Seminar in SPB, April 4, 2011 Harmonic measure of critical curves and CFT Ilya A. Gruzberg University of Chicago with E. Bettelheim, I. Rushkin, and P. Wiegmann

Seminar in SPB, April 4, 2011 Outline Stochastic geometry of critical curves Harmonic measure of critical curves and conformal maps Harmonic measure and CFT correlators Coulomb gas and curve-creating operators

Seminar in SPB, April 4, D critical models Ising model Percolation

Seminar in SPB, April 4, 2011 Critical curves Focus on one domain wall using certain boundary conditions Conformal invariance: systems in simple domains. Typically, upper half plane

Seminar in SPB, April 4, 2011 Critical curves: geometry and probabilities Fractal dimensions Multifractal spectrum of harmonic measure Crossing probability Left vs. right passage probability Many more …

Seminar in SPB, April 4, 2011 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

Seminar in SPB, April 4, 2011 Harmonic measure on a curve Electric field of a charged cluster

Seminar in SPB, April 4, 2011 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

Seminar in SPB, April 4, 2011 Conformal multifractality B. Duplantier, 2000 For critical clusters with central charge We obtain this and more using traditional CFT. Not restricted to Originally obtained by quantum gravity

Seminar in SPB, April 4, 2011 Moments of harmonic measure Global moments Local moments fractal dimension

Seminar in SPB, April 4, 2011 Harmonic measure and conformal maps Harmonic measure is conformally invariant: Multifractal spectrum is related to derivative expectation values: connection with SLE. Use CFT methods

Seminar in SPB, April 4, 2011 Various uniformizing maps (1) (2) (3)

Seminar in SPB, April 4, 2011 Correlators of boundary operators - partition function with modified BC - boundary condition (BC) changing operator - partition function

Seminar in SPB, April 4, 2011 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

Seminar in SPB, April 4, 2011 Correlators of boundary operators Insert operators (“probes” of harmonic measure) of dimension LHS: fuse RHS: statistical independence Need only -dependence in the limit

Seminar in SPB, April 4, 2011 Conformal invariance Map exterior of to by that satisfies Primary field Last factor does not depend on Put everything together:

Seminar in SPB, April 4, 2011 Mapping to Coulomb gas Stat mech models loop models height models Gaussian free field (compactified) L. Kadanoff, B. Nienhuis, J. Kondev

Seminar in SPB, April 4, 2011 Coulomb gas Parameters Phases (similar to SLE) Central charge densedilute

Seminar in SPB, April 4, 2011 Coulomb gas: fields and correlators Vertex “electromagnetic” operators Charges Holomorphic dimension Correlators and neutrality

Seminar in SPB, April 4, 2011 Curve-creating operators Magnetic charge creates a vortex in the field To create curves choose

Seminar in SPB, April 4, 2011 Curve-creating operators In traditional CFT notation - the boundary curve operator is - the bulk curve operator is

Seminar in SPB, April 4, 2011 Multifractal spectrum on the boundary One curve on the boundary KPZ formula: is the gravitationally dressed dimension! The “probe”

Seminar in SPB, April 4, 2011 Generalizations: boundary Several curves on the boundary Higher multifractailty: many curves and points

Seminar in SPB, April 4, 2011 Generalizations: bulk Several curves in the bulk

Seminar in SPB, April 4, 2011 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…