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1 SLE (18 May 2009 (version 1)) SLE Bessel [13] N Hermite N 2 Hermite N N N N N [7] Riemann Brown 2000 Schramm-Loewner Evolution (SLE) [10] [14] SLE [5, 17] 2006 Werner SLE [22] [16] : katori@phys.chuo-u.ac.jp HP 1

2 Brown Bessel Itô Karlin-McGregor Brown Schur Selberg Hermite Bru Markov Schramm-Loewner Evolution (SLE) Riemann Loewner SLE κ : Brown Bessel Itô Schramm-Loewner evolution (SLE) Virasoro 2

3 Part I : Brown Bessel (Ω, F, P) 1 Brown {B(t, ω)} t [0, ) 2 1. B(0,ω)= ω Ω ω B(t, ω) t 3. t 0 0 <t 1 < <t M,M=1, 2,..., {B(t j+1 ) B(t j )} j=0,1,...,m 1 0 t j+1 t j Gauss p(t, y x) = 1 } (y x)2 exp {, t > 0, x,y R (1.1) 2πt 2t R Brown t j, 1 j M [a j,b j ] ( ) b1 bm M 1 P B(t j ) [a j,b j ],j =1,...,M = dx 1 dx M p(t j+1 t j,x j+1 x j ) a 1 a M p(t s, y x) Brown 3 s 0 Brown B(s) Brown B(u),u<s Brown B(t),t>s 1 Ω Ω A Ω F σ (i) Ω F, (ii) A F A A c F, (iii) A 1,A 2,..., F n A n F, 3 2 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 ) 3 (1.1) t p(t, y x) =1 p(t, y x) p(0,y x) = 2 x2 δ(y x) heat kernel) Green propagator, 3 j=0 2

4 Markov τ Markov τ u u Markov Brown Markov s Markov τ Markov Markov s x t (> 0) y g(s, x; t, y) Brown g(s, x; t, y) t s g(s, x; t, y) y x d d N {1, 2, } d Brown B(t) =(B 1 (t),...,b d (t)) X (d) (t) = B(t) = B 1 (t) 2 + B 2 (t) B d (t) 2 (1.2) d Bessel d Bessel X (d) (t),d N p (d) (t, y x) ν = d 2 2 d =2(ν + 1) (1.3) p (d) (t, y x) = yν+1 1 ( x ν t exp x2 + y 2 2t p (d) (t, y 0) = y 2ν+1 exp 2 ν Γ(ν +1)tν+1 ) ( xy I ν t ( y2 2t ), t > 0, x > 0, y 0 ), t > 0, y 0 (1.4) I ν (z) Bessel Γ(z) : Γ(z) = I ν (z) = 0 n=0 e u u z 1 du Re u>0, (1.5) (z/2) 2n+ν Γ(n + 1)Γ(ν + n +1). (1.6) Bessel X (d) (t) Bessel (1.3) d =1, 2, 3, 4,... ν = 1/2, 0, 1/2, 1,... 4

5 (1.4) ν (1.4) (1.3) d 1 d Bessel Bessel (d 2) (1 d<2) [3] 1.2 Itô s>0 B(s) Brown B(u),u<s B(t) B(s) t 4 Brown X(s)dB(s) Z(t) (quadratic variation) Z t : Z t = P- lim n n (Z(t j+1 ) Z(t j )) 2 j=0 P- lim [0,t] 0 t 0 <t 1 < <t n t n 5 Z(t) Z t =0 Z(t), Ẑ(t) Z, Ẑ t 1 4 { } Z + Ẑ t Z Ẑ t 0 dz(t)dẑ(t) =d Z, Ẑ t Brown db(t)db(t) =dt dt Brown B 1 (t) B 2 (t) Brown db 1 (t)db 2 (t) =0 4 ) 5 {X n } n=1 X (Ω, F, P) n {X n } X ε>0 lim n P( X n X >ε)=0 P- lim n 5

