getdoca8be. 244KB Jun 04 2011 12:04:49 AM

❊❧❡❝t✳ ❈♦♠♠✳ ✐♥ Pr♦❜❛❜✳ ✽ ✭✷✵✵✸✮✷✽✕✹✷

❊▲❊❈❚❘❖◆■❈
❈❖▼▼❯◆■❈❆❚■❖◆❙
✐♥ P❘❖❇❆❇■▲■❚❨

❙▲❊ ❆◆❉ ❚❘■❆◆●▲❊❙
❏❯▲■❊◆ ❉❯❇➱❉❆❚

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s✱ ❇ât✳ ✹✷✺✱ ❯♥✐✈❡rs✐té P❛r✐s✲❙✉❞✱ ❋✲✾✶✹✵✺ ❖rs❛② ❝❡❞❡①✱ ❋r❛♥❝❡
❡♠❛✐❧✿

❥✉❧✐❡♥✳❞✉❜❡❞❛t❅♠❛t❤✳✉✲♣s✉❞✳❢r

❙✉❜♠✐tt❡❞

✶✸ ❏✉♥❡ ✷✵✵✷✱

❛❝❝❡♣t❡❞ ✐♥ ✜♥❛❧ ❢♦r♠

✶✵ ❋❡❜r✉❛r② ✷✵✵✸

❆▼❙ ✷✵✵✵ ❙✉❜❥❡❝t ❝❧❛ss✐✜❝❛t✐♦♥✿ ✻✵❑✸✺✱ ✽✷❇✷✵✱ ✽✷❇✹✸
❑❡②✇♦r❞s✿ ❙t♦❝❤❛st✐❝ ▲♦❡✇♥❡r ❊✈♦❧✉t✐♦♥✳ ❋❑ ♣❡r❝♦❧❛t✐♦♥✳ ❉♦✉❜❧❡ ❞♦♠✐♥♦ t✐❧✐♥❣s✳ ❯♥✐❢♦r♠
s♣❛♥♥✐♥❣ tr❡❡✳

❆❜str❛❝t
❇② ❛♥❛❧♦❣② ✇✐t❤ ❈❛r❧❡s♦♥✬s ♦❜s❡r✈❛t✐♦♥ ♦♥ ❈❛r❞②✬s ❢♦r♠✉❧❛ ❞❡s❝r✐❜✐♥❣ ❝r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s
❢♦r t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❝r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥✱ ✇❡ ❡①❤✐❜✐t ✏♣r✐✈✐❧❡❣❡❞ ❣❡♦♠❡tr✐❡s✑ ❢♦r ❙t♦❝❤❛s✲
t✐❝ ▲♦❡✇♥❡r ❊✈♦❧✉t✐♦♥s ✇✐t❤ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs✱ ❢♦r ✇❤✐❝❤ ❝❡rt❛✐♥ ❤✐tt✐♥❣ ❞✐str✐❜✉t✐♦♥s ❛r❡
✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞✳ ❲❡ t❤❡♥ ❡①❛♠✐♥❡ ❝♦♥s❡q✉❡♥❝❡s ❢♦r ❧✐♠✐t✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❡✈❡♥ts ❝♦♥✲
❝❡r♥✐♥❣ ✈❛r✐♦✉s ❝r✐t✐❝❛❧ ♣❧❛♥❡ ❞✐s❝r❡t❡ ♠♦❞❡❧s✳

✶ ■♥tr♦❞✉❝t✐♦♥
■t ❤❛❞ ❜❡❡♥ ❝♦♥❥❡❝t✉r❡❞ t❤❛t ♠❛♥② ❝r✐t✐❝❛❧ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♠♦❞❡❧s ❢r♦♠ st❛t✐st✐❝❛❧ ♣❤②s✐❝s ❛r❡
❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t ✐♥ t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t❀ ❢♦r ✐♥st❛♥❝❡✱ ♣❡r❝♦❧❛t✐♦♥✱ ■s✐♥❣✴P♦tts ♠♦❞❡❧s✱ ❋❑
♣❡r❝♦❧❛t✐♦♥ ♦r ❞✐♠❡rs✳ ❚❤❡ ❙t♦❝❤❛st✐❝ ▲♦❡✇♥❡r ❊✈♦❧✉t✐♦♥ ✭❙▲❊✮ ✐♥tr♦❞✉❝❡❞ ❜② ❖❞❡❞ ❙❝❤r❛♠♠
✐♥ ❬❙❝❤✵✵❪ ✐s ❛ ♦♥❡✲♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ r❛♥❞♦♠ ♣❛t❤s ✐♥ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❞♦♠❛✐♥s✳
❚❤❡s❡ ♣r♦❝❡ss❡s ❛r❡ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ❝❛♥❞✐❞❛t❡s ❢♦r ❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t ❝♦♥t✐♥✉♦✉s ❧✐♠✐ts ♦❢
t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❞✐s❝r❡t❡ ♠♦❞❡❧s✳ ❙❡❡ ❬❘♦❤❙❝❤✵✶❪ ❢♦r ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ❡①♣❧✐❝✐t ❝♦♥❥❡❝t✉r❡s✳
❈❛r❞② ❬❈❛✾✷❪ ✉s❡❞ ❝♦♥❢♦r♠❛❧ ✜❡❧❞ t❤❡♦r② t❡❝❤♥✐q✉❡s t♦ ♣r❡❞✐❝t ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ✭✐♥✈♦❧✈✐♥❣

❛ ❤②♣❡r❣❡♦♠❡tr✐❝ ❢✉♥❝t✐♦♥✮ t❤❛t s❤♦✉❧❞ ❞❡s❝r✐❜❡ ❝r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❝♦♥❢♦r♠❛❧ r❡❝t❛♥❣❧❡s
❢♦r ❝r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛s♣❡❝t r❛t✐♦ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✳ ❈❛r❧❡s♦♥ ♣♦✐♥t❡❞ ♦✉t
t❤❛t ❈❛r❞②✬s ❢♦r♠✉❧❛ ❝♦✉❧❞ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ❛ ♠✉❝❤ s✐♠♣❧❡r ✇❛② ❜② ❝❤♦♦s✐♥❣ ❛♥♦t❤❡r ❣❡♦♠❡tr✐❝
s❡t✉♣✱ s♣❡❝✐✜❝❛❧❧② ❜② ♠❛♣♣✐♥❣ t❤❡ r❡❝t❛♥❣❧❡ ♦♥t♦ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡

ABC ✳

❚❤❡ ❢♦r♠✉❧❛

❝❛♥ t❤❡♥ ❜❡ s✐♠♣❧② ❞❡s❝r✐❜❡❞ ❜② s❛②✐♥❣ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❝r♦ss✐♥❣ ✭✐♥ t❤❡ tr✐❛♥❣❧❡✮
❜❡t✇❡❡♥

AC

❛♥❞

BX

❢♦r

X ∈ [BC]

✐s

BX/BC ✳

❙♠✐r♥♦✈ ❬❙♠✐✵✶❪ ♣r♦✈❡❞ r✐❣♦r♦✉s❧② ❈❛r❞②✬s

❢♦r♠✉❧❛ ❢♦r ❝r✐t✐❝❛❧ s✐t❡ ♣❡r❝♦❧❛t✐♦♥ ♦♥ t❤❡ tr✐❛♥❣✉❧❛r ❧❛tt✐❝❡ ❛♥❞ ❤✐s ♣r♦♦❢ ✉s❡s t❤❡ ❣❧♦❜❛❧
❣❡♦♠❡tr② ♦❢ t❤❡ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ ✭♠♦r❡ t❤❛♥ t❤❡ ❧♦❝❛❧ ❣❡♦♠❡tr② ♦❢ t❤❡ tr✐❛♥❣✉❧❛r ❧❛tt✐❝❡✮✳
■♥ t❤❡ ♣r❡s❡♥t ♣❛♣❡r✱ ✇❡ s❤♦✇ t❤❛t ❡❛❝❤

SLEκ

✐s ✐♥ s♦♠❡ s❡♥s❡ ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞ t♦ s♦♠❡

❣❡♦♠❡tr✐❝❛❧ ♥♦r♠❛❧✐③❛t✐♦♥ ✐♥ t❤❛t t❤❡ ❢♦r♠✉❧❛s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❈❛r❞②✬s ❢♦r♠✉❧❛ ❝❛♥ ❛❣❛✐♥
❜❡ ❡①♣r❡ss❡❞ ✐♥ ❛ s✐♠♣❧❡ ✇❛②✳

