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1 Schramm-Loewner Evolution H H H H H. H γ H t t U t U t g t (z) t = g t (z) U t, g 0 (z) =z, t 0 t g t (z) t 0 g t (z) H t γ(0,t] B t κ >0 Schramm 000 U t = κb t H U t = κb t κ Schramm- Loewner Evolution (SLE κ ) SLE κ κ SLE κ κ 006 Werner SLE κ SLE κ Summer School 009 ( ) Loewner SLE (00-4 ) : katori@phys.chuo-u.ac.jp HP

2 d- (BES d ) BES d SLE κ SLE κ BES d SLE κ Schramm A 37 A A A.3 capacity A.4 Φ t (z), Φ t (z) (Φ t (U t)) b SDE A.5 SLE C (S = Z Z),. z S, n Wn ω z = =(ω(0),...,ω(n)) : ω(0) = z,ω(i) S, ω(i) ω(i ) =, i n (RW) Wn z =4 n ω Wn z 4 n C D 0 = x + y : <x<, 0 <y<, D 0 O =0( ), P = N N,,... N NO =0 NP =N RW ND 0 Ω N (D 0 ; O, P) () : Z N (D 0 ; O, P) = ω Ω N (D 0 ;O,P ) ω ω N 4 ω. (.) Z N (D 0 ; O, P) C(D 0 ; O, P)N, N (.)

3 f(n) g(n), N f(n) g(n), N C(D 0 ; O, P) D 0 H D0 (,P) O D 0 NP=N - NP ND O P= - O D O N O : [ ] D 0 N [ ] NO = O NP =N ND 0 (loop-erased RW: LERW) Ω N (D 0 ; O, P) ω =(ω(0),ω(),...) ω(i) =ω(j),i < j ω ω ω =( ω(0), ω(),...) (i) t 0 =0, ω(0) = ω(t 0 )=0 (ii) m t m = max l>t m : ω(l) = ω(t m +), ω(m) = ω(t m )= ω(t m +) ND 0 O NP Ω 0 N (D 0; O, P) RW Ω 0 N (D 0; O, P) RW ω 4 w (LERW) LERW LERW ω =(ω(0),...,ω( ω )) ν>0 ( ) i ω /N = ω(i), 0 i ω (.3) N N /ν 3

4 ω /N O, ω /N /ν P = D 0 ( /N ) ν N, P γ P t γ = lim N ω /N /ν : γ :(0,t γ ) D 0, lim t 0 γ(t) =O, lim γ(t) =P, t γ (0, ). (.4) t tγ γ d LERW = ν γ, γ(t ) γ(t ), 0 t <t t γ LERW (.4) K LERW (D 0 ; O, P) μ LERW (D 0 ;O,P ) C(D 0; O, P) μ LERW (D 0 ;O,P ) ( )=C(D 0; O, P)μ LERW (D 0 ;O,P ) ( ) (.5) K LERW (D 0 ; O, P) μ LERW (D 0 ;O,P ) (self-avoiding walk : SAW) Wn z Wn,0 z = ω Wn. z : 0 i<j n ω(i) ω(j) W z n,0 < W z n =4 n <e β < 3 W z n,0 e βn, n f(n) g(n), n log f(n) log g(n), n ω e β ω (SAW) (.) SAW ZN SAW (D 0 ; O, P) = e β ω ω Ω 0 N (D 0;O,P ) b SAW > 0 Z SAW N (D 0 ; O, P) C SAW (D 0 ; O, P)N b SAW, N (.6) LERW RW (.) (.) (.6) LERW b LERW =. SAW γ d SAW LERW d LERW (.5) SAW μ SAW (D 0 ;O,P ) ( )=CSAW (D 0 ; O, P)μ SAW (D 0 ;O,P ) ( ) e β SAW connective constant S S.638 4

5 (critical percolation model) NP _ Λ Ν + Λ Ν 0 : T H. 0 C T τ = exp(π /3), a = 3, z 0 = a T = z 0 +(i + jτ) 3a : i, j Z. T a H O NP =N,N N z T η(z) 0, Bernoulli ν p, 0 p ν p (η(z) = ) = p, ν p (η(z) =0)= p. T 0 p / p >/ p c = (.7) ( p c T (.7). Bernoulli N b per =0 5

6 N N T ND 0 =Λ N N =6 Λ N z T O NP Λ + N, Λ N η(z) =, z Λ + N, η(z) =0, z Λ N (Dobrushin ) η. Λ N ν p η 0, Λ N H ND 0 O ω 0 ( ) ( ). percolation exploration process) ν>0 (.3) N d per = ν (.4) γ 3 γ μ per (D 0 ;O,P ) ( ) (B) 3: N γ (critical Ising model) Λ N =Λ N Λ + N Λ N z Λ N σ(z), ( ). σ(z) =, z Λ + N, σ(z) =, z Λ N (Dobrushin ) 6

7 σ Λ N Λ N ( Λ + N )c ( Λ N )c Dobrushin σ, Λ N E(σ) = β>0 Gibbs z,z Λ N : z z = 3a π N,β (σ) = e βe(σ) Z N,β, Z N,β = σ(z)σ(z ) σ, Λ N e βe(σ) β ω ( + ) H ND 0 (Ising interface) β T β c = 4 log 3 = e βc = 3 d Ising (.4). γ μ Ising (D 0 ;O,P ) ( ). f D 0 C f (z) 0, z D 0 f : D 0 f(d 0 ) (.8) f, D 0 O, P f(d 0 ) f(o),f(p ) 4. γ μ (D0 ;O,P )( )=C(D 0 ; O, P)μ (D0 ;O,P )( ) (.9) (conformal covariance) (conformal invariance) (.8) f μ (D0 ;O,P )( )= f (O) b f (P ) b μ (f(d0 );f(o),f(p ))( ) (.0) b. N (.0) (boundary scaling exponent) (.0) : C(D 0 ; O, P) = f (O) b f (P ) b C(f(D 0 ); f(o),f(p )) : μ (D0 ;O,P )( )=μ (f(d0 );f(o),f(p ))( ).. (.) (.6) ND 0 /N D 0 f f (z) /N b. N b 7

8 f (P) P f O f (O) 4: f D 0 f(d 0 ) D 0 O P f(d 0 ) f(o),f(p ) O P f(o) f(p ) (domain Markov property) μ (D0 ;O,P ) γ γ(0,t],t (0,t γ ) D 0 γ(0,t] γ(t) γ(t γ )=P ( ) γ(0,t] = μ (D0 \γ(0,t];γ(t),p )( ). μ (D0 ;O,P ) 5 P P γ (t) γ (t) O O 5: [ ] γ γ(0,t] P γ(t, t γ ) [ ] D 0 γ(0,t] γ(t) P.. γ (.4) ( ) t [0,t γ ] (.8) 8