6 d Brown B(t) =(B 1 (t),b 2 (t),...,b d (t)) db j (t)db k (t) =δ jk dt dm j (t)dm k (t), 1 j, k d M(t) =(M 1 (t),m 2 (t),...,m d (t)) Z(t) = (Z 1 (t),z 2 (t),...,z d (t)) M(t) A(t) =(A 1 (t),a 2 (t),...,a d (t)) d F R d F (Z(t)) df (Z(t)) = d j=1 F x j (Z(t)) (dm j (t)+da j (t)) 15j,k5d Itô [12, 3] F (x) = d j=1 x2 j 2 F x j x k (Z(t))dM j (t)dm k (t) (1.7) (1.2) d Bessel dx (d) (t) =db(t)+ d 1 dt (1.8) 2X (d) (t) d >1 d Bessel Brown d 1 t ds 2 0 X (d) (s) 6. (1.8) t u(t, x) =1 2 1 u(t, x)+d u(t, x). (1.9) 2 x2 2x x Kolmogorov [12, 3] (1.4) d Bessel p (d) (t, y x) t =0 δ(x y) 1.3 Karlin-McGregor X 1,X 2 X 1 (τ) =X 2 (τ) τ x = X 1 (τ) =X 2 (τ) 2 N 6 (1.4) d =1 ν = 1/2 I 1/2 (z) = 2/(πz) cosh z p (1) (t, y x) = p(t, y x)+p(t, y x) p (1.1) Brown X (1) (t) = B(t) (1.2) d =1 6

7 [7] Karlin-McGregor Lindström-Gessel-Viennot Fermi Slater [4, 7, 20] 1.1 (Karlin-McGregor ) g(s, x; t, y) N x j, j =1, 2,...,N N y j, j =1, 2,...,N x 1 <x 2 < <x N y 1 <y 2 < <y N N [ ] [s, t] det g(s, x j ; t, y k ) [ ] det g(s, x j ; t, y k ) = N sgn(σ) g(s, x j ; t, y σ(j) ). (1.10) σ S N j=1 S N {1, 2,...,N} (s, x) (t, y) Ω(s, x; t, y) π j Ω(s, x j ; t, y σ(j) ),j =1, 2,...,N (1.10) σ N (σ, π 1,...,π N ) (σ, π 1,...,π N ) { } τ = sup s<u<t: π 1,...,π N u 7 v R τ π l1,π l2 1 π l1 = π l1 ( v)π l1 (v ), π l2 = π l2 ( v)π l2 (v ) 2 v π l 1 = π l1 ( v)π l2 (v ), π l 2 = π l2 ( v)π l1 (v ) j l 1,l 2 π j = π j σ (l 1,l 2 ) σ (σ, π 1,...,π N ) (σ,π 1,...,π N) (1.11) 7 sup (supremum) inf (infimum) 7

8 y σ(4) y σ(3) y σ(1) y σ(2) y σ(5) y π (k) y π (j) y π (k) y π (j) π ' 1 π ' 2 υ υ υ π 1 π 2 π 2 π 1 x j x k x j x k x 1 x 2 x 3 x 4 x 5 (a) (b) 1: (π 1,...,π N ) 2: (a) π l1 π l2 (b) π l v π l1 1 π l π l sgn(σ )= sgn(σ) (σ, π 1,...,π N ) (1.10) (1.11) (1.10) σ sgn(σ) =1 (1.10) 1.4 Brown R N W A N = {x RN : x 1 <x 2 < <x N } A N 1 Weyl W A N Brown 8 0 W A N x =(x 1,...,x N ) Brown W A N t WA N y =(y 1,...,y N ) 1.1 Karlin-McGregor [ ] f N (t, y x) = p(t, y j x k ) det (1.12) 8 (0, ) Brown p (1.1) g abs (s, x; t, y) =p(t s, y x) p(t s, y x), 0 <s<t,x,y 0 Brown [7] (1.4) d =3 ν =1/2 I 1/2 (z) = 2/(πz) sinh z p (3) (t s, y x) =(y/x)g abs (s, x; t, y) (1.12) (1.14) N 8

9 0 x Brown t W A N N N (t, x) = f N (t, y x)dy (1.13) dy = W A N N dy j j=1 T>0 (0,T] Brown Z(t) =(Z 1 (t),z 2 (t),...,z N (t)) gn T (s, x; t, y) T s x W A N N Brown t y W A N g T N (s, x; t, y) = N N(T t, y) N N (T s, x) f N(t s, y x), 0 s<t T, x, y W A N (1.14) Schur [4, 20] N N (t, x) C 2(N) C 1 (N) h N [6, 8] h N (x) ( h N (x) = det x j 1 k ( x t ), ) = 1 j<k N C 1 (N),C 2 (N) (1.6) C 1 (N) =(2π) N/2 N j=1 x t 0 (1.15) (x k x j ) (1.16) N Γ(j), C 2 (N) =2 N/2 Γ(j/2) (1.17) x =(x 1,x 2,...,x N ) x = N x 2 j (1.15) Brown Z(t) T p N (t, y x) = h N(y) h N (x) f N(t, x, y), t > 0, x, y W A N (1.18) j=1 j=1 9

