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Fractal Geometry of Critical Phenomena
John Cardy
University of Oxford
PITP Showcase Conference – May 2005
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I self-similar fractal objects are ubiquitous in natureI cloudsI coastlinesI broccoli
I deterministic mathematical fractals are well-studied
I but most natural fractals arerandomand therefore more difficultto characterize
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I self-similar fractal objects are ubiquitous in natureI cloudsI coastlinesI broccoli
I deterministic mathematical fractals are well-studied
I but most natural fractals arerandomand therefore more difficultto characterize
![Page 4: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/4.jpg)
I critical phenomenaare an important area in which randomfractals arise
I they are well-studied from other points of view (renormalizationgroup, field theory, exact solutions,. . .)
I in 2d they obey the powerful symmetry ofconformal invariance
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I critical phenomenaare an important area in which randomfractals arise
I they are well-studied from other points of view (renormalizationgroup, field theory, exact solutions,. . .)
I in 2d they obey the powerful symmetry ofconformal invariance
![Page 6: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/6.jpg)
Random fractals in general
I they are onlystatisticallyself-similar
I all physical fractals are cut-off at short scalesa and long scalesRI fractal dimensiondf :
I mean mass∼ (R/a)df
I box-counting: minimum # of discs of radiusε required to coverobject∼ (R/ε)df
I if fractal embedded inD-dimensional euclidean space,probability that a pointr lies within ε of the object∼ εD−df
![Page 7: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/7.jpg)
Random fractals in general
I they are onlystatisticallyself-similar
I all physical fractals are cut-off at short scalesa and long scalesRI fractal dimensiondf :
I mean mass∼ (R/a)df
I box-counting: minimum # of discs of radiusε required to coverobject∼ (R/ε)df
I if fractal embedded inD-dimensional euclidean space,probability that a pointr lies within ε of the object∼ εD−df
![Page 8: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/8.jpg)
Random fractals in general
I they are onlystatisticallyself-similar
I all physical fractals are cut-off at short scalesa and long scalesRI fractal dimensiondf :
I mean mass∼ (R/a)df
I box-counting: minimum # of discs of radiusε required to coverobject∼ (R/ε)df
I if fractal embedded inD-dimensional euclidean space,probability that a pointr lies within ε of the object∼ εD−df
![Page 9: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/9.jpg)
Random fractals in general
I they are onlystatisticallyself-similar
I all physical fractals are cut-off at short scalesa and long scalesRI fractal dimensiondf :
I mean mass∼ (R/a)df
I box-counting: minimum # of discs of radiusε required to coverobject∼ (R/ε)df
I if fractal embedded inD-dimensional euclidean space,probability that a pointr lies within ε of the object∼ εD−df
![Page 10: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/10.jpg)
2d critical lattice models
I prototype is Ising model:
Z =∏
r
∑
s(r)=±1
e−βE({s(r)})
wheres(r) = ±1 and e.g.E = −J∑
nn s(r)s(r ′)I critical point atβ = βc
I but, from the geometrical point of view, evenβ = 0 isinteresting:
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Percolation
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Self-avoiding Walk
SA
W in
pla
ne -
1,0
00,0
00 s
teps
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I conventional questions involvelocal correlation functions:I Ising spin-spin correlation〈s(r1)s(r2)〉I probabilityr1 andr2 are in same percolation clusterI number of SAWs (weighted byµ−length) from r1 to r2
I these all∼ |a/(r1 − r2)|2x at criticalityI x is a critical exponent which depends on what kind of quantity is
being measured but is otherwiseuniversalI but there are other more geometrical universal quantities, eg:
I probability thatr1 andr2 lie near the same Ising cluster boundaryI probability two opposite sides of a rectangle connected by a
percolation clusterI probability that a self-avoiding loop starting and ending atr1
encloses a given pointr2
I not obviously related to local correlation functions
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I conventional questions involvelocal correlation functions:I Ising spin-spin correlation〈s(r1)s(r2)〉I probabilityr1 andr2 are in same percolation clusterI number of SAWs (weighted byµ−length) from r1 to r2
I these all∼ |a/(r1 − r2)|2x at criticalityI x is a critical exponent which depends on what kind of quantity is
being measured but is otherwiseuniversalI but there are other more geometrical universal quantities, eg:
I probability thatr1 andr2 lie near the same Ising cluster boundaryI probability two opposite sides of a rectangle connected by a
percolation clusterI probability that a self-avoiding loop starting and ending atr1
encloses a given pointr2
I not obviously related to local correlation functions
![Page 15: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/15.jpg)
Methods
I exact solution of the lattice model (1950s)I limited to a few modelsI not clear what is universal
I conformal field theory/Coulomb gas methods (1980s)I very powerful, butI lack of systematic mathematical basisI often some guesswork involved
I SLE(stochastic Loewner evolution)[O Schramm; GLawler and W Werner] (2000s)
I clear mathematical basisI in some cases (eg percolation) a rigorous connection to lattice
modelI adapted to answering geometrical questions