9 (.3) f γ f(γ[t,t ]), 0 <t <t <t γ, t t f (γ(s)) d ds d γ θ :[0,t γ ] [0,t γ ] γ(t) γ(θ(t)).3 (.9) / (restriction property) D 0 D D 0 O, P D. LERW μ LERW (D ;O,P ), O P RW RW LERW LERW Radon-Nikodym dμ LERW (D ;O,P ) dμ LERW (γ) <, D D 0, D D 0 (D 0 ;O,P ) SAW dμ SAW (D ;O,P ) dμ SAW (γ) =γ(0,t γ ) D, D D 0 (.) (D 0 ;O,P ) ω ω ω ω =, ω = 0). (.) η Bernoulli η- μ per. ( μ Ising ) (locality property) D D 0 O, P D μ per (D ;O,P )(γ(0,t]) = μper (D 0 ;O,P ) (γ(0,t])γ(0,t) D, t (0,t γ ). (.) (.) γ(0,t γ ) (.) γ(0,t],t (0,t γ ). (Ω, F, P) Ω Ω A Ω F σ- ( (i) Ω F, (ii) A F A A c F, (iii) A,A,..., F n A n F, 3.) P 9

10 () Ω f F- a ω Ω:f(ω) a F [6, 4] (filtration, ) F t t 0, (i) F s F t F, 0 s<t, (ii) t F t σ (Ω, F, P; F t t 0 ) F t - (Brownian motion) B t (, BM.) (i) 0 <s<t B t B s F t - F s 0, t s ( ) b P B t B s [a, b] = exp x dx. (.) π(t s) (t s) a (ii) t B t Ω Ω s.t. P( Ω) =, ω Ω Bt (ω) t. (i) c>0 c B c t B t c B c t (d distribution.) property) d = B t c>0 (.) BM (scaling B t,b t,,b d t BM B t =(B t,b t,,b d t ) d BM B t B t BM B t = B t + B t ( ) BM.. P(B 0 =0)=, (d )BM z R d ( z C) z (d )BM z P z (B t ) P(B t + z ) P z (B 0 = z) = P ( P z ) (expectation) E ( E z ) Z t E[Z t F s ],s <t [ ] E E[Z t F s ],A = E[Z t A], A F s, s t. (.3) Z t (F t -) (martingale) Z t, t 0 E[ Z t ] <, E[Z t F s ]=Z s, s t (.4) (.3) (.4) E[Z t,a]=e[z s,a], A F s (.5) 0

11 τ F t - (stopping time) t τ t F t Z t (local martingale) F t - τ <τ < (τ j,j ) j Z t τj a b = mina, b. τ F t - F t - f E x [f(z τ+t ) F τ ]=E Zτ [f(z t )] t 0 (.6) Z t (strong Markov property).. (d )BM ( ). Z t quadratic variation) Z t : Z t = P- lim n j=0 n (Z(t j+ ) Z(t j )) P- lim n [0,t] 0 t 0 <t < <t n t 3 Z t Z t =0. Z t, Ẑt Z, Ẑ t 4 Z + Ẑ t Z Ẑ t, dz t dẑt = d Z, Ẑ t. BM db t db t = dt, dt BM.. Bt B t BM dbt db t =0, d BM B t =(Bt,B t,...,bd t ) dbi t dbj t = δ ijdt. dmt idm j t, i, j d M t = (Mt,M t,...,md t ). Z t =(Z t,z t,...,z d t ) M t, A t =(A t,a t,...,a d t ) d. F R d F (Z t ), df (Z t )= d j= F ( ) (Z t ) dm j t x + daj t + j.. j,k d F x j x k (Z t )dm j t dm k t (.7), 3 X n n= X (Ω, F, P) n X n X ε>0 lim P( Xn X >ε)=0 n P- lim n

12 . d- (BES d ) d =,, 3,, d B t =(Bt,Bt,,Bt d ) R d (B t B t ) X t = d (B j t ) (.8) j= j= X t R + x R : x > 0 F (x,x,,x d )= d x j, F = x k x k F, F x = k F x k F 3 d k= F x k = F d F d k= x k = d F (.7), B t,,b d t dx t = X t (.9) d k= ( B k t dbk t + d X t k= ) d Bt k dbt k = db k t dbl t = δ kldt, k, l d (.9) dt X t X t d (Bt k ) (dbt k ) = k= X t d (Bt k ) dt = dt B j t d j= BM, B t db t X t (stochastic differential equation, SDE) dx t = db t + d k= dt (.0) X t [6, 4, 8, 9] d (.0) SDE (d = : X t = B t ) d- (Bessel process) BES d (.0) (BM) ( ) BES d.3. X t ( ) I ν SDE (.0) (Kolmogorov backward equation) t p(t; x, y) = p(t; x, y)+d p(t; x, y) (.) x x x d, d< BES d (p(t; x, y) =p(t; y, x) ) (.) p(t; x, y) = t (xy) ν exp ( x + y ) ( xy ) I ν (.) t t

13 , ν = d I ν (z) d =(ν +) (.3) I ν (z) = n=0 Γ(z) Γ(z) = ( z ) n+ν Γ(n + )Γ(n ++ν) 0 e u u z du, Rz >0 m ν (dy) =y ν+ dy (.4) (.) m ν (dy)/dy =y ν+ BES d t 0 x>0 y 0 ( x + y ) p(t, y x) = t y ν+ x ν exp t ( xy ) I ν t (.5).3 BES d x >0 BES d X x t dx x t = d dt X x t + db t, t 0, X x 0 = x>0 (.6). x>0 x Xx x t d = X t (.7). BM (.) Y t = x Xx x t dy t = x ( db x t + d = x db x t + d = d B t + d dt. Y t d(x ) t) Xx x t x X x x t dt 3

