1. Random and complex encounters: physics meets math
Ilya A. Gruzberg
University of Chicago
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2. 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

3. Brief history of encounters
20th century. Mostly one-way: physics borrows methods from math
Examples: general relativity, quantum mechanics
Einstein
Levi-Civita
Grossmann

4. 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)

5. Stochastic conformal geometry: applications in physics
Ilya A. Gruzberg
University of Chicago
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6. 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.

7. 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

8. 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

9. 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)

10. 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

11. Random shapes in nature: growth patterns
Real patterns in nature:
Hele-Shaw flow
Mineral dendrites
Electrodeposition
Explain what Hele-Shaw flow is.
Many of these growth processes are controlled by a Laplacian field outside the cluster.

12. Random shapes in nature: turbulence
Inverse cascade in 2D Navier-Stokes turbulence
Vorticity clusters
Zero vorticity lines

13. Critical clusters Percolation clusters (site percolation)
Every site is either ON (black) with probability
or OFF (white) with probability
Percolation is something that mathematicians perceive as their own turf. For them it is a subject from probability theory. This turned out to be one of the models where the biggest progress along SLE lines has occurred. Revenge of mathematicians.

14. Critical clusters Larger critical percolation clusters
Two examples of large critical percolation clusters. Note that on the left figure there is no spanning cluster.
Crossing probability

15. Critical clusters Ising spin clusters
Critical clusters exist close to continuous phase transitions, and involve many elementary degrees of freedom.

16. DLA applet

17. Growth patterns Diffusion-limited aggregation Many different variants
T. Witten, L. Sander, 1981
Explain how this works. Stress the colors and how the whole growth is very non-local in time and space.
Many different variants

18. Viscous fingering vs. DLA
O. Praud and H. L. Swinney, 2005
Comparison of viscous fingering and DLA with time evolution represented in colors.

19. 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 …

20. Stochastic geometry: multifractal exponents
Lumpy charge distribution on a cluster boundary
Cover the curve by small discs
of radius
Charges (probabilities) inside discs
Moments
Most striking property – multifractality. Mandelbrot: “Fractal is a set, multifractal is a measure”. Explain.
Non-linear is the hallmark of a multifractal

21. Harmonic measure on DLA cluster

22. Harmonic measure: electric field
Multifractal measures: electric field of a charged cluster
Logarithmic scale.

23. 2D critical phenomena Scale invariance
A. Patashinski, V. Pokrovsky, 1964
L. Kadanoff, 1966
Scale invariance
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
Brief survey of results about 2D critical phenomena. Conformal maps will play crucial role in what follows.

24. 2D critical stat mech models
A. Belavin, A. Polyakov, A. Zamolodchikov, 1984
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
Traditional approach: correlations of local observables. New focus is on the non-local structures: clusters and their boundaries, and finite geometries.

25. Crossing probability Crossing probabilities
J. Cardy, 1992
An interesting example of a correlation function – crossing probability. Drawn here site percolation on triangular lattice.
Is there a left to right crossing of white hexagons?

26. Crossing probability
Answer later
And now?

27. 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
The red cluster boundary can be “grown” step by step in fictitious discrete time. I will explain how this so called exploration process works for the site percolation on triangular lattice on the next slide.

28. Exploration process Cluster boundary can be “grown” step-by-step
Each step is determined by local environment
Evolution on fictitious time suggest that in the continuum limit the interface can be described in terms of a differential equation.
Description in terms of differential equations in continuum

29. 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
The main idea behind the recent developments is the use of conformal maps.

30. Loewner equation Describes upper half plane with a cut along a curve
C. Loewner, 1923
Describes upper half plane with a cut along a curve
Loewner equation describes so called slit domains, in this case, the upper half plane from which a portion of a curve has been cut.
A little history, perhaps? Two ways of thinking about this.

31. SLE postulates
Conformal invariance of the measure on curves

32. 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

33. 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:
Questions: internal symmetries (Wess-Zumino, current algebras, supersymmetry, etc.), other values of the central charge?

34. SLE versus CFT Duality For describes the trace describes its perimeter
B. Duplantier, 2000
Duality
For
describes the trace
describes its perimeter
Example: percolation hull with
while external perimeter with

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

36. Self-avoiding walk

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

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

39. Double domino tilings

40. Level lines of Gaussian free field (courtesy Scott Sheffield)

41. Boundary of a percolation cluster

42. Random Peano curve

43. Calculations with SLE SLE as a Langevin equation Shift
Langevin dynamics diffusion equation
Main point of usefulness and simplicity.
Simple way of deriving crossing probabilities, various
critical exponents and scaling functions
Multifractal spectra for critical clusters

44. Crossing probability J. Cardy, 1992
Methods of CFT are used, and conformal mapping between the UHP and a rectangle.

45. Crossing probability L. Carleson S. Smirnov, 2001
“Most difficult theorem
about the identity function”
P. Jones
Simplest geometry, basis for Smirnov’s proof.

46. Conformal multifractality
Originally obtained by quantum gravity
B. Duplantier, 2000
For critical clusters with central charge
Can obtain from SLE
Can now obtain this and more using traditional CFT
D. Belyaev, S.Smirnov, 2008
Tour de force.
E. Bettelheim, I. Rushkin, IAG, P. Wiegmann, 2005
A. Belikov, IAG, I. Rushkin, 2008

47. 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

48. 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?)
Spectacular applications

49. Applications of SLE 2D quantum chaos
J. P. Keating, J. Marklof, and I. G. Williams, 2006
2D quantum chaos
E. Bogomolny, R. Dubertrand, C. Schmit, 2006
Nodal lines of chaotic wave functions are with
(expect percolation )
Spectacular applications

50. Applications of SLE 2D Ising spin glass Domain walls are with
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)
Spectacular applications

51. Applications of SLE Morphology of thin ballistically deposited films
Height isolines are (Ising?)

52. Applications of SLE Morphology of KPZ surfaces
Height isolines are (self-avoiding random walk?)

53. What about growth patterns?
Both for DLA and Hele-Shaw patterns
A lot is known but almost nothing analytically. New approach through Loewner chains. Hopefully will produce results similar to the successes of SLE.
Multifractal spectrum is known numerically
Very few analytical results

54. Continuous Loewner chains
Growth along the whole boundary of a domain

55. Discrete Loewner chains
Iterated conformal maps
M. Hastings and L. Levitov, 1998
Grow a bump on the circle, then grow again…

56. Discrete Loewner chains
M. Stepanov and
L. Levitov, 2001
Excellent tool, but mostly numerical, again.
Excellent tool for generating and studying DLA-like patterns
Variants and generalizations: interpolations between DLA
and LG, Laplacian random walks, …

57. 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

58. 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…
A lot is known but almost nothing analytically.

59. 3D DLA
From