❈♦♠❜✐♥✐♥❣ t❤✐s ✇✐t❤ t❤❡ ❝♦♥❥❡❝t✉r❡s ♦♥ ❝♦♥t✐♥✉♦✉s ❧✐♠✐ts ♦❢

✈❛r✐♦✉s ❞✐s❝r❡t❡ ♠♦❞❡❧s✱ t❤✐s ②✐❡❧❞s ♣r❡❝✐s❡ s✐♠♣❧❡ ❝♦♥❥❡❝t✉r❡s ♦♥ s♦♠❡ ❛s②♠♣t♦t✐❝s ❢♦r t❤❡s❡
♠♦❞❡❧s ✐♥ ♣❛rt✐❝✉❧❛r ❣❡♦♠❡tr✐❝ s❡t✉♣s✳ ❏✉st ❛s ♣❡r❝♦❧❛t✐♦♥ ♠❛② ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡q✉✐❧❛t❡r❛❧

✷✽

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s

✷✾

tr✐❛♥❣❧❡s✱ ✐t t✉r♥s t❤❛t✱ ❢♦r ✐♥st❛♥❝❡✱ t❤❡ ❝r✐t✐❝❛❧ ✷❞ ■s✐♥❣ ♠♦❞❡❧ ✭❛♥❞ t❤❡ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤

q = 2✮ s❡❡♠s t♦ ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ r✐❣❤t✲❛♥❣❧❡❞ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s ✭❜❡❝❛✉s❡ SLE16/3

♣❛r❛♠❡t❡r

❤✐tt✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s ✐♥ s✉❝❤ tr✐❛♥❣❧❡s ❛r❡ ✏✉♥✐❢♦r♠✑✮✳ ❖t❤❡r ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s ❝♦rr❡s♣♦♥❞ t♦

q ♣❛r❛♠❡t❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ q = 3 ❝♦rr❡s♣♦♥❞s
2π/3✳ ❙✐♠✐❧❛r❧②✱ ❞♦✉❜❧❡ ❞✐♠❡r✲♠♦❞❡❧s ♦r q = 4 P♦tts ♠♦❞❡❧s

✭❝♦♥❥❡❝t✉r❡❞ t♦ ❝♦rr❡s♣♦♥❞ t♦ κ = 4✮ s❡❡♠ t♦ ❜❡ ❜❡st ❡①♣r❡ss❡❞ ✐♥ str✐♣s ✭✐✳❡✳✱ ❞♦♠❛✐♥s ❧✐❦❡
R × [0, 1]✮✱ ❛♥❞ ❤❛❧❢✲str✐♣s ✭✐✳❡✳✱ [0, ∞) × [0, 1]✮ ❛r❡ ❛ ❢❛✈♦r❛❜❧❡ ❣❡♦♠❡tr② ❢♦r ✉♥✐❢♦r♠ s♣❛♥♥✐♥❣
❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ t❤❡

t♦ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ✇✐t❤ ❛♥❣❧❡

tr❡❡s✳

■ ✇✐s❤ t♦ t❤❛♥❦ ❲❡♥❞❡❧✐♥ ❲❡r♥❡r ❢♦r ❤✐s ❤❡❧♣ ❛♥❞ ❛❞✈✐❝❡✱ ❛s ✇❡❧❧ ❛s

❆❝❦♥♦✇❧❡❞❣♠❡♥ts✳

❘✐❝❤❛r❞ ❑❡♥②♦♥ ❢♦r ✉s❡❢✉❧ ✐♥s✐❣❤t ♦♥ ❞♦♠✐♥♦ t✐❧✐♥❣s ❛♥❞ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧s✳ ■ ❛❧s♦ ✇✐s❤
t♦ t❤❛♥❦ t❤❡ r❡❢❡r❡❡ ❢♦r ♥✉♠❡r♦✉s ❝♦rr❡❝t✐♦♥s ❛♥❞ ❝♦♠♠❡♥ts✳

✷ ❈❤♦r❞❛❧ ❙▲❊
H ❣♦✐♥❣ ❢r♦♠ 0 t♦
z ∈ H✱ t ≥ 0✱ ❞❡✜♥❡

❲❡ ✜rst ❜r✐❡✢② r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❤♦r❞❛❧ ❙▲❊ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡

∞ ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬▲❛✇❙❝❤❲❡r✵✶✱
gt (z) ❜② g0 (z) = z ❛♥❞

❘♦❤❙❝❤✵✶❪ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮✳ ❋♦r ❛♥②

∂t gt (z) =
✇❤❡r❡

√
(Wt / κ, t ≥ 0)

2
gt (z) − Wt

✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥

✜rst t✐♠❡ ♦❢ ❡①♣❧♦s✐♦♥ ♦❢ t❤✐s ❖❉❊✳ ❉❡✜♥❡ t❤❡ ❤✉❧❧

Kt

R✱

st❛rt✐♥❣ ❢r♦♠

0✳

▲❡t

τz

❜❡ t❤❡

❛s

Kt = {z ∈ H : τz < t}
(Kt )t≥0 ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ❢❛♠✐❧② ♦❢ ❝♦♠♣❛❝t s❡ts ✐♥ H❀ ❢✉rt❤❡r♠♦r❡✱ gt ✐s ❛ ❝♦♥❢♦r♠❛❧
H\Kt ♦♥t♦ H✳ ■t ❤❛s ❜❡❡♥ ♣r♦✈❡❞ ✭❬❘♦❤❙❝❤✵✶❪✱ s❡❡ ❬▲❛✇❙❝❤❲❡r✵✷❪ ❢♦r t❤❡ ❝❛s❡
κ = 8✮ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❝❡ss (γt )t≥0 ✇✐t❤ ✈❛❧✉❡s ✐♥ H s✉❝❤ t❤❛t H\Kt ✐s t❤❡
✉♥❜♦✉♥❞❡❞ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ H \ γ[0,t] ✱ ❛✳s✳ ❚❤✐s ♣r♦❝❡ss ✐s t❤❡ tr❛❝❡ ♦❢ t❤❡ ❙▲❊ ❛♥❞ ✐t

❝❛♥ r❡❝♦✈❡r❡❞ ❢r♦♠ gt ✭❛♥❞ t❤❡r❡❢♦r❡ ❢r♦♠ Wt ✮ ❜②

❚❤❡ ❢❛♠✐❧②

❡q✉✐✈❛❧❡♥❝❡ ♦❢

γt =

lim

z∈H→Wt

gt−1 (z)

D ✇✐t❤ t✇♦ ❜♦✉♥❞❛r② ♣♦✐♥ts ✭♦r ♣r✐♠❡ ❡♥❞s✮ a ❛♥❞ b✱ ❝❤♦r❞❛❧
(D,a,b)
(H,0,∞)
(H,0,∞)
SLEκ ✐♥ D ❢r♦♠ a t♦ b ✐s ❞❡✜♥❡❞ ❛s Kt
= h−1 (Kt

)✱ ✇❤❡r❡ Kt
✐s ❛s ❛❜♦✈❡✱ ❛♥❞
h ✐s ❛ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ (D, a, b) ♦♥t♦ (H, 0, ∞)✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ✉♥❛♠❜✐❣✉♦✉s ✉♣ t♦
❋♦r ❛♥② s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥

❛ ❧✐♥❡❛r t✐♠❡ ❝❤❛♥❣❡ t❤❛♥❦s t♦ t❤❡ s❝❛❧✐♥❣ ♣r♦♣❡rt② ♦❢ ❙▲❊ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ✭✐♥❤❡r✐t❡❞
❢r♦♠ t❤❡ s❝❛❧✐♥❣ ♣r♦♣❡rt② ♦❢ t❤❡ ❞r✐✈✐♥❣ ♣r♦❝❡ss