14 B t = B x t/x = d B t Y 0 = X0 x /x = x/x = x>0 BES d T x T x = inf t>0:xt x =0. (.8) SDE (.6) t<t x well-defined. (i) d = T x =, x>0 (ii) d> = lim t X x t =, x>0 (iii) d = = inf t>0 Xx t =0, x>0 x >0 BES (iv) d< = T x <, x>0 0 <x <x<x < σ = inf t >0:Xt x = x or Xt x = x φ(x) =φ(x; x,x )=P(Xσ x = x ) φ(x )=0, φ(x ) = (.9) t σ mint, σ, M t = φ(xt σ) x. ] M t = E [φ(x σ x ) Ft E[M t F s ]=M s, 0 s t (.0) M t (φ(x) ) (.7) BES d SDE (.6) t σ [ M t = φ(x)+ φ (Xs x ) db s + d ] ds t σ 0 Xs x + 0 φ (Xs x )(db s) t σ t σ [ = φ(x)+ φ (Xs x )db s + φ (Xs x )+ d ] 0 0 Xs x φ (Xs x ) ds. (f (x) = d dx f(x).) M t ( ) φ (x)+ d x φ (x) =0, x <x<x (.) 4

15 ( d dx + d ) φ (x) =0c x φ (x) =cx (d ) (.9) φ(x )=0 d φ(x) =c d = φ(x) =c x y (d ) dy = c x d (x d x d ) dy y = c(log x log x ) x x φ(x )= c x d x d φ(x) =φ(x; x,x )= x d x d log x log x log x log x d d = (i) d> d<0 (.) x = L>x (.) φ(x;0,l) lim,l) x 0 = x d x d lim x 0 L d x d x >0 BES d L>0 T x = inft >0:Xt x =0 = d = (.) =. log x log x φ(x;0,l) = lim = x 0 log L log x T x = (ii) α> x k = α k x, k =,, 3,... d > d β<0 (.) φ(x k ; x k,x k+ ) = xβ k xβ k x β k+ xβ k = αkβ α (k )β α (k+)β α (k )β = αβ α β = α β + >. Z,,, 0,,, n>0,n Z p =/(α β + ), p BES d Xt x, x>0 (iii) (.) d = x =/n < x < x = e n, n φ(x;/n, e n )= log x + log n n + log n 5 0

16 n>0 Xt x /n (iv) d<, lim x 0 x d =0(.) L T x < φ(x;0,l)= x d 0. L d d< x R + BM, B t X x t = x + B t + d t ds 0 Xs x BES d X x t x>0, t T x (.3) x<y = X x t <X y t, t<t x = T x T y. x<y T x = T y, x y q(x, y) =P(T x = T y ) (.) q(x, y) =q(,y/x) t>0 lim r P(T r <t)=0 lim q(,r) = 0 (.4) r.3 0 <x<y T x = T y 0 y Xt sup Xx t t<t x Xt x <. (.5). (.5) X y t Xx t X x t c<, 0 <t<t x X y t Xx t cx x t, 0 <t<t x X y t ( + c)x x t, 0 <t<t x Xt x =0= X y t =0, (.5) = T x = T y. T x = T y (.5) 0 ( p r = P T x = T y sup y Xt Xx t t<t x X x t ) r 6

17 τ r = inf (X y t t<t Xx t )/Xt x = r τ r x X y t /Xx t =+r BES d p r q(, +r) (.4) lim r q(, +r) =0 p = lim r p r = P ( T x = T y sup y Xt Xx t t<t x X x t ) = =0. x <y T x = T y.4 (i) 3 <d< = x<y 4, P(T x = T y ) > 0. (ii) d 3 = x<y, T x <T y. 0 <x<y ( X y t Z t = log ) Xx t Xt x, t < T x. (.6) dx x t = d dt X x t + db t, dx y t = d dt X y t + db t BM, B t. f(x, y) = log(y x)/x f x (x, y) = f(x, y)/ x f x (x, y) = y x x, f y(x, y) = y x f xx (x, y) = (y x) + x, f yy(x, y) = (y x), f xy(x, y) =f yx (x, y) = (.7) [ dz t = f x (Xt x,x y t ) db t + d ] [ dt Xt x + f y (X xt,x yt ) db t + d ] [f xx (Xt x,x y t )+f xy(xt x,x y t )+f yy(xt x,x y t ) + = Xt x db t + [( ) 3 d (Xt x + d ) X y t Xx t (Xt x) X y t t r r(t) 4 P(T x = T y) A. 0 ] dt X y t (y x) dt ] dt (.7) ds (Xs x = t. (.8) ) 7

18 dr(t)/(x x r(t) ) = dt. (.7) r(t) [ (3 dz r(t) = ) Xr(t) x db r(t) + d + d X y r(t) ] Xx r(t) X y r(t) dr(t) (X x r(t) ) (d B t ) = r(t) db s B t = 0 Xs x (Xr(t) x (db r(t)) = dr(t) ) (Xr(t) x = dt ) B t BM Z t = Z r(t) [ (3 ) d Z t = d B t + d SDE 3 (i) <d< d (3/,d) + d ε = (d d ) d X y r(t) ] Xx r(t) dt (.9) X y r(t) y =(+ε/)x σ = inf t>0:x y r(t) Xx r(t) = εxy r(t) 0 t<t x σ (X y r(t) Xx r(t) )/Xy r(t) ε (.9) ( ) 3 d + d X y r(t) ( ) Xx r(t) 3 X y r(t) d + d (d d ) = 3 d d d Z t = d B t + ( ) 3 d dt, Z 0 = Z 0 = log ε Z t Zt Z t, 0 t<t x σ d > 3/ Z t. Z t log(ε/) log ε Z t log ε log ( X y t Xx t X x t ) < log ε Xy t Xx t X x t (.5).3 ( ( P(T x = T y )=q x, + ε ) ) ( x = q, + ε ) > 0 <ε 8

19 (ii) d 3 3/ d 0 X y r(t) Xx r(t) X y r(t) > 0, 0 t<t x sup Zt = sup e e y Z t Xt = sup Xx t t<t x t<t x t<t x X x t =.3 P(T x = T y )=0 3 SLE κ 3. Ĉ C D ( ) Ĉ Ĉ \ D Ĉ, D (simply commected domain) C D = z C : z < 3. (Riemann mapping theorem) D C D ω D D D [] f(w) =0 f (w) > 0 (3.) H = z C : I(z) > 0 H A A = H A H\A A compact H-hull compact H-hull Q A Q A Q H \ A C ( 3.) f () A : H \ A D Möbius f () (z) = αz αβ, β =,α H (3.) z β f () : D H f () (0) = α ). f (3) A f (3) A : H \ A H = f () f () 9