10 X(t) =(X 1 (t),x 2 (t),...,x N (t)) N 2 h N (x) =0 x 2 j=1 j p N (t, y x) t p N(t, y x) = 1 2 N 2 x 2 j=1 j p N (t, y x)+ 1 k N k j 1 x j x k x j p N (t, y x). (1.19) (1.9) d =3 Bessel Kolmogorov N (1.9) Bessel (1.8) (1.19) Brown X(t) =(X 1 (t),...,x N (t)) dx j (t) =db j (t)+ β 2 1 k N k j 1 dt, 1 j N (1.20) X j (t) X k (t) β =2 β (1.20) Dyson [7] 1.5 Schur Selberg μ 1 μ 2 μ N 0,μ j N 0 N {0} = {0, 1, 2,...} μ =(μ 1,...,μ N ) N μ μ j {μ j } j=1 l(μ) 1 j N μ j =0 (0, 0,...,0) = μ N x =(x 1,...,x N ) s μ (x) = det [ det x μ k+n k j [ x N k j ] ] (1.21) [4, 20] Vandermonde (1.16) [ ] = (x j x k )=( 1) N(N 1)/2 h N (x). (1.22) det x N k j 1 j<k N 10

11 s μ (x) x 1,...,x N μ Schur μ = s (x) =1 s μ (x) N Z[x 1,...,x N ] Taylor 1.1 ρ ψ(x) x =0 ψ(x) = x =(x 1,...,x N ), y =(y 1,...,y N ) [ ] det ψ(x j y k ) = h N (x)h N (y) N (x j y j ) ρ j=1 x 0 y 0 μ:l(μ) N c n x n+ρ n=0 s μ (x)s μ (y) N c μj +N j. (1.23) j=1 [ ] det ψ(x j y k ) = h N (x)h N (y) N N (x j y j ) ρ c j 1 j=1 j=1 { } 1+O( x, y ) (1.24) [ ] det ψ(x j y k ) = det = [ ] (x j y k ) ρ c n (x j y k ) n n=0 N (x j y j ) ρ j=1 n=(n 1,...,n N ) N N 0 N j=1 c nj [ ] det (x j y k ) n j. n =(n 1,...,n N ) σ S N σ(n) = (n σ(1),...,n σ(n) ) f(n) n f(σ(n)) = f(n), σ S N n N N 0 f(n) det [ ] (x j y k ) n j = 1 N! = n N N 0 f(n) [ ] det (x j y k ) n σ(j) σ S N [ ] det (x j y k ) n σ(j) f(n) 0 n 1 < <n N σ S N σ S N [ ] [ det (x j y k ) n σ(j) = det x n k j ] det 1 l,m N [ y nm l ] 11

12 N j=1 c nj n [ ] det ψ(x j y k ) = N (x j y j ) ρ j=1 N 0 n 1 < <n N j=1 c nj [ det x n k j ] det 1 l,m N μ j = n j N + j, 1 j N Schur (1.21) (1.23) s μ (x) x 1,...,x N μ s (x) =1 x 0 y 0 (1.24) [ y nm l ]. [ ] 1 f N (t, y x) = det e (x j y k ) 2 /(2t) 2πt = (2πt) N/2 e ( x 2 + y 2 )/(2t) det [ ] e x jy k /t ψ(x) =e x/t = x n /(n!t n ) 1.1 (1.24) x=0 x / t 0 y / t 0 f N (t, y x) = t N 2 /2 C 1 (N) e ( x 2 + y 2 )/(2t) h N (x)h N (y) { ( x 1+O, y )} t t (1.25) C 1 (N) (1.17) N N (t, x) = f N (t, y x)dy = W A N 1 C 1 (N) h N ( x t ) { ( )} x C 2 (N) 1+O, t x t 0 C 2 (N) = W A N e x 2 /2 h N (x)dx (1.26) Selberg [11] N h N (x) 2γ =(2π) N/2 (2a) N(γ(N 1)+1)/2 R N e a x 2 j=1 Γ(1 + jγ) Γ(1 + γ) (1.27) a =1/2,γ =1/2 (1.17) (1.15) 12

