IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Harmonic measure of critical curves and CFT Harmonic measure of critical curves and CFT

Ilya A. GruzbergUniversity of Chicago

with

E. Bettelheim, I. Rushkin, and P. Wiegmann

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

2D critical models2D critical models

Ising model Percolation

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Critical curvesCritical curves

• Focus on one domain wall using certain boundary conditions

• Conformal invariance: systems in simple domains. Typically, upper half plane

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Critical curves: geometry and probabilitiesCritical curves: geometry and probabilities

• Fractal dimensions

• Multifractal spectrum of harmonic measure

• Crossing probability

• Left vs. right passage probability

• Many more …

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Harmonic measure on a curveHarmonic measure on a curve

• Probability that a Brownian particle hits a portion of the curve

• Electrostatic analogy: charge on the portion of the curve (total charge one)

• Related to local behavior of electric field: potential near wedge of angle

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Harmonic measure on a curveHarmonic measure on a curve

• Electric field of a charged cluster

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Multifractal exponentsMultifractal exponents

• Lumpy charge distribution on a cluster boundary

• Non-linear is the hallmark of a multifractal

• Problem: find for critical curves

• Cover the curve by small discs of radius

• Charges (probabilities) inside discs

• Moments

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Conformal multifractalityConformal multifractality

B. Duplantier, 2000

• For critical clusters with central charge

• We obtain this and more using traditional CFT Our method is not restricted to

• Originally obtained by quantum gravity

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Moments of harmonic measureMoments of harmonic measure

• Global moments

• Local moments

fractal dimension

• Ergodicity

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Harmonic measure and conformal mapsHarmonic measure and conformal maps

• Harmonic measure is conformally invariant:

• Multifractal spectrum is related to derivative expectation values: connection with SLE.

• Use CFT methods

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Various uniformizing mapsVarious uniformizing maps

(1) (2)(3) (4)

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Correlators of boundary operatorsCorrelators of boundary operators

- partition function with modified BC

- boundary condition (BC) changing operator

- partition function

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Correlators of boundary operatorsCorrelators of boundary operators

• Two step averaging:

1. Average over microscopic degrees of freedom in the presence of a given curve

2. Average over all curves

M. Bauer, D. Bernard

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Correlators of boundary operatorsCorrelators of boundary operators

• Insert “probes” of harmonic measure: primary operators of dimension

• LHS: fuse

• RHS: statistical independence

• Need only -dependence in the limit

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Conformal invarianceConformal invariance

• Map exterior of to by that satisfies

• Primary field

• Last factor does not depend on

• Put everything together:

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Mapping to Coulomb gasMapping to Coulomb gas

• Stat mech models loop models height models Gaussian free field (compactified)

L. Kadanoff, B. Nienhuis, J. Kondev

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Coulomb gasCoulomb gas

• Parameters

• Phases (similar to SLE)

• Central charge

densedilute

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Coulomb gas: fields and correlatorsCoulomb gas: fields and correlators

• Vertex “electromagnetic” operators

• Charges

• Holomorphic dimension

• Correlators and neutrality

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Curve-creating operatorsCurve-creating operators

• Magnetic charge creates a vortex in the field

• To create curves choose

B. Nienhuis

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Curve-creating operatorsCurve-creating operators

• In traditional CFT notation

- the boundary curve operator is

- the bulk curve operator is

with charge

with charge

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Multifractal spectrum on the boundaryMultifractal spectrum on the boundary

• One curve on the boundary

• KPZ formula: is the gravitationally dressed dimension!

• The “probe”

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Generalizations: boundaryGeneralizations: boundary

• Several curves on the boundary

• Higher multifractailty: many curves and points

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Higher multifractality on the boundaryHigher multifractality on the boundary

• Consider

• Need to find

• Here

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Higher multifractality on the boundaryHigher multifractality on the boundary

• Write as a two-step average and map to UHP:

• Exponents are dimensions of

primary boundary operators with

• Comparing two expressions for , get

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Generalizations: bulkGeneralizations: bulk

• Several curves in the bulk

IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

Open questionsOpen questions

• Spatial structure of harmonic measure on stochastic curves

• Stochastic geometry in critical systems with additional symmetries: Wess-Zumino models, W-algebras, etc.

• Stochastic geometry of growing clusters: DLA, etc: no conformal invariance…

• Prefactor in related to structure constants in CFT