20 H \ A A f () A : H \ A D z C : z = f () : D H H ) f (3) A : H \ A H ( A ) (3.) f () A f () z = β f () β [ ] lim f (3) z A (z) z =0 f (3) A (hydrodynamic condition) H \ A z R, A (/z) f (3) A /f (3) f (3) A (/z) = a z + a z + z 3 z 3 +, a j R. f (3) A (z) =b z + b 0 + b z + b z +, b j R z R f (3) A (z) a j,b j R H H Möbius f (4) (z) =d z + d 0, d > 0,d 0 R f (3) A [ ] f (4) f (3) A (z) = f (4) (f (3) A (z)) = d b z +(d b 0 + d 0 )+d b z + d b z + d b =,d b 0 + d 0 =0 d = b,d 0 = b 0 b d 0,d g A : H \ A H (3.3) lim z [ g A (z) z ] = 0 (3.4) g A (z) =z + c z + c z +, c j R (3.5) 0

21 3. γ(0) R t [0, ) γ = γ[0,t], t [0, ) γ(0, ) H Möbius t>0 z + a ( ) (t) + O z z, a (t) R, z (3.6) H \ γ(0,t] H g γ(0,t] (z) g t (z) g 0 (z) =z 3.. g t H \ γ(0,t] γ(0,t] R R 3. t (0, ) B j s,j =, BM C BM B s = B s + B s, s [0, ) (3.7) H \ γ(0,t] z BM γ(0,t] R τ t = inf s 0:B s γ(0,t] R (3.8) z g t (z) H \ γ(0,t] φ t (z) =I(z g t (z)), z H \ γ(0,t] φ t (z) =E z [φ t (B τt )], z H \ γ(0,t] φ t (z) =E z [I(B τt )] E z [I(g t (B τt ))] = E z [I(B τt )] B τt H \ γ(0,t] 3. g t (B τt ) R I(g t (z)) = I(z) E z [I(B τt )], z H \ γ(0,t] (3.9) R t = sup γ(s) γ(0) : s (0,t] (3.0)

22 γ(0,t] γ(0) R t B(γ(0),R t ) H H z H \ B(γ(0),R t ) BM BM B(γ(0),R t ) H σ ; σ = inf s 0:B s B(γ(0),R t ) R. B σ p(z,γ(0) + R t e θ ),θ (0,π) BM E z [I(B τ )] = π 0 p(z,γ(0) + R t e θ )E γ(0)+rte θ[i(b τ )]R t dθ (3.) B(γ(0),R t ) H D = z H : z γ(0) >R t p(z,γ(0) + R t e θ )= π n= [ sin(nθ)rt n Im (z γ(0)) n ], z D, θ (0,π) (3.) ( A. ). γ[0,t] γ(0) R t γ(0) /R t γ[0,t] τ t = inf s 0:B s γ(0,t] R (3.3) BM τ t /R t E γ(0)+rte θ [I(B τt )] = R t E e θ[i(b eτt )], θ (0,π) (3.9) ( ) a n+ (t) I(g t (z)) = I z + (z γ(0)) n a n (t) =R n t n= π sin((n )θ)e θ[i(b e π eτt )]dθ, n =, 3, 4, (3.4) 0 g t (3.6) g t (z) =z + n= a n+ (t), z H \ γ(0,t] (3.5) (z γ(0)) n

23 0 θ π n =, 3, sin(nθ) c n sin θ c n a n (t) Rt n π sin((n )θ) E θ[i(b e π eτt )]dθ 0 c n Rt n π sin θe θ[i(b e eτt )]dθ π 0 c n Rt n a (t), n =3, 4, 5, (3.6) 3.. (3.4) n = a (t) =R t π sin θ E θ[i(b e π eτt )]dθ (3.7) 0 a (t) = lim y ye y [I(B τt )] (3.8) ( A.3 ) γ(0,t] capacity (hcap(γ(0,t]) ) 3.3. H t = H \ γ(0,t] p Ht (z,w),z H t,w H t = γ(0,t] R E θ[i(b e eτt )] = p Ht (e θ,w)i(w)dw H t = p Ht (e θ I(w),w) eγ(0,t] I(e θ ) dw I(e θ ) = sin θ p Ht (e θ,w)dw eγ(0,t] p D (z,w) p D (z,w) I(w), z D, w D (3.9) I(z) H-excursion B s [0]: B s = B s + X s, s [0, ). (3.0) B s BM ( X s ) BES 3 (3 ) sin θ P e θ B[0, ) γ(0,t] a n (t) a n (t) =R n t π ( ) sin((n )θ) sin θ P e θ B[0, ) γ(0,t] dθ (3.) π 0 H-excursion γ(0,t] 3

24 H γ g t (x) ε >0 t + ε γ(0,t+ ε] g t+ε (z) g t+ε (z) = g γ(0,t+ε] (z) ] = [g gt(γ(t,t+ε]) g t (z) =g gt(γ(t,t+ε])(g t (z)). (3.) g t+ε (z) H \ γ(0,t+ ε] H H \ γ(0,t+ ε] g t+ε (z) g t (z) H H\g t (γ(t, t+ε]) H g t (γ(t, t+ε]) U t U 0 = γ(0) (3.5) U t = lim s t g s (γ(t)) (3.3) g t+ε (z) = g gt(γ(t,t+ε])(g t (z)) a n+ ((t, t + ε]) = g t (z)+ (g t (z) U t ) n (3.4) Rt ε = sup g t (γ(s)) U t : s [t, t + ε], (3.5) n= a n ((t, t + ε]) c n (R ε t ) n a ((t, t + ε]), n =3, 4, 5, (3.6). (3.4) g t (z) (3.5) (3.5) t t + ε /z a ((t, t + ε]) = a (t + ε) a (t). (3.7) capacity ( A.3 ) g t+ε(z) g t (z) a (t + ε) a (t) g t (z) U t c n (Rt ε)n g t (z) U t n (a (t + ε) a (t)) ε g t+ε (z) g t (z) a (t + ε) a (t) ε g t (z) U t ε c n (Rt ε ) n g t (z) U t n a (t + ε) a (t) ε ε 0 capacity a (t) = hcap(γ(0,t]) t n= n= a (t + ε) a lim ε 0 ε = da (t) dt = d hcap(γ(0,t]) (3.8) dt 4