13 2 Hermite 2.1 Bru N N Hermite H(N) N N S(N) A t A A A t A Bru H(N) Wishart ξ jk (t), 1 j, k N [6, 7] Bru 9 λ(t) =(λ 1 (t),λ 2 (t),...,λ N (t)) H(N) Ξ(t) =(ξ jk (t)) 15j,k5N λ 1 (t) λ 2 (t) λ N (t) U(t) =(u jk (t)) ( ) U(t) Ξ(t)U(t) =Λ(t) = diag λ 1 (t),λ 2 (t),,λ N (t) Ξ(t) ( ) ( ) Γ jk,lm (t)dt = U(t) dξ(t)u(t) U(t) dξ(t)u(t) (U(t) dξ(t)u(t)) jj dυ j (t) (λ 1 (t),λ 2 (t),...,λ N (t)) τ : { } τ = inf t>0:λ j (t) =λ k (t) 1 j k N jk lm 2.1 [ Bru ] ξ jk (t), 1 j, k N Ξ(t) λ(t) dλ j (t) =dm j (t)+dj j (t), t (0,τ), j =1, 2,...,N. (2.1) M(t) =(M 1 (t),m 2 (t),...,m N (t)) dm j (t)dm k (t) =Γ jj,kk (t)dt J(t) =(J 1 (t),j 2 (t),...,j N (t)) dj j (t) = 15k5N k j 1 λ j (t) λ k (t) 1 {λ j (t) λ k (t)}γ jk,kj (t)dt + dυ j (t) 1 {ω} ω ω Itô (1.7) M M N N d(m M) =(dm) M + M (d M)+(dM) (d M) 13

14 2.2 Bjk R (t), BI jk (t), 1 j, k N 2N 2 Brown 1 j, k N 1 Bjk R (t), j < k 1 Bjk 2 I (t), j < k 2 s jk (t) = Bjj(t), R j = k a jk(t) = 0, j = k 1 Bkj R (t), j > k 1 Bkj I (t), j > k 2 2 (i) GUE Ξ GUE (t) = ( s jk (t)+ ) 1a jk (t), t [0, ) (2.2) 15j,k5N H(N) t [0, ), Ξ GUE (t) H(N) H(N) U(dH) μ GUE (H, t) = t N 2 /2 ( c 1 (N) exp 12t ) TrH2, H H(N) Tr c 1 (N) =2 N/2 π N 2 /2 N N U(N) U U(N) μ GUE (H, t)u(dh) H U HU H(N) Gauss (Gaussian unitary ensemble, GUE) [11, 13] GUE x 1 <x 2 < <x N x =(x 1,x 2,,x N ) g GUE (x,t)= t N 2 /2 C 1 (N) exp ) ( x 2 h N (x) 2 (2.3) 2t [11, 13] h N (x) N N Vandermonde (1.22) C 1 (N) (1.17) Bru Ξ GUE (t) dξ jk (t)dξ lm (t) dξ jj (t)dξ jj (t) =dbjj R(t)dBR jj (t) =dt j<k dξ jk (t) =dbjk R (t)/ 2+ 1dBjk I (t)/ 2=dξ kj (t) dξ jk (t)dξ jk (t) =dt/2 dt/2 =0,dξ jk (t)dξ kj (t) =dt/2 +dt/2 =dt (j, k) (l, m), (m, l) dbjk R (t)dbr lm (t) = dbjk I (t)dbi lm (t) =0 dξ jk(t)dξ lm (t) =0 14

15 dξ jk (t)dξ lm (t) =δ jm δ kl dt, 1 j, k, l, m N dm j (t)dm k (t) ( ) ( ) Γ jj,kk (t)dt = U(t) dξ(t)u(t) U(t) dξ(t)u(t) jj kk = u lj dξ lm u mj u pk dξ pq u qk l,m p,q = l,m u lj u lk u mj u mk dt = δ jk dt U(t) U(t) U(t) =U(t)U(t) = I N N Brown Γ jk,kj (t) =1, 1 j, k N (U(t) dξ(t)u(t)) jj (dυ j (t) =0) Bru λ(t) dλ j (t) =db j (t)+ 15k5N k j 1 dt, 1 j N (2.4) λ j (t) λ k (t) τ = (2.4) Dyson (1.20) β =2 (ii) GOE ( ) Ξ GOE (t) = s jk (t), t [0, ) (2.5) 15j,k5N S(N) t [0, ) Ξ GOE (t) S(N) V(dS) μ GOE (S, t) = t N(N+1)/4 c 2 (N) exp ( 12t ) TrS2, S S(N) S(N) c 2 (N) = 2 N/2 π N(N+1)/4 N N O(N) V O(N) μ GOE (S, t)v(ds) S t VSV S(N) Gauss (Gaussian orthogonal ensemble, GOE) GOE x 1 < x 2 < < x N x =(x 1,x 2,,x N ) g GOE (x,t)= t N(N+1)/4 C 2 (N) 15 exp ) ( x 2 h N (x) (2.6) 2t

