Katori_SLE_Sep07_v2c.dvi

Transcription

1 Schramm-Loewner Evolution (SLE) and Conformal Restriction 4 September 007 (version c) SLE [6] Lawler ( [7]), 3 Werner ( []), 4 Friedrich and Werner ( [3]) d BES d SLE κ BES d SLE ( ). address : katori@phys.chuo-u.ac.jp

2 BM SLE 8/ Witt Ward A 44 A A A.3 capacity B h t(u t ) SDE 5 B. h t (z) h t (z) B : ( )

3 . (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 () Ω f F- a {ω Ω:f(ω) a} F (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. (.) a π(t s) (t s) (ii) t B t Ω Ω s.t. P( Ω) =, ω Ω Bt (ω) t. c>0 c B c t B t c B c t d = B t c>0 (.) (d distribution.) BM (scaling property) B t,b t,,bd t BM B t =(Bt,B t,,bd t ) d BM B t B t BM B t = B t + B t ( ) BM 3

4 .. 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) 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 = min{a, 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 ( ). 4

5 Z t quadratic variation Z t : Z t = sup m (Z tj+ Z tj ) j=0 sup 0 t 0 <t < <t m <t m+ t. 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. dm i t dm j t, i, j d M t = (M t,m t,...,m d 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),. d- d =,, 3,, d B t =(Bt,B t,,bd t ) Rd (B t ) X t = d (B j t ) (.8) j= 5

6 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 j= d k= F x k = F { d F (.7), B t,,bd t dx t = X t (.9) d k= ( B k t dbk t + d X t k= ) d Bt k dbk t = d k= x k } = d F db k t db l t = δ kl dt, 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 d (.0) SDE (d =.) d- (Bessel process) BES d (.0) (BM) ( ) BES d.3. X t ( ) 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 (.) 6

7 , ν = d I ν (z) d =(ν +) (.3) I ν (z) = n=0 ( z ) n+ν Γ(n + )Γ(n ++ν) (Γ(z) Γ(z) = 0 e u u z du, Re z>0.) m ν (dy) =y ν+ dy (.4) (.) m ν (dy)/dy =y ν+ BES d t 0 x>0 y 0 p(t, y x) = y ν+ t x ν exp ( x + y ) ( xy ) I ν t 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 7

8 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:X x t =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:X x t = x or X x t = x } (.9) φ(x) =φ(x; x,x )=P(X x σ = x ) (.0) φ(x )=0, φ(x ) = (.) t σ min{t, σ}, M t = φ(x x t σ ) (.). [ ] M t = E φ(xσ) x F t (.3) E[M t F s ]=M s, 0 s t (.4) 8

9 M t (φ(x) ) (.7) BES d SDE (.6) t σ [ M t = φ(x)+ φ (Xs x ) db s + d ] ds 0 Xs x + 0 t σ t σ [ = φ(x)+ φ (Xs x )db s t σ φ (X x s )(db s ) φ (Xs x )+ d Xs x φ (Xs x ). ( f (x) = d dx f(x).) M t ( ) φ (x)+ d x φ (x) =0, x <x<x (.5) ( d dx + d ) φ (x) =0c x φ (x) =cx (d ) (.) φ(x )=0 ] ds d φ(x) =c d φ(x) =c φ(x) =φ(x; x,x )= x y (d ) dy = c x d (x d x d ) dy y = c(log x log x ) x φ(x )= c x d x d x x d x d d (.6) (i) log x log x log x log x d = d> d<0 (.6) 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 = inf{t >0:X x t =0} = d = (.6) =. log x log x φ(x;0,l) = lim = x 0 log L log x 9

10 T x = (ii) α> x k = α k x, k =,, 3,... d > d β<0 (.6) 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 X x t, x>0 (iii) (.6) d = x =/n<x<x = e n, n φ(x;/n, e n )= log x + log n n + log n 0 n>0 X x t (iv) /n d<, lim x 0 x d =0(.6) 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 (.7) 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 ) (.8) 0

11 (.) q(x, y) =q(,y/x) (.9) t>0 lim r P(T r <t)=0 lim q(,r) = 0 (.30) r.3 0 <x<y {T x = T y } 0 y Xt sup Xx t t<t x Xt x <. (.3). (.3) 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 X x t =0= X y t =0, (.3) = T x = T y. T x = T y (.3) 0 ( t p r = P T x = T y sup ) Xx t r y X t<t x τ r = inf t<tx {(X y t Xx t )/Xx t = r} τ r X y t /Xx t =+r BES d p r q(, +r) X x t (.30) lim r q(, +r) =0 ( p = lim p r = P T x = T y Xt y Xx t r X x t<t x t ) = =0. x <y T x = T y

12 .4 (i) 3 <d< = x<y, 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. (.3) 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) 0 ] dt X y t (y x) dt ] dt (.33) ds (Xs x = t. (.34) ) (dr(t)/(x x r(t) ) = dt.) (.33) 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 )