25 lim ε 0 R ε t =0 g t+ε (z) g t (z) lim = g t(z) ε 0 ε t g t (z) t = da (t), a (t) = hcap(γ(0,t]). (3.9) g t (z) U t dt g 0 (z) =z (Loewner differential equation) 3.4. (3.8) a (t) = hcap(γ(0,t]) a (t) t [0] γ ( t ) capacity γ(t) =γ(a (t)) a (t) = hcap(γ((0,t])) = t (3.30) g t (z) t = g t (z) U t, g 0 (z) =z (3.3) ( (3.30) γ γ ) g t Loewner chains U t (3.5) a n (t) d dt a n(t) =P n (a (t),a (t), ), n =, 3, 4,. (3.3) a (t) = U t (3.33) P n (x,x, ) ( P = ) P n (x,x, )= m: m =n l(m) ( ) l(m) j= x mj. (3.34) m =(m,m, ),m j N,, 3, l(m) m m l(m) j= m j []. P =0, P =, n P n = x j P n j, n. (3.35) j= 5

26 d dt a (t) =, d dt a 3(t) = a (t), d dt a 4(t) = (a (t)) a (t), d dt a 5(t) = (a (t)) 3 +a (t)a (t) a 3 (t), (3.36) g 0 (z) =z a n (0) = 0, n=,, 3, a (t) = U t a n (t),n =, 3, g t (z) 3.4 SLE κ BES d Schramm [5], B t BM 5 U t = κb t, κ > 0, B 0 = 0 (3.37) t g t(z) = g t (z) κb t, g 0 (z) =z. (3.38) ( t 0 ) g t t 0 Schramm (chordal) (Schramm-Loewner evolution) [5] 6 κ SLE κ t [0, ) γ = γ(t) :t [0, ) t [0, ) H \ γ(0,t] H g t (z) g t (z) (3.3) U t = lim s t g s (γ(t)) (3.39) U t (3.37) (3.38), g t (z) (3.39) γ(t), 0 t< 3. SLE κ γ 3.5. SLE κ γ SLE κ (SLE κ path) SLE κ (SLE κ curve) 3. [0] 6

27 γ (t) H t g t 0 K t 0 U t = g t (γ (t)) = /κ B t 6: H SLE κ γ(0,t] hull K t g t H γ(t) H U t κb t SLE κ γ H t = H \ γ[0,t] K t = H \ H t (3.40) K t SLE κ γ[0,t] hull g t (z) H t H 6 H t g t K t g t SLE κ γ t hull K t g t H t z H T z = sup t 0: g t (z) well-defined g t (z) H = inf t 0:z K t (3.4) H t = K t = z H : T z >t z H : T z t (3.4) 7 SLE κ γ t>0 γ(t) g t κb t 6 g t (γ(t)) = κb t. (3.43) 5 BM (.) U d t = B κt κ BM 6 Oded Schramm [ 46 7 g t(z) H t H t pioneer point H pion t = H s SLE κ γ, γ(0) R H pion t = R γ(0,t]. 0 s t 7

28 g t H t γ(t) K t γ(t) / H t g t (γ(t)) H t H lim g t(z) = κb t (3.44) z γ(t) SLE κ γ = γ(t) :0 t< s 0 γ s γ s (t) =g s (γ(t + s)) κb s, t 0 γ s d = γ s 0 (3.45) SLE κ.3. BM (.) BES d SLE κ SLE κ 3.3 r>0 r g r t(rz) d = g t (z) (3.46) γ(t) r γ(r t) γ d = γ (3.47). g t (z) = r g r t(rz) g 0 (z) = r g 0(rz) = r rz = z g 0(z) = g 0 (z) =z B t = r B r t g t (z) dt g d t(z) = r d dt g r t(rz) = r r g r t(rz) κb r t = r g r t(rz) κ r B r t = g t (z) κ B t BM B t d = Bt g t (z) g t (z) SLE κ ĝ t (z) = g t(z) κb t κ (3.48) 8

29 ĝ t (z) dĝ t (z) = /κ ĝ t (z) dt + dw t, ĝ 0 (z) = z, W t = B t. (3.49) κ T z (3.4) SLE κ γ t = T z z H lim t Tz γ(t) = z γ(t) (3.43) κb Tz (3.48) lim ĝ t (z) =0 t T z T z z/ κ SDE (3.49) H 0 SDE (3.49) z x R 3. g t (x) R, t 0 ĝ t (x) R, t 0 SLE κ BES d dx x t = d X x t dt + dw t, X x 0 = x R \0 (3.50) κ = 4 d d = 4 κ + (3.5) T x = inft 0:Xt x =0.3 d x BM, W t x<y Xt x <X y t, t <T x T x T y.3..4 () d T x =, x >0. () d< T x <, x >0. (a) 3 <d< 0 <x<y, PT x = T y > 0. (b) d 3 0 <x<y T x <T y. SLE κ γ, κ, (i) 0 <κ 4 γ γ(0, ) H lim γ(t) =. (3.5) t (ii) 4 <κ<8 γ K t = H (3.53) t>0 γ(t) H. γ[0, ) H H (3.54) 9

30 (iii) κ 8 γ H γ[0, ) =H. (3.55) (a) (b) (c) 7: (a) 0 <κ 4 SLE (b) () H H 4 <κ<8 SLE (c) H κ 8 SLE 4 SLE κ 4. Schramm γ μ (D0 ;O,P ) / D, D C ( D, D C), z,w D, z,w D f : D D, f(z) =z, f(w) =w f (w) = μ (H;0, ) D C,D C, z,w D μ (D;z,w) H \ D D H w = f(w) = f (w) = w f(w ) w f μ (D;z, ) ( )= f (z) b μ (f(d);f(z), ) ( ), z D. (4.) Schramm U t γ μ (H;0, ) μ (H;0, ) γ(0,t] U t 30

31 [A] U t (U t+s U t ), s, t > 0 [B] (U t+s U t ) d = U t, s, t > 0 Schramm [C] U t c,c U t = c B t + c t [D] = c =0 U t = c B t c = κ capacity a (t) =t SLE κ t g t(z) = g t (z), g 0 (z) =z κb t Schramm [5] 4. (Lawler-Schramm-Werner [0]) Markov SLE κ, κ (0, ) 4. (Beffara[3]) κ 8 SLE κ γ(0, ) (Hausdorff ) d(κ) =+ κ H D = D H :, H \ D, dist(0, H \ D) > 0 D D D H Φ D ( ) Φ D (z) =z + o(),z. SLE κ γ τ D = inf t : dist(γ(t), H \ D) =0 t <τ D γ D 8 SLE γ(0,τ D ) Φ D Φ D (γ(0,τ D )) H 3

32 D = \ A γ (t) γ (τ D ) O A 8: H A D = H \ A t <τ D γ D D γ (t) Φ D 0 A Φ D (0) Φ D (A) g t g t * U t g t (A) 9: [ ] Φ D : A D = H \ A H. [ ] g t : SLE κ γ(0,t]. [ ] gt : γ (0,t]=Φ D (γ(0,t]). [ ] Φ t Φ gt(d), gt Φ D gt 3