Wt ✮✳

✸ ❆ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❙▲❊
❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❙▲❊ r❡❧✐❡s ♦♥ t❤❡ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡

gt

♦❢

H\Kt

♦♥t♦

H✳

❆s

H

❤❛s

♥♦♥✲tr✐✈✐❛❧ ❝♦♥❢♦r♠❛❧ ❛✉t♦♠♦r♣❤✐s♠s✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡ ♦t❤❡r ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣s✳ ❚❤❡ ♦r✐❣✐♥❛❧

gt

✐s ♥❛t✉r❛❧ ❛s ❛❧❧ ♣♦✐♥ts ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ s❡❡♥ ❢r♦♠ ✐♥✜♥✐t② ♣❧❛② t❤❡ s❛♠❡ r♦❧❡ ✭❤❡♥❝❡ t❤❡

❞r✐✈✐♥❣ ♣r♦❝❡ss

(Wt )

[▲❛✇❙❝❤❲❡r✵✶]

♠❛② ♣r♦✈❡ ✉s❡❢✉❧ ❢♦r ❞✐✛❡r❡♥t ♣♦✐♥ts ♦❢ ✈✐❡✇✳

✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✮✳ ❖t❤❡r ♥♦r♠❛❧✐③❛t✐♦♥s✱ s✉❝❤ ❛s t❤❡ ♦♥❡ ✉s❡❞ ✐♥

κ = 6✱ ❛♥❞ F ❜❡ t❤❡ ❝♦♥❢♦r♠❛❧
(H, 0, 1, ∞) ♦♥t♦ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ (T, a, b, c)✳ ▲❡t ht ❜❡ t❤❡ ❝♦♥❢♦r♠❛❧ ❛✉t♦✲
♦❢ (T, a, b, c) s✉❝❤ t❤❛t ht (F (Wt )) = a✱ ht (F (gt (1))) = b✱ ht (c) = c✳ ❚❤❡♥✱ ❢♦r ❛♥②

❆ ❜②✲♣r♦❞✉❝t ♦❢ ❙♠✐r♥♦✈✬s r❡s✉❧ts ✭❬❙♠✐✵✶❪✮ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❧❡t
♠❛♣♣✐♥❣ ♦❢
♠♦r♣❤✐s♠

✸✵

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
z ∈ H✱ ht (F (gt (z))) ✐s ❛ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡✳ ❖✉r ❣♦❛❧ ✐♥ t❤✐s s❡❝t✐♦♥ ✐s t♦ ✜♥❞ s✐♠✐❧❛r ❢✉♥❝t✐♦♥s
F ❢♦r ♦t❤❡r ✈❛❧✉❡s ♦❢ κ✳
❘❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ ♥♦t❛t✐♦♥s ♦❢ s❡❝t✐♦♥ ✷✳ ❋♦r t < τ1 ✱ ❝♦♥s✐❞❡r t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣
♦❢ H\Kt ♦♥t♦ H ❞❡✜♥❡❞ ❛s✿
gt (z) − Wt
g̃t (z) =
gt (1) − Wt

s♦ t❤❛t g̃t (∞) = ∞✱ g̃t (1) = 1 ❛♥❞ g̃t (γt ) = 0✱ ✇❤❡r❡ γt ✐s t❤❡ ❙▲❊ tr❛❝❡✳
◆♦t✐❝❡ t❤❛t ✐❢ F ✐s ❛♥ ❤♦❧♦♠♦r♣❤✐❝ ♠❛♣ D → C ❛♥❞ (Yt )t≥0 ✐s ❛ D✲✈❛❧✉❡❞ s❡♠✐♠❛rt✐♥❣❛❧❡✱
t❤❡♥ ✭t❤❡ ❜✐✈❛r✐❛t❡ r❡❛❧ ✈❡rs✐♦♥ ♦❢✮ ■tô✬s ❢♦r♠✉❧❛ ②✐❡❧❞s✿
dF
1 d2 F
dYt +
dhYt i
dz
2 dz 2

dF (Yt ) =

✇❤❡r❡ t❤❡ q✉❛❞r❛t✐❝ ❝♦✈❛r✐❛t✐♦♥ h., .i ❢♦r r❡❛❧ s❡♠✐♠❛rt✐♥❣❛❧❡s ✐s ❡①t❡♥❞❡❞ ✐♥ ❛ C✲❜✐❧✐♥❡❛r ❢❛s❤✐♦♥
t♦ ❝♦♠♣❧❡① s❡♠✐♠❛rt✐♥❣❛❧❡s✿
hY1 , Y2 i = (hℜY1 , ℜY2 i − hℑY1 , ℑY2 i) + i(hℜY1 , ℑY2 i + hℑY1 , ℜY2 i)

s♦ t❤❛t dhCt i = 0 ❢♦r ❛♥ ✐s♦tr♦♣✐❝ ❝♦♠♣❧❡① ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ (Ct )✳ ❚❤❡ s❡t✉♣ ❤❡r❡ ✐s s❧✐❣❤t❧②
❞✐✛❡r❡♥t ❢r♦♠ ❝♦♥❢♦r♠❛❧ ♠❛rt✐♥❣❛❧❡s ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❬❘❡✈❨♦r✾✹❪✳
■♥ t❤❡ ♣r❡s❡♥t ❝❛s❡✱ ♦♥❡ ❣❡ts✿
dg̃t (z) =




dWt
dt
2
+ (g̃t (z) − 1)
− 2g̃t (z) + κ(g̃t (z) − 1)
2
g̃t (z)
(gt (1) − Wt )
gt (1) − Wt

❋♦r ♥♦t❛t✐♦♥❛❧ ❝♦♥✈❡♥✐❡♥❝❡✱ ❞❡✜♥❡ wt = g̃t (z)✳ ❆❢t❡r ♣❡r❢♦r♠✐♥❣ t❤❡ t✐♠❡ ❝❤❛♥❣❡
Z

u(t) =

t

0

♦♥❡ ❣❡ts t❤❡ ❛✉t♦♥♦♠♦✉s ❙❉❊✿

ds
(gs (1) − Ws )2



2
dwu = (wu − 1) κ −
(1 + wu ) du + (wu − 1)dW̃u
wu
√

✇❤❡r❡ (W̃u / κ)u≥0 ✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳
▲❡t ✉s t❛❦❡ ❛√❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ t✐♠❡ ❝❤❛♥❣❡✳ ▲❡t Yt = gt (1) − Wt ❀ t❤❡♥✱ dYt = −dWt + 2dt/Yt ✱
s♦ t❤❛t (Yt / κ)t≥0 ✐s ❛ ❇❡ss❡❧ ♣r♦❝❡ss ♦❢ ❞✐♠❡♥s✐♦♥ (1 + 4/κ)✳ ❋♦r κ ≤ 4✱ t❤✐s ❞✐♠❡♥s✐♦♥ ✐s ♥♦t
s♠❛❧❧❡r t❤❛♥ 2✱ s♦ t❤❛t Y ❛❧♠♦st s✉r❡❧② ♥❡✈❡r ✈❛♥✐s❤❡s ✭s❡❡ ❡✳❣✳ ❬❘❡✈❨♦r✾✹❪✮❀ ♠♦r❡♦✈❡r✱ ❛✳s✳✱
Z

0

∞

dt
=∞
Yt2

■♥❞❡❡❞✱ ❧❡t Tn = inf{t > 0 : Yt = 2n }✳ ❚❤❡♥✱ t❤❡ ♣♦s✐t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ( TTnn+1 dt/Yt2 , n ≥
1) ❛r❡ ✐✳✐✳❞✳ ✭✉s✐♥❣ t❤❡ ▼❛r❦♦✈ ❛♥❞ s❝❛❧✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❇❡ss❡❧ ♣r♦❝❡ss❡s✮✳ ❍❡♥❝❡✿
R

