Transcript rsws1 6685

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
2D critical models
Ising model
Percolation
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
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 …
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
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure on a curve
• Electric field of a charged cluster
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Multifractal exponents
• Lumpy charge distribution on a cluster boundary
• Cover the curve by
of radius
small discs
• Charges (probabilities) inside discs
• Moments
• Non-linear
• Problem: find
is the hallmark of a multifractal
for critical curves
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
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
Moments of harmonic measure
• Global moments
fractal dimension
• Local moments
• Ergodicity
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
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Various uniformizing maps
(1)
(2)
(3)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
(4)
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
Correlators of boundary operators
• Two step averaging:
M. Bauer, D. Bernard
1. Average over microscopic degrees of freedom
in the presence of a given curve
2. Average over all curves
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
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
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
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
Coulomb gas
• Parameters
• Phases (similar to SLE)
dilute
dense
• Central charge
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
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Curve-creating operators
B. Nienhuis
• Magnetic charge creates a vortex in the
• To create
field
curves choose
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
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
Multifractal spectrum on the boundary
• One curve on the boundary
• The “probe”
• KPZ formula:
is the gravitationally dressed dimension!
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: 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 boundary
• Need to find
• Consider
• Here
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
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: bulk
• Several curves in the bulk
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
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