33 γ γ (0,t]=Φ D (γ(0,t]), t < τ D gt (z) Loewner (3.38) 9 Φ D γ (0,t] capacity ) ( ) a (γ (t) = hcap (0,t] = hcap Φ D (γ(0,t]) A.3 A.4 hcap(ra) =r hcap(a), SLE κ (3.38) (Φ t(u t )) d dt g t (z) = (Φ t(u t )) gt (z) U t (4.) U t = κb t Φ t Φ gt(d) = g t Φ D g t U t g t (γ (t)) = Φ t (U t ) 9 U t (.7) U t = κb t [ dut = Φt (U t )+ κ ] Φ t (U t )) dt + κφ t(u t )db t A.4 (A.50) Φ t (U t ) d dt Φ t(u t )= 3Φ t (U t ) du t = [ κ 3 ] Φ t (U t )dt + κφ t(u t )db t (4.3) t r(t) t = Ũ t U r(t) (4.3) r(t) 0 Φ s(u s ) ds dũ t = b(κ) Φ r(t) (U r(t)) Φ r(t) (U r(t)) κdt + κd B t (4.4) b = b(κ) = 6 κ κ. (4.5) B t r(t) 0 Φ s (U s )db s 33

34 g t (z) =gr(t) (z) (4.) d dt g t (z) = g t (z) Ũ t κ =6 b(κ) =0(4.4) dũ t = 6d B t D D g t (z) g t (z), U t d = 6Bt SLE 6, (3.38) κ =6 D γ γ (0,t]=Φ D (γ(0,t]), t<τ D SLE SLE κ μ κ (H;0, ) κ =6 κ 6 (4.4) U t, M t b (Φ t (U t)) b (.7) A.4 SDE d(φ t (U t)) b = b(φ t (U t)) b [ κ Φ t (U t ) Φ t (U t) db t + (b )κ + (Φ t (U t )) (Φ t (U t)) 8 3κ 6 Φ ] (U t ) Φ t (U dt. t) (4.6) b (4.5) b(κ) (4.6) d(φ t(u t )) b(κ) = b(κ)(φ t(u t )) b(κ) [ κ Φ t (U t ) Φ t (U t) db t 8 3κ 6 ] SΦ t (U t )dt (4.7), Sf f c M t =(Φ t(u t )) b(κ) exp Sf = f f 3 (f ) (f ) (4.8) c 6 t 0 SΦ s (U s )ds dm t = d(φ t(u t )) b(κ) exp c 6 0 (Φ t(u t )) b(κ) exp c 6 t SΦ s (U s )ds t 0 c SΦ s (U s )ds 6 SΦ t(u t )dt (4.7) dm t = b(κ) Φ t (U t ) Φ t (U t) M t κdbt c +(8 3κ)b(κ) SΦ t (U t )M t dt 6 34

35 0 c = c(κ) = (3κ 8)b(κ) = (3κ 8)(6 κ) κ dm t = b(κ) Φ t (U t) Φ t (U t) M t κdbt M t =(Φ t (U t)) b(κ) exp c(κ) 6 SΦ t (U t ) 0 t 0 SΦ s (U s )ds (4.9) (4.0) [0]. κ 8/3 c(κ) 0 M t. M t Girsanov SLE κ, (4.0) 8 SLE κ t γ(t), γ(0, ) D SLE κ γ t Φ t(u t ) κ 4 SLE κ H 3.4 (i) [] κ 4,D D dμ (D;0, ) dμ (H;0, ) (γ) =M = γ(0, ) D exp c(κ) 6 0 SΦ s (U s )ds. (4.) (4.9), κ 4 κ =8/3 c(κ) =0 (4.) γ(0, ) D (.) 4.4 SLE κ μ κ (H;0, ) κ = μ per (D 0 ;0,P ) μsaw (D 0 ;O,P ) μ 6 (H;0, ), μ8/3 (H;0, ) H D 0 Smirnov [8]. μ LERW (D 0 ;0,P ) c = μ (H;0, ), μising (D 0 ;0,P ) c = μ 3 (H;0, ). [5, 0] [9] = c. 4. (4.5) d LERW = d() = 5 4, d SAW = d(8/3) = 4 3, d Ising = d(3) = 8, d per = d(6) = 7 4, b LERW = b() =, b SAW = b(8/3) = 5 8, b Ising = b(3) =, b per = b(6) = 0. 8 (4.0) (4.) f (z) b (4.5) κ b(κ) b. 35

36 4.. ( ) SLE κ [5] 0. κ = (LERW) κ = 8 3 =. 6 (SAW) κ =3 κ =4 4 [Fortuin-Kasteleyn cluster] κ = 4 5 =4.8 3 [Fortuin-Kasteleyn cluster] κ = 6 3 =5. 3 [Fortuin-Kasteleyn cluster] κ =6 κ = 8 uniform spanning tree (UST) (4.) (q q =q =q =0 UST κ q = + cos(8π/κ), 4 κ 8.) c κ = 8. =.6 κ = 4 = SAW 3-state Potts (FK) κ =6 percolation 0 κ - - κ = LERW κ =3 Ising interface κ =4 4-state Potts (FK) κ = 6. =5.3 3 Ising (FK) κ =8 UST 0: SLE κ κ c c ) κ c 4.. (4.5) (4.9), κ c = b(5 8b) +b [3] [L n,l m ]=(n m)l n+m + c n(n ) δ n+m,0, n,m Z (4.3) L n b =0,n,L 0 b = b b (b, b ), H b = C[L,L,...] b ( ) ( ) b L L det b b L L L b 4b + c/ 6b = det =0 b L L L b b L L L L b 6b 4b( + b) 36

37 (Kac ). b 0 (4.3) [, 4]. A.5 A A. 3 <d< x 0 P(T +x = T ) > 0 x F (α, β, γ; z) =+ k= (α) k (β) k (γ) k z k k! (A.) (c) k =Γ(c + k)/γ(c) =c(c +) (c +(k )) A. 3 <d< x 0 P(T +x = T )= ( ) Γ(d ) x d 3 ( ) x F d 3,d, (d );. (A.) Γ((d ))Γ( d) +x +x.. g(x, y) =(y x)/x (.7) R t = X+x t X t Xt, x > 0 (A.3) g x (x, y) = y x, g y(x, y) = x g xx (x, y) = y x 3, g yy(x, y) =0, g xy (x, y) =g yx (x, y) = x dr t = R [ t 3 d Xt db t + d R t R t ( + R t ) t r(t) r(t) ( Rt ]( Rt X t ) dt (A.4) ) ds = t. (A.5) 0 X t r(t) B t = 0 R s Xs db s (A.6) 37