Z

0

∞

∞ Z Tn+1
X
dt
dt
≥
2 = ∞ ❛✳s✳
Yt2
Y
t
n=1 Tn

❙♦ t❤❡ t✐♠❡ ❝❤❛♥❣❡ ✐s ❛✳s✳ ❛ ❜✐❥❡❝t✐♦♥ ❢r♦♠ R+ ♦♥t♦ R+ ✐❢ κ ≤ 4✳

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s

✸✶

❲❤❡♥ κ > 4✱ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss Y ✐s s♠❛❧❧❡r t❤❛♥ 2✱ s♦ t❤❛t τ1 < ∞ ❛❧♠♦st
s✉r❡❧②✳ ■♥ t❤✐s ❝❛s❡✱ ✉s✐♥❣ ❛ s✐♠✐❧❛r ❛r❣✉♠❡♥t ✇✐t❤ t❤❡ st♦♣♣✐♥❣ t✐♠❡s Tn ❢♦r n < 0✱ ♦♥❡ s❡❡s
t❤❛t
Z τ1
0

dt
=∞
Yt2

❍❡♥❝❡✱ ✐❢ κ > 4✱ t❤❡ t✐♠❡ ❝❤❛♥❣❡ ✐s ❛✳s✳ ❛ ❜✐❥❡❝t✐♦♥ [0, τ1 ) → R+ ✳
❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❢♦r ❛❧❧ κ > 0✱ t❤❡ st♦❝❤❛st✐❝ ✢♦✇ (g̃u )u≥0 ❞♦❡s ❛❧♠♦st s✉r❡❧② ♥♦t ❡①♣❧♦❞❡ ✐♥
✜♥✐t❡ t✐♠❡✳
❲❡ ♥♦✇ ❧♦♦❦ ❢♦r ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s F s✉❝❤ t❤❛t (F (wu ))u≥0 ❛r❡ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡s✳ ❆s
❜❡❢♦r❡✱ ♦♥❡ ❣❡ts✿


2
κ
dF (wu ) = F ′ (wu )(κ −
(1 + wu )) + F ′′ (wu )(wu − 1) (wu − 1)du + F ′ (wu )(wu − 1)dW̃u
wu
2

❍❡♥❝❡ ✇❡ ❤❛✈❡ t♦ ✜♥❞ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ H s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿


2
κ
F ′ (w) κ − (1 + w) + F ′′ (w) (w − 1) = 0
w
2

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉❝❤ t❤❛t
✇❤❡r❡

F ′ (w) ∝ wα−1 (w − 1)β−1 ,


α
β

=
=

❋♦r κ = 4✱ F (w) = log(w) ✐s ❛ s♦❧✉t✐♦♥✳

1 − κ4
8
κ −1

✹ Pr✐✈✐❧❡❣❡❞ ❣❡♦♠❡tr✐❡s
■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❛tt❡♠♣t t♦ ✐❞❡♥t✐❢② t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ♠❛♣ F ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡

κ ♣❛r❛♠❡t❡r✳
•

❈❛s❡ 4 < κ < 8

❯s✐♥❣ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛ ❬❆❤❧✼✾❪✱ ♦♥❡ ❝❛♥ ✐❞❡♥t✐❢② F ❛s t❤❡ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈✲
❛❧❡♥❝❡ ♦❢ (H, 0, 1, ∞) ♦♥t♦ ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ (Tκ , a, b, c) ✇✐t❤ ❛♥❣❧❡s â = ĉ = απ =
(1 − κ4 )π ❛♥❞ b̂ = βπ = ( κ8 − 1)π ✳ ❙♣❡❝✐❛❧ tr✐❛♥❣❧❡s t✉r♥ ♦✉t t♦ ❝♦rr❡s♣♦♥❞ t♦ s♣❡✲
❝✐❛❧ ✈❛❧✉❡s ♦❢ κ✳ ❚❤✉s✱ ❢♦r κ = 6✱ ♦♥❡ ❣❡ts ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡✱ ❛s ✇❛s ❢♦r❡s❡❡❛❜❧❡
❢r♦♠ ❙♠✐r♥♦✈✬s ✇♦r❦ ✭❬❙♠✐✵✶❪✮✳ ❋♦r κ = 16
3 ✱ ❛ ✈❛❧✉❡ ❝♦♥❥❡❝t✉r❡❞ t♦ ❝♦rr❡s♣♦♥❞ t♦ ❋❑
♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ q = 2 ❛♥❞ t♦ t❤❡ ■s✐♥❣ ♠♦❞❡❧✱ ♦♥❡ ❣❡ts ❛♥ ✐s♦r❡❝t❛♥❣❧❡ tr✐❛♥❣❧❡✳
❙✐♥❝❡ F (H) ✐s ❜♦✉♥❞❡❞✱ t❤❡ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡s F (g̃t∧τ1 (z)) ❛r❡ ❜♦✉♥❞❡❞ ✭❝♦♠♣❧❡①✲✈❛❧✉❡❞✮
♠❛rt✐♥❣❛❧❡s✱ s♦ t❤❛t ♦♥❡ ❝❛♥ ❛♣♣❧② t❤❡ ♦♣t✐♦♥❛❧ st♦♣♣✐♥❣ t❤❡♦r❡♠✳ ❲❡ t❤❡r❡❢♦r❡ st✉❞②
✇❤❛t ❤❛♣♣❡♥s ❛t t❤❡ st♦♣♣✐♥❣ t✐♠❡ τ1,z = τ1 ∧ τz ✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ♣♦ss✐❜❧❡ ❝❛s❡s✱ ❡❛❝❤
❤❛✈✐♥❣ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ τ1 < τz ✱ τ1 = τz ❛♥❞ τ1 > τz ✳ ❈❧❡❛r❧②✱ limtրτz (gt (z) − Wt ) =
0✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ (gt (z) − Wt ) ✐s ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ ③❡r♦ ✐❢ t st❛②s ❜♦✉♥❞❡❞
❛✇❛② ❢r♦♠ τz ✳ ❘❡❝❛❧❧ t❤❛t
g̃t (z) =

gt (z) − Wt
gt (1) − Wt

✸✷

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
❙♦✱ ❛s t ր τ1,z ✱ g̃t (z) → ∞ ✐❢ τ1 < τz ❛♥❞ g̃t (z) → 0 ✐❢ τz < τ1 ✳ ■♥ t❤❡ ❝❛s❡ τ1 = τz = τ ✱
t❤❡ ♣♦✐♥ts 1 ❛♥❞ z ❛r❡ ❞✐s❝♦♥♥❡❝t❡❞ ❛t t❤❡ s❛♠❡ ♠♦♠❡♥t✱ ✇✐t❤ γτ ∈ ∂H✳ ❆s t ր τ ✱
t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛s✉r❡ ♦❢ (−∞, 0) s❡❡♥ ❢r♦♠ z t❡♥❞s t♦ 0❀ ✐♥❞❡❡❞✱ t♦ r❡❛❝❤ (−∞, 0)✱ ❛
❇r♦✇♥✐❛♥ ♠♦t✐♦♥ st❛rt✐♥❣ ❢r♦♠ z ❤❛s t♦ ❣♦ t❤r♦✉❣❤ t❤❡ str❛✐ts [γt , γτ ] t❤❡ ✇✐❞t❤ ♦❢ ✇❤✐❝❤
t❡♥❞s t♦ ③❡r♦✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛s✉r❡s ♦❢ (0, 1) ❛♥❞ (1, ∞) s❡❡♥ ❢r♦♠
z st❛② ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ 0✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t g̃t (z) t❡♥❞s t♦ 1✱ ❛s ✐s ❡❛s✐❧② s❡❡♥ ❜②
♠❛♣♣✐♥❣ H t♦ str✐♣s✳
◆♦✇ ♦♥❡ ❝❛♥ ❛♣♣❧② t❤❡ ♦♣t✐♦♥❛❧ st♦♣♣✐♥❣ t❤❡♦r❡♠ t♦ t❤❡ ♠❛rt✐♥❣❛❧❡s F (g̃t∧τ1,z (z))✳ ❚❤❡
♠❛♣♣✐♥❣ F ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ t♦ H✱ ❤❡♥❝❡✿
F (z) = F (0)P(τz < τ1 ) + F (1)P(τz = τ1 ) + F (∞)P(τz > τ1 )