16 C 2 (N) (1.17) [11, 13] GUE λ(t) dλ j (t) =db j (t) k5N k j 1 λ j (t) λ k (t) dt, 1 j N τ = Dyson (1.20) β = (i) GUE λ(t) =(λ 1 (t),...,λ N ) Brown X(t) =(X 1 (t),...,x N (t)) β =2 Dyson 3.1 Brown X(t) =(X 1 (t),...,x N (t)) GUE λ(t) =(λ 1 (t),...,λ N (t)) Dyson (1.20) N Brown β =2 Dyson (1.20) log log- Dyson 1 Coulomb (1.20) β β =1/k B T T =1/(2k B ) β 2 2 N 16

17 Part II : Schramm-Loewner Evolution (SLE) C S S = Z 1Z, Z S { z S n } Wn z = ω =(ω(0),...,ω(n)) : ω(0) = z, ω(j) S, ω(j) ω(j 1) =1, 1 j n (RW) Wn z =4n { ω Wn z 4 n C D 0 = x + } 1y : 1 <x<1, 0 <y<2, D 0 2 O =0 P =2 1 N N N NO =0 NP =2N 1 RW ND 0 Ω N (D 0 ; O, P) Z N (D 0 ; O, P) = 4 ω. (4.1) ω Ω N (D 0 ;O,P ) ω ω N Z N (D 0 ; O, P) C(D 0 ; O, P)N 2 0. f(n) g(n),n f(n)/g(n) 1,N C(D 0 ; O, P) D 0 Poisson H D0 (,P) O D 0 (loop-erased RW: LERW) Ω N (D 0 ; O, P) ω =(ω(0),ω(1),...) ω(j) =ω(k),j < k ω ω ω =( ω(0), ω(1),...) t 0 =0, ω(0) = ω(t 0 )=0 m 1 t m = max { } l>t m 1 : ω(l) = ω(t m 1 +1), ω(m) = ω(t m )= ω(t m 1 +1) Schramm-Loewner Evolution 17

18 ND 0 0 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 ( ) j ω 1/N = 1 ω(j), 0 j ω (4.2) N 1/ν N ω 1/N O ω /N 1/ν P =2 1 D 0 1/N ν N P γ P t γ = lim N ω /N 1/ν : γ :(0,t γ ) D 0, lim t 0 γ(t) =O, lim γ(t) =P, t γ (0, ). (4.3) t tγ γ d LERW =1/ν γ γ(t 1 ) γ(t 2 ), 0 t 1 <t 2 t γ LERW (4.3) K LERW (D 0 ; O, P) 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 ) ( ) (4.4) μ LERW (D 0 ;O,P ) (self-avoiding walk : SAW) Wn z { Wn,0 z = ω Wn }. z : 0 j<k n ω(j) ω(k) 2 <e β < 3 Wn,0 z e βn,n 11 f(n) g(n),n log f(n) log g(n),n ω e β ω (SAW) (4.1) SAW ZN SAW (D 0 ; O, P) = e β ω ω Ω 0 N (D 0;O,P ) 11 e β SAW connective constant S

19 b SAW > 0 Z SAW N (D 0 ; O, P) C SAW (D 0 ; O, P)N 2b SAW, N (4.5) LERW b LERW =1 SAW γ d SAW LERW d LERW (4.4) SAW μ SAW (D 0 ;O,P ) ( )=CSAW (D 0 ; O, P)μ SAW (D 0 ;O,P ) ( ) (critical percolation model) NP _ Λ Ν + Λ Ν 0 3: T H 1 0 C τ = exp(2π { 1/3) T = z 0 +(j+kτ) 3a : } j, k Z. a =2/3 z 0 = a 1 T a H O NP =2N 1,N N z T η(z) {0, 1} Bernoulli ν p, 0 p 1 ν p (η(z) = 1) = p, ν p (η(z) =0)=1 p. T 1 0 p 1/2 19

