統計力学模型とＳＬＥ 中央大学理工学部 香取眞理
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1 1 Loewner
2 (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 (restriction property) µ : NG, Ising (locality property) µ : NG, [ ( ) ] µ µ µ µ µ µ LERW SAW per Ising SAW per : : OK OK 2
3 2. Bessel Brown, martingale, Ito 2.2 d Bessel BESd 2.3 BESd 3. Schramm-Loewner Evolution (SLE) 3.1 Riemann Loewner 3.4 SLE BESd
4 4. SLE 4.1 Schramm Markov µ (SLE6) 8/ 3 µ (SLE8/3) 4.3 3κ c = µ µ ( 8)(6 κ ) 2κ per SAW µ κ, κ (0, ) 4 SLE ( ) Virasoro ( c) ( SLE : 0 < κ κ LERW, SAW (percolation), (Ising model) (Abelian sandpile model) 4) ( SLE : 4 < κ < 8) κ ( SLE : κ = κ 8)
5 5
6 6
7 O. Schramm, Scaling limits of loop erased random walks and uniform spanning trees, Israel J. Math. 118 (2000) G. F. Lawler, Conformally Invariant Processes in the Plane, American Math. Soc., R. Friedrich and W. Werner, Conformal restriction, highestweight representations and SLE, Commun. Math. Phys. 243 (2003) S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy s formula, scaling limits, C.R.Acad.Sci.Paris, Ser.I Math. 333 (2001) S. Smirnov, Conformal invariance in random cluster models, I. Holomorphic fermions in the Ising model, arxiv: V. Beffara, The dimension of the SLE curves, Ann. Probab. 36 (2008) SLE 62 (no.7) (2007)
8
9 9
10 10
11 11
12 12 RW (complex
13 (loop erased RW: LERW) 13
14 (loop erased RW: LERW) 14
15 (loop erased RW: LERW) 15
16 (loop erased RW: LERW) 16
17 (loop erased RW: LERW) 17
18 (LERW) 18
19 (LERW) 19
20 (self avoiding walk : SAW) 20 a1 n=200 a2 n=800
21 (SAW) 21
22 (critical percolation model) 22
23 (percolation exploration process) 23
24 24 (b1) (b2)
25 Ising (critical Ising model) 25
26 Ising (Ising Interface) 26
27 1.2 Markov 27
28 28
29 29
30 Markov 30
31 31
32 1.3 (restriction property) 32
33 1.3 (restriction property) 33
34 (locality property) 34
35 (locality property) 35
36 .1,, 36
37 37
38 38
39 39
40 40
41 41
42 42
43 43
44 44
45 . d BESd 45
46 46
47 47
48 48
49 49
50 . BESd 50
51 51
52 52
53 53
54 54
55 55
56 56
57 57
58 58
59 59
60 A.1 60
61 61
62 62
63 63
64 64
65 Schramm Loewner SLE.1 Riemann 65
66 66
67 67
68 68
69 69
70 70
71 . 71
72 72
73 73
74 74
75 75
76 76
77 77
78 78
79 79
80 80
81 81
82 . Loewner 82
83 83
84 84
85 85
86 86
87 87
88 . SLE BESd 88
89 89
90 Oded Schramm (December 10, 1961 in Jerusalem, Israel September 1, 2008, Washington State, USA) 90
91 91
92 92
93 93
94 94
95 95
96 96
97 97
98 98
99 99
100 SLE 3 3 ) 100
101 101 (c) (b) (t) R (swallowed) Ut (c) x1, x2, Ut
102 ) ( ˆ 2 ) ( ˆ + = = = + = + = κ κ κ κ d d d x X t dt d t db x t dx x g t dt t db x t dg c c
103 103 κ = 1.0 κ = 2.0 Tom Kennedy s Home Page
104 104 κ = 3.0 κ = 4.0 Tom Kennedy s Home Page
105 105 = 5.0 κ = 6.0 κ Tom Kennedy s Home Page
106 106 κ = 7.0 = 8.0 κ Tom Kennedy s Home Page
107 SLE.1 Schramm 107
108 108
109 109
110 110
111 111
112 112
113 113
114 114 A
115 115
116 116
117 117
118 118
119 119
120 (locality property) 120
121 121
122 122
123 123
124 124
125 125
126 126
127 127
128 128
129 129
130 130
131 [1] SLE Ut=Bt filtration, Bessel flow SLE [2] SLE SLE SLE( ), multiple SLE, [3] SAW, Ising Potts O(N) DLA, height models, Coulomb gas,, universality class ) [4] (Cardy s formula, Schramm Zhou s 2pt martingale) CLE (conformal loop ensemble) 131
Katori_Bussei_SS09_HP1.dvi
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
Katori_SLE_Feb2010_s1.dvi
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
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-.................................
( ) Loewner SLE 13 February
( ) Loewner SLE 3 February 00 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 (009 8 7-9 ) . d- (BES d ) d B t = (Bt, B t,, Bd t ) (d
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187 1 p p 188 2 t 3 y 1 1 p 2 3 4 5 p p 1 i 2 189 190 1 i 1 i o p d d dt 1 2 3 4 5 6 9 0 191 192 d c d b db d 1 i 1 193 194 2 d d d r d b sla sla 1 o p i o o o op 195 u u 1 u t 1 i u u 1 i 196 1 2 bd t
2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W
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SGC -83 2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ 1 3 4 Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W [14] c = 1 CFT [8] Rational CFT [15], [56]
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