❚❤✉s✿
Pr♦♣♦s✐t✐♦♥ ✶✳

w = F (z)
τz < τ1 ✱ τz = τ1 ✱ τz > τ1 ✳

❚❤❡ ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ♦❢
❡✈❡♥ts

✐♥ t❤❡ tr✐❛♥❣❧❡

Tκ

❛r❡ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t❤❡

❉❡✜♥❡ T 0 = {w ∈ Tκ : τz < τ1 }✱ T 1 = {w ∈ Tκ : τz = τ1 }✱ T ∞ = {w ∈ Tκ : τz > τ1 }✱
✇❤✐❝❤ ✐s ❛ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ ♦❢ TκS
✳ ❚❤❡s❡ t❤r❡❡ s❡ts ❛r❡ ❛✳s✳ ❜♦r❡❧✐❛♥❀ ✐♥❞❡❡❞✱ T ∞ =
0
F (H\Kτ1 ) ✐s ❛✳s✳ ♦♣❡♥✱ ❛♥❞ T = t 0}✳ ❚❤❡♥ F (∞) = 0 ❛♥❞
F (0) = 1✳ ▼♦r❡♦✈❡r✱ ℜF (g̃t (z)) ✐s ❛ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡✳ ■♥ t❤❡ ❝❛s❡ κ ≥ 8✱ ✐t ✐s ❦♥♦✇♥
t❤❛t τ1 < ∞✱ τz < ∞✱ ❛♥❞ τ1 6= τz ❛✳s✳ ✐❢ z 6= 1 ✭s❡❡ ❬❘♦❤❙❝❤✵✶❪✮✳ ❍❡♥❝❡✱ ✐❢ τ = τ1 ∧ τz ✱
g̃τ (z) ❡q✉❛❧s 0 ♦r ∞✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r τz < τ1 ♦r τz > τ1 ✳ ❆♣♣❧②✐♥❣ t❤❡ ♦♣t✐♦♥❛❧
st♦♣♣✐♥❣ t❤❡♦r❡♠ t♦ t❤❡ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡ ℜF (g̃t (z))✱ ♦♥❡ ❣❡ts✿
R

1

P(τz < τ1 ) = ℜF (z)

✸✹

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②

✶

✵

∞
❋✐❣✉r❡ ✸✿

•

F (H)✱

κ = 8✿

❝❛s❡

❤❛❧❢✲str✐♣

❈❛s❡ κ > 8
■♥ t❤✐s ❝❛s❡✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡

D=
❚❤❡♥

F (H)



F

s♦ t❤❛t ✐t ♠❛♣s

z : ℑz > 0, 0 < arg(z) <



(H, 0, 1, ∞)

4
1−
κ



♦♥t♦

(D, 1, ∞, 0)

4
π, π < arg(z − 1) < π
κ

✇❤❡r❡



✐s ♥♦t ❜♦✉♥❞❡❞ ✐♥ ❛♥② ❞✐r❡❝t✐♦♥✱ ♣r❡✈❡♥t✐♥❣ ✉s ❢r♦♠ ✉s✐♥❣ t❤❡ ♦♣t✐♦♥❛❧

st♦♣♣✐♥❣ t❤❡♦r❡♠✳

✶

✶

(1 − 4/κ)π
✵

∞
❋✐❣✉r❡ ✹✿

•

F (H)✱

❝❛s❡

κ>8

❈❛s❡ κ < 4
■❢

κ ≥ 8/3✱

♦♥❡ ❝❛♥ ❝❤♦♦s❡

D=
❋♦r

κ = 8/3✱

F

s♦ t❤❛t ✐t ♠❛♣s

(H, 0, 1, ∞)

♦♥t♦

(D, ∞, 0, ∞)✱

✇❤❡r❡





4
4
z : ℑz < 1, −
− 1 π < arg(z) < π
κ
κ

♦♥❡ ❣❡ts ❛ s❧✐t ❤❛❧❢✲♣❧❛♥❡✳ ❋♦r

κ < 8/3✱

t❤❡ ♠❛♣

F

❝❡❛s❡s t♦ ❜❡ ✉♥✐✈❛❧❡♥t✳

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s

✸✺

✵

∞
✶

(8/κ − 1)π

❋✐❣✉r❡ ✺✿ F (H)✱ ❝❛s❡

8
3

≤κ≤4

✺ ❘❛❞✐❛❧ ❙▲❊
▲❡t D ❜❡ t❤❡ ✉♥✐t ❞✐s❦✳ ❘❛❞✐❛❧ ❙▲❊ ✐♥ D st❛rt✐♥❣ ❢r♦♠ ✶ ✐s ❞❡✜♥❡❞ ❜② g0 (z) = z ✱ z ∈ D ❛♥❞
t❤❡ ❖❉❊s✿
∂t gt (z) = −gt (z)
√

gt (z) + ξ(t)
gt (z) − ξ(t)

✇❤❡r❡ ξ(t) = exp(iWt ) ❛♥❞ Wt / κ ✐s ❛ r❡❛❧ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡ ❤✉❧❧s (Kt ) ❛♥❞
t❤❡ tr❛❝❡ (γt ) ❛r❡ ❞❡✜♥❡❞ ❛s ✐♥ t❤❡ ❝❤♦r❞❛❧ ❝❛s❡ ✭❬❘♦❤❙❝❤✵✶❪✮✳ ❉❡✜♥❡ g̃t (z) = gt (z)ξt−1 ✱ s♦ t❤❛t
g̃t (0) = 0✱ g̃(γt ) = 1✱ ✇❤❡r❡ (γt ) ✐s t❤❡ ❙▲❊ tr❛❝❡✳ ❖♥❡ ♠❛② ❝♦♠♣✉t❡✿
dg̃t (z) = −g̃t (z)

1
g̃t (z) + 1
dt + g̃t (z)(−idWt − κdt)
g̃t (z) − 1
2

❚❤❡ ❛❜♦✈❡ ❙❉❊ ✐s ❛✉t♦♥♦♠♦✉s✳ ❆s ❜❡❢♦r❡✱ ♦♥❡ ❧♦♦❦s ❢♦r ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s F s✉❝❤ t❤❛t
(F (g̃t (z)))t≥0 ❛r❡ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡s✳ ❆ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ✐s✿


z+1 κ
κ
F ′ (z) −z
− z − F ′′ (z)z 2 = 0
z−1
2
2

✐✳❡✳✱

F ′′ (z)
=
F ′ (z)




4 1
2
1
−1
−
.
κ
z
κz−1

▼❡r♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❞❡✜♥❡❞ ♦♥ D ❡①✐st ❢♦r κ = 2/n✱ n ∈ N∗ ✳ ❋♦r κ = 2✱
F (z) = (z − 1)−1 ✐s ❛♥ ✭✉♥❜♦✉♥❞❡❞✮ s♦❧✉t✐♦♥✳

✻ ❘❡❧❛t❡❞ ❝♦♥❥❡❝t✉r❡s
■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❢♦r♠✉❧❛t❡ ✈❛r✐♦✉s ❝♦♥❥❡❝t✉r❡s ♣❡rt❛✐♥✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s ❧✐♠✐ts ♦❢ ❞✐s❝r❡t❡
❝r✐t✐❝❛❧ ♠♦❞❡❧s ✉s✐♥❣ t❤❡ ♣r✐✈✐❧❡❣❡❞ ❣❡♦♠❡tr✐❡s ❢♦r ❙▲❊ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳
✻✳✶

❋❑ ♣❡r❝♦❧❛t✐♦♥ ✐♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s

❋♦r ❛ s✉r✈❡② ♦❢ ❋❑ ♣❡r❝♦❧❛t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ r❛♥❞♦♠✲❝❧✉st❡r ♠♦❞❡❧✱ s❡❡ ❬●r✐✾✼❪✳ ❲❡ ❜✉✐❧❞ ♦♥ ❛
❝♦♥❥❡❝t✉r❡ st❛t❡❞ ✐♥ ❬❘♦❤❙❝❤✵✶❪ ✭❈♦♥❥❡❝t✉r❡ ✾✳✼✮✱ ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ t❤❡ ❞✐s❝r❡t❡ ❡①♣❧♦r❛t✐♦♥

✸✻

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
♣r♦❝❡ss ❢♦r ❝r✐t✐❝❛❧ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r q ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ t❤❡ tr❛❝❡ ♦❢ SLEκ
❢♦r q ∈ (0, 4)✱ ✇❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❤♦❧❞s✿
κ=

4π
√
cos−1 (− q/2)
√

❚❤❡♥ t❤❡ ❛ss♦❝✐❛t❡❞ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ Tκ ❤❛s ❛♥❣❧❡s â = ĉ = cos−1 ( q/2)✱ b̂ = π − 2â✳ ▲❡t Γn
❜❡ ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ tr✐❛♥❣❧❡ Tκ ♦♥ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡ ✇✐t❤ ♠❡s❤ n1 ❀ ❛❧❧ ✈❡rt✐❝❡s
♦♥ t❤❡ ❡❞❣❡s (a, b] ❛♥❞ [b, c) ❛r❡ ✐❞❡♥t✐✜❡❞✳ ▲❡t Γ†n ❜❡ t❤❡ ❞✉❛❧ ❣r❛♣❤✳ ❚❤❡ ❞✐s❝r❡t❡ ❡①♣❧♦r❛t✐♦♥
♣r♦❝❡ss β r✉♥s ❜❡t✇❡❡♥ t❤❡ ♦♣❡♥❡❞ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ (a, b] ∪ [b, c) ✐♥ Γn ❛♥❞ t❤❡ ❝❧♦s❡❞
❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ (a, c) ✐♥ Γ†n ✳

