Random and complex encounters: physics meets math

1. Random and complex encounters:physics meets math Ilya A. Gruzberg University of Chicago TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

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 isthe 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. Mineral dendrites • Electrodeposition Random shapes in nature: growth patterns • Real patterns in nature: • Hele-Shaw flow

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

13. Critical clusters • Percolation clusters (site percolation) Every site is either ON (black) with probability or OFF (white) with probability

14. Critical clusters • Larger critical percolation clusters • Crossing probability

15. Critical clusters • Ising spin clusters

16. DLA applet

17. Growth patterns • Diffusion-limited aggregation T. Witten, L. Sander, 1981 • Many different variants

18. Viscous fingering vs. DLA O. Praud and H. L. Swinney, 2005

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 • 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

23. 2D critical phenomena 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

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

25. Crossing probability • Crossing probabilities J. Cardy, 1992 Is there a left to right crossing of white hexagons?

26. Crossing probability 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

28. 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

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

30. Loewner equation C. Loewner, 1923 • Describes upper half plane with a cut along a curve

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:

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

35. Self-avoiding walk

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

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

38. Double domino tilings

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

40. Boundary of a percolation cluster

41. Random Peano curve

42. 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

43. Crossing probability J. Cardy, 1992

44. Crossing probability L. Carleson S. Smirnov, 2001 “Most difficult theorem about the identity function” P. Jones

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

46. 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

47. Applications of SLE D. Bernard, G. Boffetta, A. Celani, G. Falkovich, 2006 • 2D turbulence • Zero vorticity contours are (percolation?) • Temperature isolines are (Gaussian free field?)

48. Applications of SLE 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 )

49. 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)

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