis of Loewner chains and applications to growth phenomena: Laplacian growth, diffusion-limited aggregation, etc.

• 3D random shapes: conformal maps are not useful…

3D DLA

• From http://math.mit.edu/~chr/research/3d-dla.htm