1. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of critical curves and CFT Ilya A. Gruzberg University of Chicago with E. Bettelheim, I. Rushkin, and P. Wiegmann

2. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 2D critical models Ising model Percolation

3. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Critical curves Focus on one domain wall using certain boundary conditions Conformal invariance: systems in simple domains. Typically, upper half plane

4. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Critical curves: geometry and probabilities Fractal dimensions Multifractal spectrum of harmonic measure Crossing probability Left vs. right passage probability Many more …

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

6. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure on a curve Electric field of a charged cluster

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

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

9. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Moments of harmonic measure Global moments Local moments fractal dimension Ergodicity

10. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure and conformal maps Harmonic measure is conformally invariant: Multifractal spectrum is related to derivative expectation values: connection with SLE. Use CFT methods

11. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Various uniformizing maps (1) (2) (3) (4)

12. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Correlators of boundary operators - partition function with modified BC - boundary condition (BC) changing operator - partition function

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

14. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Correlators of boundary operators Insert “probes” of harmonic measure: primary operators of dimension LHS: fuse RHS: statistical independence Need only -dependence in the limit

15. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Conformal invariance Map exterior of to by that satisfies Primary field Last factor does not depend on Put everything together:

16. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Mapping to Coulomb gas Stat mech models loop models height models Gaussian free field (compactified) L. Kadanoff, B. Nienhuis, J. Kondev

17. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Coulomb gas Parameters Phases (similar to SLE) Central charge densedilute

18. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Coulomb gas: fields and correlators Vertex “electromagnetic” operators Charges Holomorphic dimension Correlators and neutrality

19. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Curve-creating operators Magnetic charge creates a vortex in the field To create curves choose B. Nienhuis

20. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Curve-creating operators In traditional CFT notation - the boundary curve operator is - the bulk curve operator is with charge

21. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Multifractal spectrum on the boundary One curve on the boundary KPZ formula: is the gravitationally dressed dimension! The “probe”

22. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Generalizations: boundary Several curves on the boundary Higher multifractailty: many curves and points

23. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Higher multifractality on the boundary Consider Need to find Here

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

25. IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Generalizations: bulk Several curves in the bulk

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