❈❛r❞②✬s ❋♦r♠✉❧❛
▲❡t τ ❜❡ t❤❡ ✜rst t✐♠❡ β ❤✐ts (b, c)✳ ❚❤❡♥✱ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t② ✭✐✳❡✳ ❛s t❤❡ ♠❡s❤ t❡♥❞s t♦
③❡r♦✮✱ t❤❡ ❧❛✇ ♦❢ βτ ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦✇❛r❞s t❤❡ ✉♥✐❢♦r♠ ❧❛✇ ♦♥ (b, c)✳
❈♦♥❥❡❝t✉r❡ ✶✳

❑❡♥②♦♥ ❬❑❡♥✵✷❪ ❤❛s ♣r♦♣♦s❡❞ ❛♥ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧ ❢♦r ❛♥② ✐s♦r❛❞✐❛❧ ❧❛tt✐❝❡✱ ✐♥ ♣❛rt✐❝✉❧❛r
√
−1
(− q/2)
❢♦r ❛♥② r❡❝t❛♥❣✉❧❛r ❧❛tt✐❝❡✳ ▲❡t κ✱ q ❛♥❞ α ❜❡ ❛s ❛❜♦✈❡✱ ✐✳❡✳ 4 < κ < 8✱ 4π
κ = cos
❛♥❞ α = 1 − κ4 ✳ ❈♦♥s✐❞❡r t❤❡ r❡❝t❛♥❣✉❧❛r ❧❛tt✐❝❡ Z cos απ + iZ sin απ ✳ ❚❤❡♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s
❤♦♠♦t❤❡t✐❝ t♦ Tκ ♥❛t✉r❛❧❧② ✜t ✐♥ t❤❡ ❧❛tt✐❝❡ ✭s❡❡ ✜❣✉r❡ ✻✮✳ ▲❡t Γ = (V, E) ❜❡ t❤❡ ✜♥✐t❡ ❣r❛♣❤
r❡s✉❧t✐♥❣ ❢r♦♠ t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ❧❛tt✐❝❡ t♦ ❛ ✭❧❛r❣❡✮ Tκ tr✐❛♥❣❧❡✱ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❜♦✉♥❞❛r②
❝♦♥❞✐t✐♦♥s✳ ❆ ❝♦♥✜❣✉r❛t✐♦♥ ω ∈ {0, 1}E ♦❢ ♦♣❡♥ ❡❞❣❡s ❤❛s ♣r♦❜❛❜✐❧✐t②✿
e (ω) ev (ω)
νv

pΓ (ω) ∝ q k(ω) νhh

✇❤❡r❡ k(ω) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❛♥❞ eh ✭r❡s♣✳ ev ✮
✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦♣❡♥ ❤♦r✐③♦♥t❛❧ ✭r❡s♣✳ ✈❡rt✐❝❛❧✮ ❡❞❣❡s✳ ❚❤❡ ✇❡✐❣❤ts νh ✱ νv ❛r❡ ❣✐✈❡♥ ❜② t❤❡
❢♦r♠✉❧❛s✿
νv =

√

νh =

q
νv

q

sin(2α2 π)
sin(α(1 − 2α)π)

απ

❋✐❣✉r❡ ✻✿ ❘❡❝t❛♥❣❧❡ ❧❛tt✐❝❡✱ ❞✉❛❧ ❣r❛♣❤ ❛♥❞ ❛ss♦❝✐❛t❡❞ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡
❋♦r t❤✐s ♠♦❞❡❧✱ ♦♥❡ ♠❛② ❝♦♥❥❡❝t✉r❡ ❈❛r❞②✬s ❢♦r♠✉❧❛ ❛s st❛t❡❞ ❛❜♦✈❡✳ ◆♦t❡ t❤❛t ❢♦r q = 2✱
16
3 ✱ ♦♥❡ r❡tr✐❡✈❡s t❤❡ ✉s✉❛❧ ❝r✐t✐❝❛❧ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ♦♥ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡✳
▲❡t ✉s ♥♦✇ ❢♦❝✉s ♦♥ t❤❡ ✐♥t❡❣r❛❧ ✈❛❧✉❡s ♦❢ t❤❡ q ♣❛r❛♠❡t❡r✳ ■t ✐s ❦♥♦✇♥ t❤❛t ❢♦r t❤❡s❡ ✈❛❧✉❡s
t❤❡r❡ ❡①✐sts ❛ st♦❝❤❛st✐❝ ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ❛♥❞ t❤❡ P♦tts ♠♦❞❡❧ ✭✇✐t❤ ♣❛r❛♠❡t❡r
q ✮ ✭s❡❡ ❬●r✐✾✼❪✮✳

κ=

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s

✸✼

• q=1

■♥ t❤✐s ❝❛s❡ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✐s s✐♠♣❧② ♣❡r❝♦❧❛t✐♦♥✱ κ = 6✱ ❛♥❞ t❤❡ ♣r✐✈✐❧❡❣❡❞ ❣❡♦♠❡tr② ✐s
t❤❡ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡✳ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❈❛r❧❡s♦♥✬s ♦❜s❡r✈❛t✐♦♥ ♦♥ ❈❛r❞②✬s ❢♦r♠✉❧❛✳

• q=2

✵

✇
❢

✶

❢

∞

❋✐❣✉r❡ ✼✿ ❉✐s❝r❡t❡ ❡①♣❧♦r❛t✐♦♥ ♣r♦❝❡ss ❢♦r ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✭q = 2✱ κ =

16
3 ✮

❍❡r❡ κ = 16
3 ✱ ❛♥❞ Tκ ✐s ❛♥ ✐s♦r❡❝t❛♥❣❧❡ tr✐❛♥❣❧❡✳ ❆s t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥
❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r q = 2 ❛♥❞ t❤❡ ■s✐♥❣ ♠♦❞❡❧ ✭P♦tts ♠♦❞❡❧ ✇✐t❤ q = 2✮✱ t❤✐s
s✉❣❣❡sts t❤❛t t❤❡ ✐s♦r❡❝t❛♥❣❧❡ tr✐❛♥❣❧❡ ♠❛② ❜❡ ♦❢ s♦♠❡ s✐❣♥✐✜❝❛♥❝❡ ❢♦r t❤❡ ■s✐♥❣ ♠♦❞❡❧✳
• q=3

❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❣❡♦♠❡tr② ✐s t❤❡ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ T 245 ✱ ✇❤✐❝❤ ❤❛s ❛♥❣❧❡s â = ĉ = π6 ✱
b̂ = 2π
3 ✳ ❚❤❡ ♣♦ss✐❜❧❡ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ t❤❡ q = 3 P♦tts ♠♦❞❡❧ ✐s ♥♦t ❝❧❡❛r✱ ❛s t❤✐s ♠♦❞❡❧
✐s ♥♦t ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥ ❡①♣❧♦r❛t✐♦♥ ♣r♦❝❡ss✳