13 B t BM Z t = Z r(t) [ (3 ) d Z t = d B t + d + d X y r(t) ] Xx r(t) dt (.35) SDE 3 (i) <d< d (3/,d) ε = (d d ) d y =(+ε/)x X y r(t) 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 (.35) ( ) 3 d + d X y r(t) Xx r(t) X y r(t) d Z t = d B t + ( ) 3 d + d (d d ) = 3 d d ( ) 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 ( X y t log ) Xx t Xt x < log ε Xy t Xx t Xt x <ε (.3).3 ( ( P(T x = T y )=q x, + ε ) ) ( x = q, + ε ) > 0 (ii) d 3 3/ d 0 X y r(t) Xx r(t) X y r(t) > 0, 0 t<t x 3

14 y Z sup Zt = sup e e t Xt = sup Xx t t<t x t<t x t<t x X x t =.3 P(T x = T y )=0.4 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! (.36) (c) k =Γ(c + k)/γ(c) =c(c +) (c +(k )).5 3 <d< x 0 P(T +x = T )= ( ) Γ(d ) x d 3 ( F d 3,d, (d ); Γ((d ))Γ( d) +x ) x. (.37) +x.. g(x, y) =(y x)/x R t = X+x t X t Xt, x > 0 (.38) 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 (.7) dr t = R [ t 3 d Xt db t + d R t R t ( + R t ) ]( Rt X t ) dt (.39) t r(t) r(t) ( ) Rt ds = t. (.40) 0 X t 4

15 r(t) B t = 0 B t BM R t = R r(t) d R t = = [ 3 d d R t [ d + d R t R s Xs R t + db s (.4) ] R t ( R dt + db t t +) ] dt + db t (.4) ψ(x) =P(T +x = T )=q(, +x) (.43) M t = ψ( R t ) (.44) BES d M t E[M t F s ]=M s, 0 s<t. (.45) (.7) (.4) [ dm t = ψ ( R t )db t + ψ ( R d t ) + d R t ] dt + R t + ψ ( R t )dt. (.46) =0 ψ(x) [ d ψ (x)+ x x = u u + d x + u = x +x x u ψ(u) =ψ(x) (.47) ] ψ (x) =0. (.47) u( u) ψ { (u)+ ( d) (3 d)u} ψ (u) = 0 (.48) { } u( u)f + γ (α + β +)u F αβf = 0 (.49) 5

16 α =0, β = d, γ = ( d) (.50) (.49) u =0 F (α, β, γ; u) u γ F ( γ + α, γ + β, γ; u) (.50) 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) ψ(0) = ψ(0) = P(T = T )= ψ() = ψ( ) = lim x P(T +x = T ) = 0 (.5) c = c = Γ(d ) F (d 3,d, (d ); ) = Γ((d ))Γ( d) (.5) 6

17 . C, H = {z C :Im(z) > 0} i= γ(0) R t [0, ) γ = γ[0,t], t [0, ) ( ) γ(0, ) H Möbius t>0 z + a(t) z ( ) + O z, a(t) R, z (.) H \ γ(0,t] H ( A. ). g γ(0,t] (z) g t (z) g 0 (z) =z.. g t H \ γ(0,t] γ(0,t] R R. t (0, ) Bs,j j =, BM C BM B s = Bs +ibs, s [0, ) (.) H \ γ(0,t] z BM γ(0,t] R τ t = inf{s 0:B s γ(0,t] R} (.3) z g t (z) H \ γ(0,t] φ t (z) = Im (z g t (z)), z H \ γ(0,t] (.4) 7

18 φ t (z) =E z [φ t (B τt )], z H \ γ(0,t] (.5) φ t (z) =E z [Im (B τt )] E z [Im (g t (B τt ))] = E z [Im (B τt )] B τt H \ γ(0,t]. g t (B τt ) R Im (g t (z))=im(z) E z [Im (B τt )], z H \ γ(0,t] (.6) R t = sup{ γ(s) γ(0) : s (0,t]} (.7) 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 i θ ),θ (0,π) BM E z [Im (B τ )] = π 0 p(z,γ(0) + R t e i θ )E γ(0)+rtei θ [Im (B τ )]R t dθ (.8) B(γ(0),R t ) H D = {z H : z γ(0) >R t } p(z,γ(0) + R t e i θ )= π n= [ sin(nθ)rt n Im (z γ(0)) n ], z D, θ (0,π) (.9) ( A. ). γ[0,t] γ(0) R t γ(0) 8

19 /R t γ[0,t] τ t = inf{s 0:B s γ(0,t] R} (.0) BM τ t /R t E γ(0)+rtei θ [Im (B τt )] = R t E ei θ [Im (B eτt )], θ (0,π) (.) (.6) ( ) a n+ (t) Im (g t (z))=im z + (z γ(0)) n a n (t) =R n t n= (.) π sin((n )θ)e ei θ [Im (B eτt )]dθ, n =, 3, 4, (.3) π 0 g t (.) g t (z) =z + n= a n+ (t), z H \ γ(0,t] (.4) (z γ(0)) n 0 θ π n =, 3, sin(nθ) c n sin θ c n a n (t) Rt n π sin((n )θ) E ei θ [Im (B eτt )]dθ π 0 c n Rt n π sin θe ei θ [Im (B π eτt )]dθ 0 c n Rt n a (t), n =3, 4, 5, (.5).. (.3) n = a (t) =R t π sin θ E ei θ [Im (B π eτt )]dθ (.6) 0 a (t) = lim y Ei y [Im (B τt )] (.7) 9