38 B t BM R t = R r(t) d R t = = [ 3 d d R t [ d + d R t ] R t ( R dt + db t t +) ] dt + db t R t + (A.7) ψ(x) =P(T +x = T )=q(, +x) (A.8) M t = ψ( R t ) (A.9) BES d (.7) M t E[M t F s ]=M s, 0 s<t. (A.0) (.7) (A.7) dm t = ψ ( R t )db t + ψ ( R t ) [ d R t + d ] dt + R t + ψ ( R t )dt. =0 ψ(x) [ d ψ (x)+ + d ] ψ (x) =0. x x + (A.) (A.) x = u u u = x +x x u ψ(u) =ψ(x) (A.) u( u) ψ (u)+ ( d) (3 d)u ψ (u) = 0 (A.3) u( u)f + γ (α + β +)u F αβf = 0 (A.4) α =0, β = d, γ = ( d) (A.5) (A.4) u =0 F (α, β, γ; u) u γ F ( γ + α, γ + β, γ; u) (A.5) α =0 u d 3 F (d 3,d, (d ); u) c,c ψ(u) =c + c u d 3 F (d 3,d, (d ); u) 38

39 ψ(0) = ψ(0) = P(T = T )= ψ() = ψ( ) = lim P(T +x = T ) = 0 x c = c = Γ(d ) F (d 3,d, (d ); ) = Γ((d ))Γ( d) (A.6) (A.7) A. D BM cb t τ D = inf t 0:B t / D (A.8) BM D D A. D F : D R D D F D(= D D) R u D z D u(z) =E z [F (B τd )] (A.9) D D z D D hm(z,d; ) hm(z,d; V )=P z [B τd V ], V D (A.0) (harmonic measure) (A.9) u(z) = F (w)hm(z,d; dw) (A.) D u(z) = F (w)h D (z,w) dw D D (A.) H D (z,w) (Poisson kernel) w D H D (z,w) z D w 0 D z D w 0 D H D (z,w) δ(w w 0 ) : lim H D (z,w) =δ(w w 0 ), z w 0 w 0,w D. 39

40 ( )BM A.3 f : D D D = D D z D, V D hm(f(z),d ; f(v ))=hm(z,d; V ) (A.3) (A.) H D (f(z),f(w)) = f (w) H D (z,w), z D, w D. (A.4) (half-infinite strip) D = z = x + y : x>0, 0 <y<π (A.5) D = q : q (0,π) H D (z, q),z D, q (0,π) q (0,π) H D (z, q) z x H D(x + y, q)+ y H D(x + y, q) =0, x+ y D (separation of variables) (q ) H D (x + y, q) =X(x)Y (y) c = X (x)y (y)+x(x)y (y) =0 X (x) X(x) = Y (y) Y (y) = c X (x) =cx(x), Y (y) = cy (y) (A.6) (A.7) (A.7) Y (y) =a sin( cy)+b cos( cy) y =0,π Y (y) =0 b =0, c = n, n =,, 3, (A.6) Y (y) =a sin(ny), n =,, 3, X (x) =n X (x) 40

41 x X(x) = const. e nx H D (x + y, q) = a n (q) q a n (q)e nx sin(ny) n= lim H D(x + y, q) =δ(y q) x 0 (A.8) a n (q) = π sin(nq) lim x 0 H D(x + y, q) = π sin(nq) sin(ny) n= = π δ(x) = π n= n= e n(q+y) + π e nx n= e n(q y) q, y > 0 (A.8) H D (x + y, q) = π e nx sin(ny) sin(nq), x+ y D, q (0,π) (A.9) n= r R, R > 0 z = x + y ζ = α + β : ζ = f(z) =r + Re z. (A.30) D D = ζ = α + β : ζ r >R,β>0 = H \B(r, R) (A.3) x : x>0 α : α>r+ R x + π : x>0 α : α<r R q : q (0,π) r + Re q ; q (0,π) 4

42 (A.30) e x+ y = ζ r R, ζ r ex = R, e y = ζ r ζ r e x e y = n R ζ r e nx e ( R ny = ζ r [( ) R n ] e nx sin(ny) = I ζ r [ ] = R n I (ζ r) n q (0,π) ) n f( q) =r + Re q, q D f ( q) =Re q f ( q) = R (A.4) H D (ζ,r + Re q ) = R [ R n I π (ζ r) n n= = [ sin(nq)r n I π n= (ζ r) n ] sin(nq) ] (A.3) A.3 capacity (3.4) z g A (z) φ A (z) =I(z g A (z)) H \ A A. A. τ = τ H\A = inf t : B t R A (A.33) (A.34) 4

43 ( )BM R compact H-hull A φ A (z) φ A (z) =E z [φ A (B τ )] (A.35) (A.33) φ A (z) =E z [I(B τ )] E z [I(g A (B τ ))] = E z [I(B τ )] B τ H \ A g A I(g A (B τ ))=0 I(g A (z)) = I(z) E z [I(B τ )] (A.36) z = y, y > 0 I( y) =y (3.5) (LHS) = I I(g A ( y)) = y E y [I(B τ )] ( y + c y + O y c y ( )) y = y c ( ) y + O y ( ) + O y = y E y [I(B τ )]. c = lim y ye y [I(B τ )] A Q capacity (half-plane capacity) hcap(a) A Q hcap(a) lim y ye y [I(B τh\a )], A Q. (A.37) g A (z) =z + hcap(a) z ( ) + O z, z (A.38) capacity C ( ) S = z : z S y C S + y = z + y : z S S r>0 ( ) rs A.4 r>0,x R A Q hcap(ra) =r hcap(a), hcap(a + x) = hcap(a). (A.39) (A.40) 43

44 . g A g ra (z) =rg A (z/r), g A+x (z) =g A (z x)+x (A.4) (A.4) (A.38) (A.4) = z + hcap(ra) z ( ) + O z z (A.4) = r r + hcap(a) ( + O z/r ) = z + r hcap(a) z + O z/r ( z ) /z (A.39) (A.4) = z + hcap(a + z) z + O ( ) z (A.4) = (z x)+ hcap(a) z x = z + hcap(a) + O z ( + O ) ( z z x ) + x (A.40) A, B Q A B g A H \ A H A B g A (B \ A) Q H \ g A (B \ A) H g ga (B\A) A.5 A, B Q,A B g B = g ga (B\A) g A, A B Q (A.43) hcap(b) = hcap(a) + hcap(g A (B \ A)) (A.44). (A.38) g A (z) = z + hcap(a) z ( ) + O z g ga (B\A)(z) = z + hcap(g A(B \ A)) z 44 + O, ( ) z, z.

