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

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

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 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.

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 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.

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

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

DLA applet

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

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

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 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

Harmonic measure on DLA cluster

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

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.

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.

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?

Crossing probability Answer later 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 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.

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

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.

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.

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

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

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 Main point of usefulness and simplicity. Simple way of deriving crossing probabilities, various critical exponents and scaling functions Multifractal spectra for critical clusters

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

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

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

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?) Spectacular applications

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

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

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 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

Continuous Loewner chains Growth along the whole boundary of a domain

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

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, …

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… A lot is known but almost nothing analytically.

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