20 ( A.3 ) γ(0,t] capacity (hcap(γ(0,t]) ).3. H t = H \ γ(0,t] p Ht (z,w),z H t,w H t = γ(0,t] R E ei θ [Im (B eτt )] = p Ht (e i θ,w)im (w)dw H t = p Ht (e i θ Im (w),w) eγ(0,t] Im (e i θ ) dw Im (ei θ ) = sin θ p Ht (e i θ,w)dw eγ(0,t] Im (w) p D (z,w) p D (z,w), z D, w D (.8) Im (z) H-excursion B s [7]: B s = B s +ix s, s [0, ). (.9) B s BM X s BES 3 (3 ) sin θ P ei θ( B[0, ) γ(0,t] ) a n (t) a n (t) =R n t π sin((n )θ) sin θ P θ( ) ei B[0, ) γ(0,t] dθ (.0) π 0 H-excursion γ(0,t].. 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)). (.) 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]) 0

21 U t U 0 = γ(0) (.4) U t = lim s t g s (γ(t)) (.) 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) n= { } Rt ε = sup g t (γ(s)) U t : s [t, t + ε], (.4) a n ((t, t + ε]) c n (R ε t )n a ((t, t + ε]), n =3, 4, 5, (.5). (.3) g t (z) (.4) (.4) t t + ε /z a ((t, t + ε]) = a (t + ε) a (t). (.6) 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)) n= 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= a (t + ε) a lim ε 0 ε = da (t) dt = d hcap(γ(0,t]) (.7) dt lim ε 0 R ε t =0 g t+ε (z) g t (z) lim = g t(z) ε 0 ε t

22 g t (z) t = da (t), a (t) = hcap(γ(0,t]). (.8) g t (z) U t dt g 0 (z) =z (Loewner evolution).4. (.7) a (t) = hcap(γ(0,t]) a (t) t [7] γ ( t ) capacity γ(t) =γ(a (t)) a (t) = hcap(γ((0,t])) = t (.9) g t (z) t = g t (z) U t, g 0 (z) =z (.30) ( (.9) γ γ ) g t Loewner chains U t (.4) a n (t) d dt a n(t) =P n (a (t),a (t), ), n =, 3, 4,. (.3) a (t) = U t (.3) P n (x,x, ) ( P = ) P n (x,x, )= m: m =n l(m) ( ) l(m) j= x mj. (.33) 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. (.34) j=

23 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), (.35) g 0 (z) =z a n (0) = 0, n=,, 3, a (t) = U t a n (t),n =, 3, g t (z).3 SLE κ BES d Schramm [0], U t = κb t, κ > 0, B 0 = 0 (.36) B t BM t g t(z) = g t (z) κb t, g 0 (z) =z. (.37) ( t 0 ) {g t } t 0 (chordal) (Schramm-Loewner evolution) κ SLE κ (.37).).. t [0, ) γ = {γ(t) :t [0, )} t [0, ) H \ γ(0,t] H g t (z) g t (z) (.30) U t = lim s t g s (γ(t)) (.38) U t (.36) (.37), g t (z) (.38) γ(t), 0 t< 3

24 . SLE κ γ.5. SLE κ γ SLE κ (SLE κ path) SLE κ (SLE κ curve). [7] SLE κ γ H t = H \ γ[0,t] K t = H \ H t (.39) K t SLE κ γ[0,t] hull g t (z) H t H 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 } (.40) H t = {z H : T z >t} K t = {z H : T z t} (.4) SLE κ γ t>0 γ(t) g t κb t g t (γ(t)) = κb t. (.4) 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 (.43) z γ(t) 4

25 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 (.44) SLE κ. BM (.) BES d SLE κ SLE κ. r>0 γ(t) r γ(r t) r g r t(rz) d = g t (z) (.45) γ d = γ (.46). 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 (.47) 5

26 ĝ t (z) dĝ t (z) = /κ ĝ t (z) dt + dw t, ĝ 0 (z) = z κ, W t = B t. (.48) T z (.40) SLE κ γ t = T z z H lim t Tz γ(t) = z γ(t) (.4) κb Tz (.47) lim ĝ t (z) =0 t T z T z z/ κ SDE (.48) H 0 SDE (.48) z x R. 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} (.49) κ = 4 d d = 4 κ + (.50) T x = inf{t 0:Xt x =0} d x BM, W t x <y X x t <X y t, t <T x T x T y..4 () d T x =, x >0. () d< T x <, x >0. (a) 3 <d< 0 <x<y, P{T x = T y } > 0. (b) d 3 0 <x<y T x <T y. SLE κ γ, κ, 3..3 (i) 0 <κ 4 γ γ(0, ) H lim γ(t) =. (.5) t 6