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Methods
I exact solution of the lattice model (1950s)I limited to a few modelsI not clear what is universal
I conformal field theory/Coulomb gas methods (1980s)I very powerful, butI lack of systematic mathematical basisI often some guesswork involved
I SLE(stochastic Loewner evolution)[O Schramm; GLawler and W Werner] (2000s)
I clear mathematical basisI in some cases (eg percolation) a rigorous connection to lattice
modelI adapted to answering geometrical questions
![Page 17: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/17.jpg)
Methods
I exact solution of the lattice model (1950s)I limited to a few modelsI not clear what is universal
I conformal field theory/Coulomb gas methods (1980s)I very powerful, butI lack of systematic mathematical basisI often some guesswork involved
I SLE(stochastic Loewner evolution)[O Schramm; GLawler and W Werner] (2000s)
I clear mathematical basisI in some cases (eg percolation) a rigorous connection to lattice
modelI adapted to answering geometrical questions
![Page 18: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/18.jpg)
I describing the fractal soup of all these curves is too hard
I focus on just one of them, picked out by the boundary conditions
I this curve can be ‘grown’ dynamically on the lattice:
I SLEdescribes the continuous version of this process
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I describing the fractal soup of all these curves is too hard
I focus on just one of them, picked out by the boundary conditions
I this curve can be ‘grown’ dynamically on the lattice:
I SLEdescribes the continuous version of this process
![Page 20: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/20.jpg)
I describing the fractal soup of all these curves is too hard
I focus on just one of them, picked out by the boundary conditions
I this curve can be ‘grown’ dynamically on the lattice:
I SLEdescribes the continuous version of this process
![Page 21: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/21.jpg)
Loewner evolution
I instead of the curve itself, consider the (unique) conformalmappinggt(z) which sends the region outside the curve to thehalf-plane
ta
gt
I gt sends the growing tip toat
I it satisfies a simple equation
dgt(z)dt
=2
gt(z)− at
I so instead of thinking about random curves we can think aboutrandom (continuous) functionsat
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Loewner evolution
I instead of the curve itself, consider the (unique) conformalmappinggt(z) which sends the region outside the curve to thehalf-plane
ta
gt
I gt sends the growing tip toat
I it satisfies a simple equation
dgt(z)dt
=2
gt(z)− at
I so instead of thinking about random curves we can think aboutrandom (continuous) functionsat
![Page 23: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/23.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 24: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/24.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 25: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/25.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 26: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/26.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 27: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/27.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 28: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/28.jpg)
StochasticLoewner evolution
I if the measure on the curves is conformally invariant,at must be1d Brownian motion[Schramm]
I diffusion constantκ: 〈a2t 〉 = κt
I different values ofκ gives different universality classes of 2dcritical behavior, eg:
I κ = 83 self-avoiding walks
I κ = 3 Ising domain wallsI κ = 4 level lines at roughening transitionI κ = 6 percolation cluster boundaries
I problems in 2d critical behavior reduced to calculations in 1dBrownian motion
![Page 29: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/29.jpg)
Fractal dimension
gt
df = 1 + κ/8 (κ ≤ 8)
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I many more analytic results which characterize the fractalgeometry of 2d critical phenomena, eg:
I crossing probabilitiesI multifractal exponentsI as well as all the more conventional exponentsν, η, etc
I outlook: extensions to other 2d random fractalsI diffusion limited aggregationI turbulenceI broccoli?
![Page 31: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/31.jpg)
I many more analytic results which characterize the fractalgeometry of 2d critical phenomena, eg:
I crossing probabilitiesI multifractal exponentsI as well as all the more conventional exponentsν, η, etc
I outlook: extensions to other 2d random fractalsI diffusion limited aggregationI turbulenceI broccoli?
![Page 32: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/32.jpg)
I many more analytic results which characterize the fractalgeometry of 2d critical phenomena, eg:
I crossing probabilitiesI multifractal exponentsI as well as all the more conventional exponentsν, η, etc
I outlook: extensions to other 2d random fractalsI diffusion limited aggregationI turbulenceI broccoli?
![Page 33: Fractal Geometry of Critical Phenomena€¦ · John Cardy University of Oxford PITP Showcase Conference – May 2005. I self-similar fractal objects are ubiquitous in nature I clouds](https://reader036.vdocuments.us/reader036/viewer/2022081407/6056d0f5f17e72183545199d/html5/thumbnails/33.jpg)
Some References for Physicists
I W. Kager and B. Nienhuis,A guide to stochastic Loewnerevolution and its applications,math-ph/0312251
I J. Cardy,SLE for theoretical physicists,cond-mat/0503313