45 ] ( ) [g ga (B\A) g A (z) =g ga (B\A) g A (z) ( = g ga (B\A) z + hcap(a) ( )) + O z z = z + hcap(a) ( ) hcap(g A (B \ A)) + O z z + z + hcap(a) + O z = z + ( ) ( ) hcap(a) + hcap(g A (B \ A)) + O z z ( z ( ) ) + O z (A.44) g B (z) =z + hcap(b) z ( ) + O z, (A.44) A.4 Φ t (z), Φ t (z) (Φ t (U t)) b SDE SLE κ g t f t = g t f t (g t (z)) = z t (z t ) SLE κ d dt f t(g t (z)) + f t(g t (z)) d dt g t(z) =0 d dt g t(z) = g t (z) U t d dt f t(g t (z)) + f t (g t(z)) =0 g t (z) U t g t (z) z g t d dt f t(z) = f t (z), f 0 (z) =z (A.45) z U t Φ t Φ gt(h) (A.46) 9 Φ t = g t Φ H f t, h 0 =Φ H. (A.47) 45

46 d dt Φ t(z) = d dt g t (Φ H (f t (z))) = ġt (Φ H (f t (z))) + (gt ) (Φ H (f t (z))) Φ H(f t (z)) f t (z). (g t (z) ġ t (z) = d dt g t (z) z (gt ) (z) = d dz g t (z).) (4.) (A.45) d dt Φ t(z) = (Φ H (U t)) g t (Φ H(f t (z))) U t [ ] +(gt ) (Φ H (f t (z))) Φ H (f t(z)) f t (z) z U t Φ t t t =0 t =0 f 0 (z) =z,f 0 (z) =,g 0 (z) =z,(g 0 ) (z) = d dt h 0(z) = = (Φ H (U 0)) Φ H (z) Φ H (U 0 ) Φ H (z) z U 0 (h 0 (U 0)) h 0 (z) h 0 (U 0 ) h 0 (z) z U 0 t 0 d dt Φ t(z) = (Φ t (U t)) Φ t (z) Φ t (U t ) Φ t (z) z U t (A.48) z d dz Φ t(z) = (Φ t (U t)) Φ t (z) (Φ t (z) Φ t (U t )) + Φ t (z) (z U t ) Φ t (z) z U t (A.49) (A.48) (A.49) z U t d dt Φ d t(u t ) = lim z U t dt Φ t(z) = 3Φ t (U t) (A.50) d dt Φ t(u t ) = lim z U t b (.7) d dt Φ t(z) = (Φ t (U t)) Φ t (U t) 4 3 Φ t (U t ) (A.5) d(φ t(u t )) b = b(φ t(u t )) b Φ t (U t )du t + b(b )(Φ t(u t )) b (Φ t (U t )) + b(φ t(u t )) b Φ t (U t ) (du t ) +b(φ t(u t )) b d dt Φ t(u t )dt. 46

47 (A.5) du t = κdb t, (du t ) = κdt (RHS) = b(φ t (U t) b [ κ Φ t (U t ) + Φ t (U t) db t t (U t )) (b )κ(φ (Φ t (U t)) + κφ t (U t ) Φ t (U t) + Φ t (U t) = b(φ t (U t)) b [ κ Φ t (U t ) Φ t (U t) db t + (4.6) (b )κ + ( (Φ t (U t )) Φ t (U t) (Φ t (U t )) (Φ t (U t)) Φ ) Φ ( κ 4 3 ) t (U t) t (U t ) Φ t (U t) ] dt ] dt A.5 SLE g t (z) a n (t),n=,, 3, SLE κ a (t) = U t = κb t (A.5) a n (t) (a (t) =t ). a (t) (3.36) a n (t), n x =(x,x, ), a(t) =(a (t),a (t), ) Q(x) x n n Q(a(t)), (.7) dq(a(t)) = κdb t + dt κ x x + P n (x(t)) Q(x(t)) x n n x=a(t) (da (t)) = κdt (3.3) A = κ x + n P n (x) x n (A.53) AM(x) =0 M(x) M(a(t)) x n n a (t) (a (t)) κa (t) (a (t)) 3 3κa (t)a (t) a 3 (t)+a (t)a (t) (A.54) SLE []. Bauer Bernard, c =(3κ 8)(6 κ)/κ ( ) b =(6 κ)/κ ( ) [] : Summer School 009 Loewner SLE 47

48 [] L. V. Ahlfors, Complex Analysis, 3rd ed., (McGraw-Hill, 979). [] M. Bauer and D. Bernard, SLE matringales and the Virasoro algebra, Phys. Lett. B 557 (003) [3] V. Beffara, The dimension of the SLE curves, Ann. Probab. 36 (008) [4] R. Friedrich and W. Werner, Conformal restriction, highest-weight representations and SLE, Commun. Math. Phys. 43 (003) 05-. [5] W. Kager and B. Nienhuis, A guide to stochastic Löwner evolution and its application, J. Stat. Phys. 5 (004) [6] I. S. E. (00). [7], SLE, Vol.6, No.7 (007) [8] 6 ), (006) [9], Vol. 6, No.3 (009) [0] G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005). [] G. Lawler, Schramm-Loewner Evolution (SLE), arxiv: [math.pr]. [] G. Lawler, O. Schramm, and W. Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 6 (003) [3] G. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 3 (004) [4] (999). [5] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 8 (000) -8. [6],, No.545 (007 ), pp.46-5,. [7] SLE Schramm-Loewner Evolution,, No.546 (008 ), pp.7-, [8] S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy s formula, scaling limit, C. R. Acad. Sci. Paris Sér. I Math. 333 (00) [9] S. Smirnov, Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, arxiv:

49 [0] :,,, 006. [],, No.546 (008 ), pp.3-9. [] W. Werner, Conformal restriction and related questions, Probability Surveys (005) [3] 006). 49

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Katori_SLE_Sep07_v2c.dvi Schramm-Loewner Evolution (SLE) and Conformal Restriction 4 September 007 (version c) SLE [6] Lawler ( [7]), 3 Werner ( []), 4 Friedrich and Werner ( [3]) 3........................ 3. d-.................................