27 (ii) 4 <κ<8 γ K t = H (.5) t>0 γ(t) γ[0, ) H H (.53) H. (iii) κ 8 γ H γ[0, ) =H. (.54). (i) 0 <κ 4 d BES d T x =, x >0 H x<0 ), γ(0, ) R = γ(0, ) H SLE κ (.44), s 0 γ s (0, ) R = γ(t ) γ(t ), t <t γ(t )=γ(t ) t <t s (t,t ) s γ s (0, ) R (.5) b = lim inf t γ(t) SLE κ γ P(b r) r>0 P(b >0) = b = r (0, ) σ r = min{t : γ(t) r} r σ r < g σr () κb σr < const. r inf t ĝ t () = 0. d>( κ<4) inf t ĝ t () > 0 r>0 σ r = SEL κ γ P(b >0) = κ =4 (.5) [7] p.5 Proposition 6. (ii) z H R T z < z / t<tz K t z ( T z hull ) (swallowed) z z B z (T w = T z,w B)..4(i) 4 <κ<8( 3/ <d<) T x = T x> SLE κ γ(t ) T x = T x x> 7

28 ε = dist(,γ[0,t ]) > 0 H B(,ε) (.53) T, B H K T B u>0 ε >0 P(B(0,ε) H K T ) u t = t ε,u P(B(0,ε) H K t ) u SLE κ ε (.5) (iii).4(ii) 8 κ< ( <d 3/) x>0 x γ[0, ) x R γ[0, ) (.54) ( [7] 7 ).6. ( ) [4]. κ = loop-erased random walk (LERW) κ = 8 3 =. 6 self-avoiding walk (SAW) κ = 4 5 κ =4 4 =4.8 3 κ = 6 3 =5. 3 κ =6 κ = 8 uniform spanning tree (UST) (.55) (q q =q =q =0 UST κ q = + cos(8π/κ), 4 κ 8.) SLE κ = (LERW) [9], κ =6 Smirnov []. Smirnov SLE (arxiv: math-ph ). 8

29 .4 SLE g t (z) a n (t),n =,, 3, SLE κ a (t) = U t = κb t (.56) a n (t) (a (t) =t ). a (t) (.35) a n (t), n x =(x,x, ), a(t) =(a (t),a (t), ) Q(x) {x n } n Q(a(t)), 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) A = κ x + n P n (x) x n (.57) 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) (.58) SLE []. Bauer Bernard c =(3κ 8)(6 κ)/κ ( ) h =(6 κ)/κ ( ) [] 9

30 3 3. BM C BM B t = B t +ib t, i=. (3.) (B 0 =0.) 0 D (D C, D C.) D D B t D T = inf{t 0:B t / D} (3.) t T BM D Ψ D D ( D C) Φ:D D. (3.3) BM Φ(0) D T BM T = T 0 BM, B t,t [0,T] t r(t) = BM, B r(t) Φ (B s ) ds (3.4) t 0 Φ (B s ) ds Φ(B t )= B R t 0 Φ (B s) ds. (3.5), B Φ(0), T D BM BM, B BM, B BM D C,D C A, B D,A B A B D BM P BM D;A,B A, B D D D D D 30

31 P BM D;A,B BM ω ω D BM P BM {ω D } P BM D;A,B D;A,B {ω D } = P BM D ;A,B, D D, (restriction property) []. A, B D D,A B. (3.6) P BM D;A,B A, B D D D D P BM D ;A,B (i) D P BM {ω D }. D;A,B (ii) A, B D D Φ ( A ) Φ P BM D;A,B Φ PBM D;A,B. BM P D;A,B {ω D } =Φ P D;A,B (3.7) 3. D C,D C D A D B (B A) {P D;A,B } (conformal restriction property) (i) D D, A, B D D,A B P D;A,B {ω D } = P D ;A,B. (3.8) (ii) D Φ Φ P D;A,B = P Φ(D);Φ(A),Φ(B) mod. (3.9) A B BM {P D;A,B } (D; A, B) D = H ( ), A =0,B = 3

32 {P D;A,B } PH;0, γ PH;0, ( γ 0 H γ.) (R) λ>0 λγ = γ mod. (R) H H 0 (H \ H.) H Φ H : H H, Φ H (0) = 0, Φ H (z) z (z ) Φ H. A.) γ {γ H} =Φ (γ). H (H 0 ) γ ( ) (D; A, B),D C,D C,A,B D Ψ:H D, Ψ(0) = A, Ψ( ) =B Ψ(γ) P D;A,B {P D;A,B } (R), (R) γ H γ H H, 0, H Φ H : H H, Φ H (0) = 0, Φ H (z) z (z ) H P(γ H) Φ H P(γ H) =f(φ H ) 3

