Harmonic measure of critical curves and CFT

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

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

3. Critical 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

4. Critical 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

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

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

7. 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 • Problem: find for critical curves IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

8. Conformal multifractality • Originally obtained by quantum gravity B. Duplantier, 2000 • For critical clusters with central charge • We obtain this and more using traditional CFT • Our method is not restricted to IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

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

10. Harmonic 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

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

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

13. Correlators of boundary operators • Average over microscopic degrees of freedom • in the presence of a given curve • 2. Average over all curves M. Bauer, D. Bernard • Two step averaging: IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007

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

15. Conformal 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

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

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

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

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

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

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

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

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

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

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

26. Open questions • Spatial structure of harmonic measure on stochastic curves • Prefactor in • related to structure constants in CFT • Stochastic geometry in critical systems with additional • symmetries: Wess-Zumino models, W-algebras, etc. • Stochastic geometry of growing clusters: DLA, etc: • no conformal invariance… IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007