Transcript ann2010 9484

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
Galilei
Newton
•
Even in 19th century mathematicians knew physics well (F. Klein)
•
By the turn of 20th century – clear separation of physics and math
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:
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.
• Not “conformal geometry”. From Wikipedia:
In mathematics, conformal geometry is the study of the set of anglepreserving (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:
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2001: Conformal Field Theory and Applications
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2003: Symplectic Geometry and Physics
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2004: Multiscale Geometry and Analysis in High Dimensions
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2007: Random Shapes
Workshop:
•
2009: Laplacian Eigenvalues and Eigenfunctions
Random shapes in nature: growth patterns
• Real patterns in nature:
• Hele-Shaw flow
• Mineral dendrites
• Electrodeposition
Random shapes in nature: turbulence
• Inverse cascade in 2D Navier-Stokes turbulence
• Vorticity clusters
• Zero vorticity lines
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
• Many different variants
T. Witten, L. Sander, 1981
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
• Cover the curve by
of radius
small discs
• Charges (probabilities) inside discs
• Moments
• Non-linear
is the hallmark of a multifractal
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 scales
The basis for renormalization group approach
• Conformal invariance
• In 2D conformal maps = analytic functions
A. Polyakov, 1970
2D critical stat mech models
• Traditional approach:
A. Belavin, A. Polyakov, A. Zamolodchikov, 1984
(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
Crossing probability
• Crossing probabilities
J. Cardy, 1992
Is there a left to right crossing of white hexagons?
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
C. Loewner, 1923
• Describes upper half plane with a cut along a curve
SLE postulates
• Conformal invariance of the measure on curves
Schramm-Loewner evolution
O. Schramm, 1999
• Conformal invariance leads uniquely to Loewner equation driven
by a Brownian motion:
• Noise strength
is an important parameter:
SLE generates conformally-invariant measures on random curves
that are continuum limits of critical cluster boundaries
SLE versus CFT
• SLE describes all critical 2D systems with
CFT correlators = SLE martingales
M. Bauer and D. Bernard, 2002
R. Friedrich and W. Werner, 2002
• Relation between noise strength and central charge:
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
• SLE as a Langevin equation
• Shift
• Langevin dynamics
diffusion equation
• Simple way of deriving crossing probabilities, various
critical exponents and scaling functions
• 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
• Originally obtained by quantum gravity
B. Duplantier, 2000
• For critical clusters with central charge
• Can obtain from SLE
D. Belyaev, S.Smirnov, 2008
• Can now obtain this and more using traditional CFT
E. Bettelheim, I. Rushkin, IAG, P. Wiegmann, 2005
A. Belikov, IAG, I. Rushkin, 2008
Applications of SLE
•
Numerical applications based on effective “zipper” algorithms that test
ensembles of curves for conformal invariance
D. Marshall
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Each curve is discretized and “unzipped”, giving the driving function
•
These functions are tested as Brownian motions (Gaussianity,
independence of increments
T. Kennedy
Applications of SLE
• 2D turbulence
D. Bernard, G. Boffetta, A. Celani, G. Falkovich, 2006
• Zero vorticity contours are
• Temperature isolines are
(percolation?)
(Gaussian free field?)
Applications of SLE
• 2D quantum chaos
J. P. Keating, J. Marklof, and I. G. Williams, 2006
E. Bogomolny, R. Dubertrand, C. Schmit, 2006
• Nodal lines of chaotic wave functions are
· ¼ 5:3 (6:05; 5:92)
with
(expect percolation
· = 6
)
Applications of SLE
• 2D Ising spin glass
C. Amoruso, A. K. Hartmann, M. B. Hastings, M. A. Moore, 2006
D. 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?)
What about growth patterns?
Both for DLA and Hele-Shaw patterns
Multifractal spectrum
Very few analytical results
is known numerically
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
M. Stepanov and
L. Levitov, 2001
Excellent tool for generating and studying DLA-like patterns
Variants and generalizations: interpolations between DLA
and LG, Laplacian random walks, …
Anderson transitions in 2D and conformal restriction
E. Bettelheim, IAG, A.W. W. Ludwig, in progress
• 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
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