33 γ γ H Φ H (γ), γ γ H H γ H f(φ H ) f(φ H Φ H ) = P(γ Φ H = P(γ Φ H = P(γ Φ H Φ H (H)) Φ H (H) γ Φ H (H))P(γ Φ H (H)) Φ H (H) γ H)P(γ H) = P(Φ H (γ) Φ H (H) γ H)P(γ H) = P(γ Φ H (H))P(γ H) = f(φ H ) f(φ H ) (3.0) [8]. 3. γ h>0 P(γ H) =(Φ H (0))h, H H, H H (3.) h (restriction exponent) 3.. BM h = 3.4 SLE 8/3 γ SLE κ κ 4 γ 0 h>0 s.t. P(γ H) =(Φ H(0)) h H H, H H. (3.) t< g t : H \ γ(0,t] H t ( ) g t (γ(t)) = U t = κb t γ = γ[0, ) (3.) H H ) P (g t (γ[t, )) U t H γ[0,t] =(Φ H (0))h (3.3) 33

34 γ = γ[0, ) H γ[t, ) H g t g t (γ[t, )) g t (H), (3.3) H = g t (H) U t ) γ[0,t] P(γ H γ[0,t]) = P (g t (γ[t, )) U t g t (H) U t = (Φ g t(h) U t (0)) h =(Φ g t(h) (U t)) h (3.4) (.) [ ] E P(γ H γ[0,t]) F s = P(γ H γ[0,s]) s t (3.5) (Φ g t(h) (U t)) h H H h t =Φ gt(h) (.7) SDE [ κ d(h t(u t )) h = h(h t(u t )) h h t (U { t) (h )κ + h t (U t) db (h t t + (U t)) (h t (U t)) 8 3κ h } ] (U t ) 6 h t (U dt. t) (3.6) B SLE κ κ h (h )κ +=0, 8 3κ =0 κ = 8 3, h = 5 8 (3.7) =0(h t(u t )) h =(Φ g (U t(h) t)) h 3. SLE κ κ = 8 3 h = 5 8 (3.6) h = h(κ) = 6 κ κ d(h t (U t)) h = h(h t (U t)) h { κ h t (U t ) h t (U t) db t 8 3κ 6 ] S ht (U t )dt (3.8) (3.9) 34

35 , S f f S f = f f 3 (f ) (f ) (3.0) λ { λ M t =(h t(u t )) h exp 6 t 0 } S hs (U s )ds (3.) { λ dm t = d(h t (U t)) h exp 6 (3.9) t 0 } { λ S hs (U s )ds +(h t (U t)) h exp 6 t 0 } λ S hs (U s )ds 6 S h t (U t )dt dm t = h h t (U t ) κm t h t (U t) db t + { } λ (8 3κ)h S ht (U t )M t dt (3.) 6 λ = λ(κ) =(8 3κ)h(κ) = (8 3κ)(6 κ) κ (3.3) M t κ 8/3 3.. () h c h = { 5 c ± } ( c)(5 c) 6 c = h(5 8h) +h (3.4) (3.8) h (3.4) c = {(6 κ)/κ}{5 8(6 κ)/κ} + (6 κ)/κ = (3κ 8)(6 κ) κ (3.3) λ c λ = c (3.5) 35

36 4 Witt 4. Ward- γ H x >0,ε >0 E ε (x) ={γ [x, x +iε ] } (4.) 0 <x <x < <x n E εj (x j ), j n γ ( 3.) h>0 ( P γ H \ n ) [x j,x j +iε j ] = j= ( ) h Φ H\ S n j= [x j,x j +i ε j ] (0) ( ) ( h P E ε (x ) E εn (x n ) = Φ H\ n j= [x j,x j +i ε j ] (0)) (4.) f(x,ε; ; x n,ε n )=P(E ε (x ) E εn (x n )) (4.3) n = ( ) h f(x, ε) =P(E ε (x)) = Φ H\[x,x+i ε ] (0) (4.4) P(E ε (x ) E ε (x )) = P(E ε (x )) + P(E ε (x )) P(E ε (x ) E ε (x )) = f(x,ε )+f(x,ε ) f(x,ε ; x,ε ) ( ) h ) h f(x,ε ; x,ε ) = Φ H\[x,x+i ε ] (0) (Φ H\[x,x+i ε ] (0) ( h + Φ H\[x,x+i ε ] [x,x+i ε ] (0)) (4.5) inclusion-exclusion f(x,ε; ; x n,ε n ) Φ H\ k j= [x j,x j +i ε j ],k =,,,n h 36

37 Φ H\ k j= [x j,x j +i ε j ] Schwarz-Christoffel k = ( ) x h f(x, ε) =P(E ε (x)) = (4.6) x +ε f(x, ε) = ) h/ (+ ε x ε h x + O(ε4 ) B (h) (x) = lim ε 0 ε f(x, ε) (4.7) B (h) (x) = h x (4.8) B n (h) (x,,x n ) = lim ε 0,,ε ε ε n f(x,ε ; ; x n,ε n ) (4.9) n BM H 0. ( h =.) H-excursion B t = B t +ix t (B t BM, X t BES 3 ) B n () (x,,x n )= σ S n n j= (x σ(j) x σ(j ) ) (4.0) S n {,,n} B n (h) n B (h) 0 Ward- (Ward-Takahashi identities) 4. n =0,,,,x,x,,x n R + = {x R : x>0} B (h) n+ (x, x,,x n ) = h x B(h) n j= n (x,,x n ) {( x j x + x ) x j } (x j x) B n (h) (x,,x n ) (4.) 37

