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11
Stochastic Loewner Evolution and Statistical Mechanics
Part 1/2 KATORI, Makoto (Chuo University)
香取眞理 [かとりまこと](中央大学)
第53回函数論シンポジウム
2010年11月21-23日
名城大学名鉄サテライト
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22
from http://www.emis.math.ca/EMIS/mirror/IMU/medals/2006/
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3
from http://www.icm2010.org.in/prize-winners-2010/fields-medal
The 2010 Fields medalsStanislav Smirnov
for the proof of conformal invariance of percolation and the planar Ising model in statistical physics.
Section de Mathématiques, Université de Genève
http://www.icm2010.org.in/wp-content/icmfiles/medalists/stan.jpg
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441. Statistical Mechanics Models and Measures on Continuous Path Space
1.1 Scaling limits of planar lattice models
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551. Statistical Mechanics Models and Measures on Continuous Path Space
1.1 Scaling limits of planar lattice models
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661. Statistical Mechanics Models and Measures on Continuous Path Space
1.1 Scaling limits of planar lattice models
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771. Statistical Mechanics Models and Measures on Continuous Path Space
1.1 Scaling limits of planar lattice models
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881. Statistical Mechanics Models and Measures on Continuous Path Space
1.1 Scaling limits of planar lattice models
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99
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1010
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1111
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1212
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1313
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1414
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1515
[0th example] RW on Square Latticed
Complex BMcont. limit
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1616[1st
example]
Loop‐Erased RW: LERW
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1717[1st
example]
Loop‐Erased RW: LERW
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1818[1st
example]
Loop‐Erased RW: LERW
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1919[1st
example]
Loop‐Erased RW: LERW
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2020[1st
example]
Loop‐Erased RW: LERW
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2121[1st
example]
Loop‐Erased RW: LERW
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2222Continuum Limit of Loop‐Erased RW: LERW
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2323Continuum Limit of Loop‐Erased RW: LERW
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2424[2nd
example] Self‐Avoiding Walk : SAW (自己回避ウォーク)]
Figures (a1)
n = 200 steps,(a2)
n = 800 steps
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2525Continuum Limit of SAWs
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2626[3rd
example] critical percolation model(臨界浸透模型)
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2727[3rd
example] critical percolation model(臨界浸透模型)
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2828percolation exploration process(浸透探索過程)
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2929Continuum limit of percolation exploration process
Figures (b1) on 35×35 triangular lattice
(b2) on 100 ×100 triangular lattice
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3030[4th
example] critical Ising
model (臨界 Ising
模型)
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3131
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3232Ising
Interface (Ising
界面)
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33331.2 Conformal Invariance and Domain
Markov Property
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3434The following two properties are expected. ①
conformal covariance and conformal invariance
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3535
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3636② Domain
Markov Property
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3737In some
special cases the measures can have the additional properties. 1.3 Restriction Property and Locality Property
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3838In some
special cases the measures can have the additional properties. 1.3 Restriction Property and Locality Property
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3939③ Restriction Property
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4040④
Locality property
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4141④
Locality Property
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4242④
Locality Property
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43
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44
Ensembles of Discrete Paths in
Statistical Mechanics Models on Planar Lattice
[1] Loop-Erased Random Walks ⇒
SLE2 (κ=2)① conformal ② domain Markov
[2] Self-Avoiding Walks ⇒
SLE8/3 (κ=8/3)① conformal ② domain Markov ③ restriction
[3] Percolation Exploration Process ⇒
SLE6 (κ=6)① conformal ② domain Markov ④ locality
[4] Ising Interface ⇒
SLE3 (κ=3)① conformal ② domain Markov
Conformally CovariantMeasures of
Continuous Paths(boundary scaling exponent b)
cont.limit(scaling limit)
ν =1/d
Stochastic Loewner Evolution�and Statistical Mechanics�Part 1/2�KATORI, Makoto (Chuo University)�香取眞理 [かとりまこと](中央大学)スライド番号 2スライド番号 3 1. Statistical Mechanics Models and Measures on � Continuous Path Space�1.1 Scaling limits of planar lattice modelsスライド番号 5スライド番号 6スライド番号 7スライド番号 8スライド番号 9スライド番号 10スライド番号 11スライド番号 12スライド番号 13スライド番号 14スライド番号 15[1st example] Loop-Erased RW: LERWスライド番号 17スライド番号 18スライド番号 19スライド番号 20スライド番号 21スライド番号 22スライド番号 23[2nd example] Self-Avoiding Walk : SAW (自己回避ウォーク)] Continuum Limit of SAWs[3rd example] critical percolation model(臨界浸透模型)[3rd example] critical percolation model(臨界浸透模型)percolation exploration process(浸透探索過程)Continuum limit of percolation exploration process[4th example] critical Ising model�(臨界 Ising 模型)スライド番号 31Ising Interface (Ising 界面)1.2 Conformal Invariance � and Domain Markov PropertyThe following two properties are expected.�① conformal covariance and conformal invarianceスライド番号 35② Domain Markov Property In some special cases the measures can have the additional properties.�1.3 Restriction Property and Locality PropertyIn some special cases the measures can have the additional properties.�1.3 Restriction Property and Locality Property③ Restriction Property④ Locality property④ Locality Property④ Locality Propertyスライド番号 43Ensembles of�Discrete Paths in�Statistical Mechanics Models�on Planar Lattice