20 1 p >1/2 p c =1/2 p c T p c =1/2 Bernoulli N 1 b per =0 N N T ND 0 =Λ N 3 N =6 Λ N z T O NP Λ + N Λ N η(z) =1, z Λ + N, η(z) =0, z Λ N Dobrushin Λ N ν p η {0, 1} T ND 0 H ND 0 O ω 0 1 ν>0 (4.2) N d per =1/ν (4.3) γ γ μ per (D 0 ;O,P ) ( ) Ising (critical Ising model) Λ N =Λ N Λ + N Λ N z Λ N σ(z) { 1, 1} z Λ ± N σ(z) =±1 Dobrushin o Λ N =Λ N ( Λ + N )c ( Λ N )c σ { 1, 1} Λ N E(σ) = 1 2 z,z Λ N : z z = 3a σ(z)σ(z ) k B β =1/(k B T ) β>0 Gibbs π N,β (σ) = e βe(σ) Z N,β, Z N,β = σ { 1,1} o Λ N e βe(σ) T 0 Ising ω 1 +1 H ND 0 β T Ising β c = (log 3)/4 = d Ising (4.3) γ μ Ising (D 0 ;O,P ) ( ) 20

21 4.2 Markov f D 0 C f (z) 0, z D 0 f : D 0 f(d 0 ) (4.6) f D 0 O, P f(d 0 ) f(o),f(p ) 4.1 γ μ (D0 ;O,P )( )=C(D 0 ; O, P)μ (D0 ;O,P )( ) (4.7) (conformal covariance) (conformal invariance) (4.6) f μ (D0 ;O,P )( )= f (O) b f (P ) b μ (f(d0 );f(o),f(p ))( ) (4.8) b 4.1 N (4.8) (boundary scaling exponent) (4.8) 12 C(D 0 ; O, P) = f (O) b f (P ) b C(f(D 0 ); f(o),f(p )), (4.9) μ (D0 ;O,P )( ) = μ (f(d0 );f(o),f(p ))( ). (4.10) Markov (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 ) Markov γ (4.3) t [0,t γ ] (4.6) (4.2) f(z) =z/n,n N f (z) =1/N f (O) b = f (P ) b = N b (4.9) (4.5) 21

22 (4.2) f γ f(γ[t 1,t 2 ]), 0 <t 1 <t 2 <t γ, t2 t 1 f (γ(s)) d ds d γ θ :[0,t γ ] [0,t γ ] γ(t) γ(θ(t)) 4.3 (4.7) / Markov (restriction property) D 0 D 1 D 0 O, P D LERW μ LERW (D 1 ;O,P ) O P RW RW LERW LERW Radon-Nikodym dμ LERW (D 1 ;O,P ) dμ LERW (D 0 ;O,P ) < 1, D 1 D 0, D 1 D 0 SAW dμ SAW (D 1 ;O,P ) dμ SAW (D 0 ;O,P ) = 1{γ(0,t γ ) D 1 }, D 1 D 0 (4.11) 1{ω} ω (4.11) η Bernoulli η μ per μ Ising (locality property) D 1 D 0 O, P D 1 μ per (D 1 ;O,P )(γ(0,t]) = μper (D 0 ;O,P ) (γ(0,t])1{γ(0,t) D 1}, t (0,t γ ). (4.12) (4.11) γ(0,t γ ) (4.12) γ(0,t],t (0,t γ ) 22

23 5 Schramm-Loewner Evolution (SLE) 5.1 Riemann D, D C D, D C, z, w D, z,w D Riemann f : D D, f(z) =z, f(w) =w f (w) =1 H = {z C :Imz>0} μ (H;0, ) D C,D C, z, w D μ (D;z,w) H \ D D H w = f(w) = f (w) =1 w f(w ) w f μ (D;z, ) ( )= f (z) b μ (f(d);f(z), ) ( ), z D. (5.1) 5.2 Loewner γ :(0, ) H lim γ(t) =U 0 R lim γ(t) = t 0 t H t γ(0,t] H \ γ(0,t] t (0, ) H \ γ(0,t] H f lim f (z) =1 g t z a(t) > 0 g t (z) =z + a(t) + O( z 2 ), z z a(t) γ(0,t] (half-plane capacity) g t (z) (chordal) Loewner d dt g t(z) = da(t)/dt g t (z) U t, g 0 (z) =z. (5.2) U t = g t (γ(t)) R t U t U t a(t) Loewner (5.2) g t : H t H,t (0, ) H t H H t pioneer point H pion = H s 0 s<t γ(0) R γ :(0, ) H H pion t = R γ(0,t] g t γ 23