38 . E = E ε (x ) E ε (x n ) ( γ ε n ) x R δ [x, x +iδ ] γ A = P(E E δ (x)) = f(x,ε; ; x n,ε; x, δ) A = P( E γ [x, x +iδ ] = ) P(E) =A + A P(γ [x, x +iδ ] = ) (4.), δ 0 ε 0 A ε n δ B (h) n+ (x,,x n,x) ϕ(z) Φ H\[x,x+i δ ] (z) = (z x) +δ x +δ (4.3) γ γ [x, x +iδ ] = ϕ(z) :H \ [x, x +iδ ] H γ γ n { } A = P γ ϕ([x j,x j +iε ]) j= f(ϕ(x ),εϕ (x ); ; ϕ(x n ),εϕ (x n )) n ε n ϕ (x j ) B (h) (ϕ(x ),,ϕ(x n )), ε 0 j= ϕ([x j,x j +iε ]) [ϕ(x j ),ϕ(x j +iε )] [ϕ(x j ),ϕ(x j )+iεϕ (x j ) ], ε 0 38

39 , (4.3) δ 0 ( ϕ(z) =z + δ z x + ) + o(δ ), x ϕ (z) = δ (z x) + o(δ ) γ h P(γ [x, x +iδ ] = ) = ( ) h ϕ (0) = hδ x + o(δ ) (4.) ε 0,δ 0 ε n B n (h) (x,,x n ) = ε n δ B (h) n+ (x,,x n,x) n ( + ε n δ (x j x) j= ) B (h) ( x + δ ( ) ( hδ x + o(δ ). x x + x δ δ (4.) ) (,,x n + δ x n x + )) x N Z {,,, 0,,, } L N = { x +N j (N +)x N j x j j } (4.4) (Witt ) [L N, L M ]=(N M)L N+M. (4.5) Ward- ( 4.) B (h) n+ (x, x,,x n )= h x B(h) n (x,,x n )+ x N L N B n (h) (x,,x n ) (4.6) N 39

40 4. γ = γ[0, ] γ(0) = 0,γ( ) = SLE κ n {,, 3, }, 0 <x < <x n E = E ε (x ) E εn (x n ) 0 <t< γ[0,t] P(E γ[0,t]) g t γ g t (z) =g t (z) κb t (4.7) dg t (z) = κdb t + dt (4.8) g t (z) dg t(z) = g t (z) dt (4.9) (g t (z)) g t (γ) =g t (γ[0, ]) 0 SEL κ ((.44).) SLE κ n } P(E γ[0,t]) = P {γ g t ([x j,x j +iε j ]) (4.0) j= E εj (x j ) [x j,x j +iε j ], gt g t ([x j,x j + i ε j ]) ( ) εj g t ([x j,x j +iε j ]) [gt (x j ), g t (x j )+iε j g t (x j) ] SLE κ 4. lim ε 0,,ε n 0 ε ε n P(E γ[0,t]) ( = lim ε 0,,ε ε ε n P n }) {γ g t ([x j,x j +iε j ]) n 0 j= = lim ε 0,,ε ε ε n f(g n 0 t(x ), g t (x )ε ; ; g t (x n ), g t (x n)ε n ) = (g t(x )) (g t(x n )) B n (h) (g t (x ),, g t (x n )) (4.) 40

41 E[P(E γ[0,t]) F s ]=P(E γ[0,s]), 0 s<t { } d (g t(x )) (g t(x n )) B n (h) (g t (x ),, g t (x n )) { = (g t (x )) (g t (x n)) } n g t (x j) dg t B(h) n (g t(x ),, g t (x n )) j= { + (g t (x )) (g t (x n)) } + κ dt yj B (h) n j= {( κdb t + ) g t (x j ) dt B n (h) y (y,,y n ) j n (y,,y n ) yj =g t (x j ), j n }. yj =g t (x j ), j n (4.8) (4.9) SDE { } n 4 yj + + κ y j y j yj B n (h) (y,,y n ) yj =g t (x j ), j n j= (4.) (4.4) κ L B(h) n L B n (h) = 0 (4.3) {L N B n (h) } N Z {l N B} N Z Witt [l N,l M ]=(N M)l N+M. (4.4) (4.3) l B l B ( κ l l ) B =0 l ( κ l l ) B =0 l ( κ l l ) B =0 (4.4) ( κ ) { κ } l l l B = (l l +6l l +6l 0 ) (l l +4l 0 ) B ( κ ) { κ } l l l B = (l l +4l l 0 +l ) (l l +3l ) B 4