✻✳✷

❯❙❚ ✐♥ ❤❛❧❢✲str✐♣s

■t ✐s ♣r♦✈❡❞ ✐♥ ❬▲❛✇❙❝❤❲❡r✵✷❪ t❤❛t t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ t❤❡ ✉♥✐❢♦r♠ s♣❛♥♥✐♥❣ tr❡❡s ✭❯❙❚✮
P❡❛♥♦ ❝✉r✈❡ ✐s t❤❡ SLE8 ❝❤♦r❞❛❧ ♣❛t❤✳ ▲❡t Rn,L ❜❡ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡ [0, n] × [0, nL]✱ ✇✐t❤ t❤❡
❢♦❧❧♦✇✐♥❣ ❜♦✉♥❞❛r✐❡s ❝♦♥❞✐t✐♦♥s✿ t❤❡ t✇♦ ❤♦r✐③♦♥t❛❧ ❛r❝s ❛s ✇❡❧❧ ❛s t❤❡ t♦♣ ♦♥❡ ❛r❡ ✇✐r❡❞✱ ❛♥❞
t❤❡ ❜♦tt♦♠ ♦♥❡ ✐s ❢r❡❡✳ ■♥ ❢❛❝t✱ ❛s ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ❧✐♠✐t ❛s L ❣♦❡s t♦ ✐♥✜♥✐t②✱ ♦♥❡ ♠❛② ❛s
✇❡❧❧ ❝♦♥s✐❞❡r t❤❛t t❤❡ t♦♣ ❛r❝ ✐s ❢r❡❡✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ♥❡❛t❡r✳ ❲❡ ❝♦♥s✐❞❡r t❤❡
✉♥✐❢♦r♠ s♣❛♥♥✐♥❣ tr❡❡ ✐♥ Rn,L ✳ ▲❡t w ❜❡ ❛ ♣♦✐♥t ♦❢ t❤❡ ❤❛❧❢✲str✐♣ {z : 0 < ℜz < 1, ℑz > 0}✱
❛♥❞ wn ❛♥ ✐♥t❡❣r❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ nw✳ ▲❡t a ∈ [0, n] ❜❡ t❤❡ ✉♥✐q✉❡ tr✐♣❧❡ ♣♦✐♥t ♦❢ t❤❡
♠✐♥✐♠❛❧ s✉❜tr❡❡ T ❝♦♥t❛✐♥✐♥❣ (0, 0)✱ (n, 0) ❛♥❞ wn ✱ ❛♥❞ ❧❡t b ❜❡ t❤❡ ♦t❤❡r tr✐♣❧❡ ♣♦✐♥t ♦❢ t❤❡
♠✐♥✐♠❛❧ s✉❜tr❡❡ ❝♦♥t❛✐♥✐♥❣ (0, 0)✱ (n, 0)✱ wn ❛♥❞ (0, nL)✳ ❖♥❡ ❝❛♥ ❢♦r♠✉❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❛s②
❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ t❤❡ ❯❙❚✿

✸✽

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②

▲❡♠♠❛ ✶✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐ts ❤♦❧❞✿

lim lim PRn,L (b

❜❡❧♦♥❣s t♦ t❤❡ ♦r✐❡♥t❡❞ ❛r❝

[0, a] ∪ [a, wn ]

✐♥

T ) = ℜw

lim lim PRn,L (b

❜❡❧♦♥❣s t♦ t❤❡ ♦r✐❡♥t❡❞ ❛r❝

[0, a] ∪ [a, wn ]

✐♥

T ) = ℜw

n→∞ L→∞

L→∞ n→∞

b ❜❡❧♦♥❣s t♦ t❤❡
[0, a] ∪ [a, wn ]✱ ♦r t♦ t❤❡ ✭♦r✐❡♥t❡❞✮ ❛r❝ [wn ; a] ∪ [a, 1]✳ ❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡
P(τF −1 (w) > τ1 ) = ℜw ❢♦r ❛ ❝❤♦r❞❛❧ SLE8 ❣♦✐♥❣ ❢r♦♠ 0 t♦ 1 ✐♥ t❤❡ ❤❛❧❢✲str✐♣ ✭✐♥

▲❡t ✉s ❝❧❛r✐❢② t❤❡ ❛❧t❡r♥❛t✐✈❡ ✭✉♣ t♦ ❡✈❡♥ts ♦❢ ♥❡❣❧✐❣✐❜❧❡ ♣r♦❜❛❜✐❧✐t②✮✿ ❡✐t❤❡r
✭♦r✐❡♥t❡❞✮ ❛r❝
❝♦♠♣✉t❡❞

❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ❡❛r❧✐❡r ❝♦♥✈❡♥t✐♦♥s✱ s✉❜s❝r✐♣ts r❡❢❡r t♦ ♣♦✐♥ts ✐♥ t❤❡ ❤❛❧❢✲♣❧❛♥❡✱ ♥♦t ✐♥ t❤❡

❤❛❧❢✲str✐♣✮✳ ❆s t❤✐s ♣❛t❤ ✐s ✐❞❡♥t✐✜❡❞ ❛s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ t❤❡ ❯❙❚ P❡❛♥♦ ❝✉r✈❡ ✭st❛rt ❢r♦♠

0 ❛♥❞ ❣♦ t♦ 1 ✇✐t❤ t❤❡ ❯❙❚ r♦♦t❡❞ ♦♥ t❤❡ ❜♦tt♦♠ ❛❧✇❛②s ♦♥ ②♦✉r r✐❣❤t✲❤❛♥❞✮✱ t❤❡ ❡✈❡♥t
{τ1 < τF −1 (w) } ❛♣♣❡❛rs ❛s ❛ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❛♥ ❡✈❡♥t ✐♥✈♦❧✈✐♥❣ ♦♥❧② t❤❡ s✉❜tr❡❡ T ✳ ■❢ ♦♥❡
r❡♠♦✈❡s t❤❡ ❛r❝ ❥♦✐♥✐♥❣ a t♦ iLn✱ wn ✐s ❡✐t❤❡r ♦♥ t❤❡ ❧❡❢t ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦r ♦♥ t❤❡ r✐❣❤t
♦♥❡ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r wn ✐s ✏✈✐s✐t❡❞✑ ❜② t❤❡ ❡①♣❧♦r❛t✐♦♥ ♣r♦❝❡ss ❜❡❢♦r❡ ♦r ❛❢t❡r t❤❡ t♦♣
❛r❝✱ ✉♣ t♦ ❡✈❡♥ts ♦❢ ♥❡❣❧✐❣✐❜❧❡ ♣r♦❜❛✐❧✐t②✳

✭✵✱▲♥✮

✭✵✱▲♥✮

❜

wn

wn

❜
✵

❛

♥

✵

❛

♥

❋✐❣✉r❡ ✽✿ ❚❤❡ ❛❧t❡r♥❛t✐✈❡

†
■♥ ❢❛❝t✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ t❤❡ ❧❡♠♠❛ ✇✐t❤♦✉t ✉s✐♥❣ t❤❡ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ❢♦r ❯❙❚✳ ■♥❞❡❡❞✱ ❧❡t wn
√
2
❜❡ ❛ ♣♦✐♥t ♦♥ t❤❡ ❞✉❛❧ ❣r✐❞ st❛♥❞✐♥❣ ❛t ❞✐st❛♥❝❡
2 ❢r♦♠ wn ✳ ❚❤❡♥✱ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱

PRn,L (b

❜❡❧♦♥❣s t♦ t❤❡ ❛r❝

[0, a] ∪ [a, wn ]

✐♥

T)

−PR† (wn† ✐s ❝♦♥♥❡❝t❡❞ t♦ t❤❡ r✐❣❤t✲❤❛♥❞ ❜♦✉♥❞❛r② ✐♥ t❤❡ ❞✉❛❧ tr❡❡)
n,L

→0

❆❝❝♦r❞✐♥❣ t♦ ❲✐❧s♦♥✬s ❛❧❣♦r✐t❤♠ ❬❲✐❧✾✻❪✱ t❤❡ ♠✐♥✐♠❛❧ s✉❜tr❡❡ ✐♥ t❤❡ ❞✉❛❧ tr❡❡ ❝♦♥♥❡❝t✐♥❣

wn†

t♦ t❤❡ ❜♦✉♥❞❛r② ❤❛s t❤❡ ❧❛✇ ♦❢ ❛ ❧♦♦♣✲❡r❛s❡❞ r❛♥❞♦♠ ✇❛❧❦ ✭▲❊❘❲✮ st♦♣♣❡❞ ❛t ✐ts ✜rst ❤✐t ♦❢
t❤❡ ❜♦✉♥❞❛r②✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❤✐tt✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ ❜♦✉♥❞❛r② ♦r t❤❡ ❧❡❢t✲❤❛♥❞ ❜♦✉♥❞❛r②
❢♦r ❛ ▲❊❘❲ ❡q✉❛❧s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ❢♦r ❛ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦✳ ❚❤❡ ❝♦♥t✐♥✉♦✉s
❧✐♠✐t ❢♦r ❛ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❤❡s❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ r❡✢❡❝t❡❞
♦♥ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❤❛❧❢✲str✐♣❀ ❛s t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛s✉r❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ ❜♦✉♥❞❛r② ♦❢ t❤❡
†
✇❤♦❧❡ s❧✐t {0 < ℜz < 1} s❡❡♥ ❢r♦♠ wn ✐s ℜw + o(1)✱ t❤✐s ♣r♦✈❡s t❤❡ ❧❡♠♠❛✳