24 5.3 SLE κ γ a(t) =at, a > 0 U t Loewner (5.2) g t γ μ (H;0, ) / Markov U t γ U t U t 1.1 Brown B(t) Loewner d dt g t(z) = a g t (z) U t, g 0 (z) =z, U t = B(t) (5.3) Schramm Schramm-Loewner evolution [15] 13 Schramm κ (0, ) a = 2 κ SLE κ SLE κ g t SLE κ γ SLE κ μ κ (H;0, ) SLE κ lim γ(t) =B(0) = 0 lim γ(t) = t 0 t 5.1 (Lawler-Schramm-Werner [9]) 14 Markov SLE κ, κ (0, ) (5.3) h t (z) =g t (z) U t = g t (z) B(t) z H x dh t (x) =db(t)+ (1.8) a h t (x) dt, h 0(x) =x>0 (5.4) a = d 1 2 d =2a +1 d = 4 κ +1 κ = 4 d 1 (5.5) 13 Oded Schramm (ICM 2006) Wendelin Werner [16] John Charles Fields 1998 Andrew Wiles Fermat ICM 2006 Gauss 24

25 0 0 0 (a) (b) (c) 4: (a) 0 <κ 4 SLE (b) H H 4 <κ<8 SLE (c) H κ 8 SLE h t (x) x>0 (2a +1)=(4/κ +1) Bessel Bessel (Bessel flow) 15 SLE κ κ (phase) 4 (5.5) d =2 κ =4,d =3/2 κ =8 5.2 (Lawler-Schramm-Werner [9]) (a) 0 <κ 4 SLE κ γ γ(0, ) H (b) κ>4 γ γ(0, ) R (c) κ 8 γ H γ[0, ) =H. 5.3 (Beffara[1]) κ 8 SLE κ γ(0, ) Hausdorff 16 d(κ) =1+ κ x>0 d Bessel T x (i) d 2 P(T x = ) =1, x >0. (ii) 1 d<2 P(T x < ) =1, x >0. (iii) 3/2 <d<2 0 <x<y P(T x = T y ) > 0. (iv) 1 d 3/2 0 <x<y P(T x <T y )=1. 16 d R d V ε >0 ε U 1,U 2,... diam(u n ) sup{ x y : x, y U n } <ε,n 1, V U n. α>0 n 1 25

26 5.4 H { } D = D H :, H \ D, 0 D D D D H Φ D Φ D (z) =z + o(1),z. SLE κ γ { } τ D = inf t : γ H \ D t <τ D γ D SLE γ(0,τ D ) Φ D Φ D (γ(0,τ D )) H = H R γ Φ D (γ(0,τ D )) gt (z) Loewner (5.3) Φ D Itô gt (z) U t (5.3) du t = b Φ t (U t ) dt + db(t). (5.6) Φ t(u t ) b(κ) = 3a 1 2 = 6 κ 2κ. (5.7) κ =6 b =0 D D g t (z) g t (z) t<τ D, U t = B(t) Brown SLE, (5.3) 5.4 SLE κ μ κ (H ;0, ) κ =6 κ 6 (5.6) U t Girsanov [12, 3] M t 17 f Schwarz H α ε (V ) = inf n 1(diam(U n )) α inf ε H α (V ) = lim ε 0 H α ε (V ) Hausdorff α V Hausdorff dim H (V ) = inf{α : H α (V )=0} = sup{α : H α (V )= }. 17 j τ j τ 1 <τ 2 < j Z(t τ j ) a b = min{a, b} Z(t) 26