42 . h l N B =0, N =,, 3,, l 0 B = hb (4.5) l ( κ l l ) B =(3κ 8)hB l ( κ l l ) B =(κh + κ 6)l B (4.6) κ = 8 3, h = 6 κ κ = 5 8 (4.7), 3. SLE 8/3 Witt (4.8) 4.. c Witt [ l N, l M ]=(N M) l N+M + c N(N )δ N+M,0 (4.9) γ ( P D;A,B ) BM (h =) SLE 8/3 (h =5/8) 3. SLE κ κ =8/3 c =0 SLE κ h ( ) κ h = 6 κ κ h κ>0 3. P(γ H) =(Φ H (0))h Φ H (0) < P(γ H) h γ h H H \ H ). 4

43 κ<8/3,h > 5/8 SLE 8/3 0 <κ<8/3 λ(κ) = c(κ) = (6 κ)(8 3κ) κ Werner λ Brownian loops Poisson cloud SLE κ (streee-energy tensor) [3, ] > 0 43

44 A A. Ĉ D ( ) Ĉ Ĉ \ D Ĉ, D (simply commected domain) C D = {z C : z < } A. (Riemann mapping theorem) D C D ω D D D f(w) =0 f (w) > 0 (A.) [, 5] H A A = H A H\A A compact H-hull compact H-hull Q A Q A Q H \ A C ( A.) : H \ A D f () A Möbius f () (z) = αz αβ, β =,α H (A.) z β f () : D H f () (0) = α ). f (3) A f (3) A : H \ A H = f () f () H\A A f () A : H\A D {z C : z =} f () : D H H ) f (3) A : H \ A H ( A ) (A.) f () A f () z = β f () β f (3) A (3) lim [f z A (z) z] =0 ( (hydrodynamic condition) ) 44

45 f (3) A H \ A z R, /f (3) A (/z) 0 0 analytic 0 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 (A.3) lim [g A(z) z] = 0 z (A.4) g A (z) =z + c z + c z +, c j R (A.5) A. D BM τ D = inf{t 0:B t / D} (A.6) 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.7) 45

46 D D z D D hm(z,d; ) hm(z,d; V )=P z [B τd V ], V D (A.8) (harmonic measure (in D from z)) (A.7) u(z) = F (w)hm(z,d; dw) (A.9) D D u(z) = F (w)h D (z,w) dw D (A.0) 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 z w 0 H D (z,w) =δ(w w 0 ), w 0,w D. ( )BM A.3 f : D D D = D D z D, V D hm(f(z),d ; f(v )) = hm(z,d; V ) (A.) (A.0) (half-infinite strip) H D (f(z),f(w)) = f (w) H D (z,w), z D, w D. (A.) D = {z = x +iy : x>0, 0 <y<π} (A.3) D = {i q : q (0,π)} H D (z,i q),z D, q (0,π) q (0,π) H D (z,i q) z x H D(x +iy, i q)+ y H D(x +iy, i q) =0, x+iy D (separation of variables) (q ) H D (x +iy, i q) =X(x)Y (y) 46

47 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.4) (A.5) (A.5) Y (y) =a sin( cy)+b cos( cy) y =0,π Y (y) =0 b =0, c = n, n =,, 3, Y (y) =a sin(ny), n =,, 3, (A.4) X (x) =n X (x) x X(x) = const. e nx H D (x +iy, i q) = a n (q)e nx sin(ny) n= a n (q) q lim H D(x +iy, i q) =δ(y q) x 0 (A.6) a n (q) = π sin(nq) lim x 0 H D(x +iy, i q) = π sin(nq) sin(ny) n= = π n= e i n(q+y) + π n= e i n(q y) δ(x) = π n= e i nx 47

48 q, y > 0 (A.6) H D (x +iy, i q) = π e nx sin(ny) sin(nq), x+iy D, q (0,π) (A.7) n= r R, R > 0 z = x +iy ζ = α +iβ : ζ = f(z) =r + Re z. (A.8) D D = {ζ = α +iβ : ζ r >R,β>0} = H \B(r, R) (A.9) (A.8) {x : x>0} {α : α>r+ R} {x +iπ : x>0} {α : α<r R} {i q : q (0,π)} {r + Re i q ; q (0,π)} e x+i y = ζ r R, ζ r ex = R, ei y = ζ r ζ r e x e i y = R ζ r n ( R e nx e i ny = ζ r [( ) n ] R e nx sin(ny) = Im ζ r [ ] = R n Im (ζ r) n q (0,π) ) n f(i q) =r + Re i q, i q D f (i q) =Re i q f (i q) = R 48

49 (A.) H D (ζ,r + Re i q ) = R { [ R n Im π (ζ r) n n= = [ sin(nq)r n Im π n= (ζ r) n ]} sin(nq) ] (A.0) A.3 capacity (A.4) z g A (z) analytic φ A (z) = Im (z g A (z)) (A.) H \ A A. A. τ = τ H\A = inf{t : B t R A} (A.) ( )BM R compact H-hull A φ A (z) φ A (z) =E z [φ A (B τ )] (A.3) (A.) φ A (z) =E z [Im (B τ )] E z [Im (g A (B τ ))] = E z [Im (B τ )] B τ H \ A g A Im (g A (B τ ))=0 Im (g A (z))=im(z) E z [Im (B τ )] (A.4) z =iy, y > 0 Im (i y) =y Im (g A (i y)) = y E i y [Im (B τ )] (A.5) ( (LHS) = Im i y + c ( )) i y + O y = y c ( ) y + O y y c y ( ) + O y = y E i y [Im (B τ )]. 49