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s
✻✳✸

✸✾

❉♦✉❜❧❡ ❞♦♠✐♥♦ t✐❧✐♥❣s ✐♥ ♣❧❛♥❡ str✐♣s

❋♦r ❛♥ ❡❛r❧② ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ❞♦✉❜❧❡ ❞♦♠✐♥♦ t✐❧✐♥❣ ♠♦❞❡❧✱ s❡❡ ❬❘❛❣❍❡♥❆r♦✾✼❪✳ ■t ✐s ❝♦♥❥❡❝t✉r❡❞
t❤❛t t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ t❤❡ ♣❛t❤ ❛r✐s✐♥❣ ✐♥ t❤✐s ♠♦❞❡❧ ✐s t❤❡ SLE4 tr❛❝❡ ✭s❡❡ ❬❘♦❤❙❝❤✵✶❪✱
Pr♦❜❧❡♠ ✾✳✽✮✳ ❇✉✐❧❞✐♥❣ ♦♥ ❑❡♥②♦♥✬s ✇♦r❦ ❬❑❡♥✾✼✱ ❑❡♥✵✵❪✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t
♦❢ ❛ ♣❛rt✐❝✉❧❛r ❞✐s❝r❡t❡ ❡✈❡♥t ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ SLE4 ❝♦♥❥❡❝t✉r❡✳

❋✐❣✉r❡ ✾✿ ❉♦✉❜❧❡ ❞♦♠✐♥♦ t✐❧✐♥❣s ❛♥❞ ❛ss♦❝✐❛t❡❞ ♣❛t❤
❈♦♥s✐❞❡r t❤❡ r❡❝t❛♥❣❧❡ Rn,L = [−nL, nL + 1] × [0, 2n + 1] ✭✐t ✐s ✐♠♣♦rt❛♥t t❤❛t t❤❡ r❡❝t❛♥❣❧❡
❤❛✈❡ ♦❞❞ ❧❡♥❣t❤ ❛♥❞ ✇✐❞t❤✮✳ ❘❡♠♦✈❡ ❛ ✉♥✐t sq✉❛r❡ ❛t t❤❡ ❝♦r♥❡r (−nL, 0) ♦r (nL + 1, 0) t♦ ❣❡t
t✇♦ ❚❡♠♣❡r❧❡②❛♥ ♣♦❧②♦♠✐♥♦s ✭❢♦r ❣❡♥❡r❛❧ ❜❛❝❦❣r♦✉♥❞ ♦♥ ❞♦♠✐♥♦ t✐❧✐♥❣s✱ s❡❡ ❬❑❡♥✵✵❪✮✳ ▲❡t γ
❜❡ t❤❡ r❛♥❞♦♠ ♣❛t❤ ❣♦✐♥❣ ❢r♦♠ (−nL, 0) t♦ (nL + 1, 0)✱ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ s✉♣❡r♣♦s❡❞ ✉♥✐❢♦r♠
❞♦♠✐♥♦ t✐❧✐♥❣s ♦♥ t❤❡ t✇♦ ♣♦❧②♦♠✐♥♦s✳ ▲❡t w ❜❡ ❛ ♣♦✐♥t ♦❢ t❤❡ str✐♣ {z : 0 < ℑz < 1}✱ ❛♥❞
wn ❛♥ ✐♥t❡❣r❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ 2nw ✐♥ Rn,L ✳
Pr♦♣♦s✐t✐♦♥ ✷✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❤♦❧❞s✿

lim lim PRn,L (wn ❧✐❡s ❛❜♦✈❡ γ) = ℑz

L→∞ n→∞

❲❡ ✉s❡ ❛ s✐♠✐❧❛r ❛r❣✉♠❡♥t t♦ t❤❡ ♦♥❡ ❣✐✈❡♥ ✐♥ ❬❑❡♥✾✼❪✱ ✹✳✼✳ ▲❡t R1 ✱ R2 ❜❡ t❤❡ t✇♦
♣♦❧②♦♠✐♥♦s✱ ❛♥❞ h1 ✱ h2 t❤❡ ❤❡✐❣❤t ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ t✇♦ ♣♦❧②♦♠✐♥♦s ✭t❤❡s❡ r❛♥❞♦♠
✐♥t❡❣❡r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ✉♣ t♦ ❛ ❝♦♥st❛♥t✮✳ ■t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ♦♥❡ ♠❛② ❝❤♦♦s❡
h1 ✱ h2 s♦ t❤❛t h = h1 − h2 = 0 ♦♥ t❤❡ ❜♦tt♦♠ s✐❞❡✱ ❛♥❞ h = 4 ♦♥ t❤❡ t❤r❡❡ ♦t❤❡r s✐❞❡s✳ ▲❡t x
❜❡ ❛♥ ✐♥♥❡r ❧❛tt✐❝❡ ♣♦✐♥t✳ ❚❤❡♥✿

Pr♦♦❢✳

E(h(x)) = 4P(x ❧✐❡s ❛❜♦✈❡ γ)

■♥❞❡❡❞✱ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t✇♦ ❞♦♠✐♥♦s t✐❧✐♥❣s✳ ❚❤✐s ✉♥✐♦♥ ❝♦♥s✐sts ♦❢ t❤❡ ♣❛t❤ γ ✱
❞♦✉❜❧❡❞ ❞♦♠✐♥♦s ❛♥❞ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡♥ x ✐s s❡♣❛r❛t❡❞ ❢r♦♠ t❤❡ ❜♦tt♦♠ s✐❞❡ ❜② ❛ ❝❡rt❛✐♥
♥✉♠❜❡r ♦❢ ❝❧♦s❡❞ ❝②❝❧❡s✱ ❛♥❞ ♣♦ss✐❜❧② γ ✳ ❈♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡ ✉♥✐♦♥✱ ❡❛❝❤ ❝❧♦s❡❞ ❝②❝❧❡ ❛❝❝♦✉♥ts

✹✵

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
❢♦r ±4 ✇✐t❤ ❡q✉❛❧ ♣r♦❜❛❜✐❧✐t② ✐♥ h(x)✳ ▼♦r❡♦✈❡r✱ ❝r♦ss✐♥❣ γ ❢r♦♠ ❜❡❧♦✇ ✐♥❝r❡❛s❡s h ❜② 4✳ ❚❤✐s
②✐❡❧❞s t❤❡ ❢♦r♠✉❧❛✳
❆s n ❣♦❡s t♦ ✐♥✜♥✐t②✱ t❤❡ ❛✈❡r❛❣❡ ❤❡✐❣❤t ❢✉♥❝t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s ✭❬❑❡♥✵✵❪✱
❚❤❡♦r❡♠ ✷✸✮✳ ❚❤❡♥ t❛❦❡ t❤❡ ❧✐♠✐t ❛s L ❣♦❡s t♦ ✐♥✜♥✐t② t♦ ❝♦♥❝❧✉❞❡ ✭♦♥❡ ♠❛② ♠❛♣ ❛♥② ✜♥✐t❡
r❡❝t❛♥❣❧❡ RL t♦ t❤❡ ✇❤♦❧❡ s❧✐t✱ ✜①✐♥❣ ❛ ❣✐✈❡♥ ♣♦✐♥t x❀ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❝♦♥✈❡r❣❡ t♦ t❤❡
❛♣♣r♦♣r✐❛t❡ ❝♦♥❞✐t✐♦♥s✱ ♦♥❡ ❝♦♥❝❧✉❞❡s ✇✐t❤ P♦✐ss♦♥✬s ❢♦r♠✉❧❛✮✳

✼ SLE(κ, ρ) ♣r♦❝❡ss❡s ❛♥❞ ❣❡♥❡r❛❧ tr✐❛♥❣❧❡s
■♥ t❤✐s s❡❝t✐♦♥ ✇❡ q✉✐❝❦❧② ❞✐s❝✉ss ❤♦✇ ❛♥② tr✐❛♥❣❧❡ ♠❛② ❜❡ ❛ss♦❝