27 Sf(z) = f (z) f (z) 3f (z) 2 2f (z) 2 c = 2b(3 4a) a = (3κ 8)(6 κ) 2κ (5.8) { t } a M t = exp c 0 12 SΦ s(u s )ds Φ t(u t ) b (5.9) dμ (H ;0, ) dm t = b Φ t (U t) Φ t(u t ) M tdb(t) 18 κ = 2 4,D D a { dμ } (D;0, ) a (γ) =M = 1{γ(0, ) D} exp c 0 12 SΦ s(u s )ds (5.10) (5.8) κ 4 κ =8/3 c =0 (5.10) 1{γ(0, ) D} (4.11) 5.5 SLE κ μ κ (H ;0, ) κ = μ per (D 0 ;0,P ) μsaw (D 0 ;O,P ) μ6 (H;0, ), μ 8/3 (H;0, ) H D 0 Smirnov [18, 21] μ LERW (D 0 ;0,P ) = μ2 (H;0, ) [15, 9] μising (D 0 ;0,P ) = μ3 (H ;0, ) [19] = 5.3 (5.7) 19 d LERW = d(2) = 5 4, d SAW = d(8/3) = 4 3, d Ising = d(3) = 11 8, d per = d(6) = 7 4, b LERW = b(2) = 1, b SAW = b(8/3) = 5 8, b Ising = b(3) = 1 2, b per = b(6) = (i) t γ(t) γ(0, ) D Φ t (U t) 1,t (ii) SΦ t (U t ) 0 κ 4 (i) [9] 19 t γ(t) (5.1) (5.9) (5.1) (5.7) b 27

28 a (a 0 =0) 0 a [a, a ]=1 L n (n =1, 2, 3, ) L n (n = 1, 2, 3, ), L 0 [L n,l m ]=(n m)l n+m + c n(n2 1) δ n+m,0, n,m Z (6.1) 12 c Virasoro [22] (L n b =0,n 1) b L n,n 1 L 0 b = b b b b n k,...,n 1 ; b = L nk L n1 b, 0 <n 1 n 1 n k Verma L 0 L 0 n k,...,n 1 ; b =(b + n n k ) n k,...,n 1 ; b. L 0 b k j=1 n j 2 L 1 L 1 b L 2 b ( ) ( b L 2 L 2 b b L 2 L 1 L 1 b 4b + c/2 6b det = det b L 1 L 1 L 2 b b L 1 L 1 L 1 L 1 b 6b 4b(1 + 2b) ) =0 (6.2) Kac ( ) 1 χ 2,1 = 2 L 2b +1 1L 1 L 2 b (6.3) 3 2 null vector L n,n 1 2 b 0 (6.2) c = 2b(5 8b) 1+2b 28 (6.4)

29 (5.7) (5.8) κ b c (6.4) L 1 x, L 2 1 x x 20 (6.3) b x + a 2 x x, a = 2 2b +1 = κ 3 (2a+1) Bessel (5.4) (generator) G (2a+1) Kolmogorov (1.9) t u = G(d) u. SLE κ (5.8) Virasoro (5.7) 2 [2] : [1] V. Beffara : The dimension of the SLE curves, Ann. Probab. 36, (2008); [2] R. Friedrich and W. Werner : Conformal restriction, highest-weight representations and SLE, Commun. Math. Phys. 243, (2003); [3] I. S. E. (2001). [4] : 13, No.4, ([296]-[307]) (2003). [5] SLE, 62, No.7, (2007). [6] M. Katori and H. Tanemura : Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, J. Math. Phys. 45, (2004); { } (n 1)Δ 20 L n L n = (x z) n 1 (x z) n 1 x z =0, Δ=0 29

30 [7] 6 ), (2006) [8] M. Katori and H. Tanemura : Noncolliding Brownian motion and determinantal processes, J. Stat. Phys., 129, (2007); [9] G. F. Lawler : Conformally Invariant Processes in the Plane, (American Mathematical Society, 2005). [10] I, II (2004). [11] M. L. Mehta : Random Matrices, 3rd ed. (Elsevier Academic Press, London, 2004). [12] (1999). [13] (2005). [14] 2005). [15] O. Schramm : Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118, (2000); [16],, No.545 ( ), pp.46-51,. [17] SLE Schramm-Loewner Evolution,, No.546 ( ), pp.7-12, [18] S. Smirnov : Critical percolation in the plane: Conformal invariance, Cardy s formula, scaling limits, C. R. Acad. Sci. Paris, Sér. I Math. 333, (2001); Smirnov web page pdf file smirnov/papers/index.html. [19] S. Smirnov : Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, [20] :,,, [21],, No.546 ( ), pp [22] 2006). 30

統計力学模型とＳＬＥ中央大学理工学部　香取眞理

統計力学模型とＳＬＥ中央大学理工学部　香取眞理 1 Loewner 0. 1. 1.1 (loop erased RW : LERW) (self avoiding walk : SAW) (critical percolation model) Ising (critical Ising model) 1.2 Markov (conformal cov./inv.) Markov (domain Markov property) 1.3 LERW