50 c = lim y yei y [Im (B τ )] A Q capacity (half-plane capacity) hcap(a) A Q hcap(a) lim y yei y [Im (B τh\a )], A Q. (A.5) g A (z) =z + hcap(a) z ( ) + O z, z (A.6) 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.7) (A.8). g A g ra (z) =rg A (z/r), g A+x (z) =g A (z x)+x (A.9) (A.30) (A.6) (A.9) = z + hcap(ra) z ( ) + O z { z (A.9) = r r + hcap(a) ( )} + O z/r z/r ( ) = z + r hcap(a) + O z z /z (A.7) (A.30) = z + hcap(a + z) z + O ( ) z (A.30) = (z x)+ hcap(a) z x = z + hcap(a) z + O ( + O ) ( z z x ) + x 50

51 (A.8) 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) g B = g ga(b\a) g A, A B Q (A.3) A.5 A, B Q,A B hcap(b) = hcap(a) + hcap(g A (B \ A)) (A.3). (A.6) g A (z) = z + hcap(a) z ( ) + O z g ga(b\a)(z) = z + hcap(g A(B \ A)) z + O, ( z ), z. ] ( ) [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.3) g B (z) =z + hcap(b) z (A.3) ( ) + O z, 5

52 B h t (U t) SDE B. h t (z) h t(z) γ 0 <κ 4 SLE κ H H, 0, H γ H Φ H : H H, Φ H (0) = 0, Φ H (z) z (z ) t SLE κ γ[0,t] H Φ H (γ[0,t]) Φ H (H) =H Φ H (γ[0,t]) SLE κ H SLE κ gt gt : H \ Φ H (γ[0,t]) H. g t d dt g t (z) =(Φ H (U t)) gt (z) U t, Ut Φ H(U t ) (B.) SLE κ g t (z) (Φ H (U t)) ( A.3 A.4 hcap(ra) =r hcap(a) ) SLE κ g t f t = gt 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) z g t g t (z) U t d dt f t(g t (z)) + f t (g t(z)) =0 g t (z) U t d dt f t(z) = f t (z) z U t, f 0 (z) =z (B.) h t =Φ gt(h) (B.3) 5

53 h t = g t Φ H f t, h 0 =Φ H. (B.4) d dt h 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 (g t ) (z) = d dz g t (z).) (B.) (B.) d dt h (Φ H t(z) = (U t)) [ ] gt (Φ H (f t (z))) Ut +(gt ) (Φ H (f t (z))) Φ H(f t (z)) f t(z) z U t h 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 z d dt h t(z) = (h t(u t )) h t (z) h t (U t ) h t(z) z U t d dz h t (z) = (h t(u t )) h t(z) (h t (z) h t (U t )) + h t(z) (z U t ) h t (z) z U t (B.5) (B.6) z U t d dt h d t(u t ) = lim z U t dt h t(z) = 3h t (U t) d dt h t(u t ) = lim z U t d dt h t(z) = (h t (U t )) h t(u t ) 4 3 h t (U t ) (B.5) (B.6) (B.7) (B.8) B. d(h t(u t )) h = h(h t(u t )) h h t (U t )du t + { } h(h )(h t (U t)) h (h t (U t)) + h(h t (U t)) h h t (U t) (du t ) +h(h t (U t)) h d dt h t (U t)dt. 53

54 (B.8) du t = κdb t, (du t ) = κdt (RHS) = h(h t (U t) h [ κ h t (U t) + h t (U t) db t { (h t (U t )) (h )κ (h t(u t )) + κh t (U t ) h t(u t ) + h t(u t ) = h(h t(u t )) h [ κ h t (U t ) h t(u t ) db t + (3.6) { (h )κ + ( (h t (U t )) h t(u t ) (h t (U t )) (h t(u t )) h ( κ 4 3 )} ] t (U t) dt ) h t (U t ) h t(u t ) } ] dt 54

55 [] 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] R. Friedrich and W. Werner, Conformal restriction, highest-weight representations and SLE, Commun. Math. Phys. 43 (003) 05-. [4] W. Kager and B. Nienhuis, A guide to stochastic Löwner evolution and its application, J. Stat. Phys. 5 (004) [5], Loewner 006 ). http : // u.ac.jp/j/katori/ pdf file [6], SLE, Vol.6, No.7 (007) [7] G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005). [8] G. Lawler, O. Schramm, and W. Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 6 (003) [9] G. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 3 (004) [0] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 8 (000) -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) [] W. Werner, Conformal restriction and related questions, Probability Surveys (005)