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

❤✐tt✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s ✐♥ s✉❝❤ tr✐❛♥❣❧❡s ❛r❡ ✏✉♥✐❢♦r♠✑✮✳ ❖t❤❡r ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡s ❝♦rr❡s♣♦♥❞ t♦ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ t❤❡ q ♣❛r❛♠❡t❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ q = 3 ❝♦rr❡s♣♦♥❞s t♦ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ ✇✐t❤ ❛♥❣❧❡ 2π/3✳ ❙✐♠✐❧❛r❧②✱ ❞♦✉❜❧❡ ❞✐♠❡r✲♠♦❞❡❧s ♦rq = 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♣❛♥♥✐♥❣

tr❡❡s✳

❆❝❦♥♦✇❧❡❞❣♠❡♥ts✳ ■ ✇✐s❤ t♦ t❤❛♥❦ ❲❡♥❞❡❧✐♥ ❲❡r♥❡r ❢♦r ❤✐s ❤❡❧♣ ❛♥❞ ❛❞✈✐❝❡✱ ❛s ✇❡❧❧ ❛s ❘✐❝❤❛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❞❛❧ ❙▲❊

❲❡ ✜rst ❜r✐❡✢② r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❤♦r❞❛❧ ❙▲❊ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡H❣♦✐♥❣ ❢r♦♠₀t♦

∞✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬▲❛✇❙❝❤❲❡r✵✶✱ ❘♦❤❙❝❤✵✶❪ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮✳ ❋♦r ❛♥②z∈H✱ _t≥0✱ ❞❡✜♥❡

gt(z)❜②g0(z) =z ❛♥❞

∂tgt(z) =

2 gt(z)−Wt

✇❤❡r❡ (Wt/√κ, t ≥0) ✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ R✱ st❛rt✐♥❣ ❢r♦♠ 0✳ ▲❡t τz ❜❡ t❤❡

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

Kt={z∈H : τz< t}

❚❤❡ ❢❛♠✐❧②(Kt)t≥0✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ❢❛♠✐❧② ♦❢ ❝♦♠♣❛❝t s❡ts ✐♥H❀ ❢✉rt❤❡r♠♦r❡✱gt✐s ❛ ❝♦♥❢♦r♠❛❧

❡q✉✐✈❛❧❡♥❝❡ ♦❢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 ✐♥Hs✉❝❤ t❤❛tH\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

γt= lim z∈H→Wt

gt−1(z)

❋♦r ❛♥② s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥D✇✐t❤ t✇♦ ❜♦✉♥❞❛r② ♣♦✐♥ts ✭♦r ♣r✐♠❡ ❡♥❞s✮a❛♥❞b✱ ❝❤♦r❞❛❧ SLEκ✐♥D❢r♦♠at♦b✐s ❞❡✜♥❡❞ ❛sKt(D,a,b)=h−1(K

(H,0,∞)

t )✱ ✇❤❡r❡K

(H,0,∞)

t ✐s ❛s ❛❜♦✈❡✱ ❛♥❞

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

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

✸ ❆ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❙▲❊

❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❙▲❊ r❡❧✐❡s ♦♥ t❤❡ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡ gt ♦❢ H\Kt ♦♥t♦ H✳ ❆sH ❤❛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) ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✮✳ ❖t❤❡r ♥♦r♠❛❧✐③❛t✐♦♥s✱ s✉❝❤ ❛s t❤❡ ♦♥❡ ✉s❡❞ ✐♥

[▲❛✇❙❝❤❲❡r✵✶] ♠❛② ♣r♦✈❡ ✉s❡❢✉❧ ❢♦r ❞✐✛❡r❡♥t ♣♦✐♥ts ♦❢ ✈✐❡✇✳

❆ ❜②✲♣r♦❞✉❝t ♦❢ ❙♠✐r♥♦✈✬s r❡s✉❧ts ✭❬❙♠✐✵✶❪✮ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❧❡tκ= 6✱ ❛♥❞F❜❡ t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ♦❢(H_,0,1,∞)♦♥t♦ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡(T, a, b, c)✳ ▲❡t_ht ❜❡ t❤❡ ❝♦♥❢♦r♠❛❧ ❛✉t♦✲

✸✵ ❊❧❡❝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) =

gt(z)−Wt

gt(1)−Wt

s♦ t❤❛tg˜t(∞) =∞✱g˜t(1) = 1 ❛♥❞˜gt(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(Yt) =

dF dzdYt+

1 2

d2_F

dz2dhYti

✇❤❡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, Y2i= (hℜY1,ℜY2i − hℑY1,ℑY2i) +i(hℜY1,ℑY2i+hℑY1,ℜY2i)

s♦ t❤❛t dhCti= 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✿

d˜gt(z) =

₂

˜ gt(z)−

2˜gt(z) +κ(˜gt(z)−1)

_dt

(gt(1)−Wt)2

+ (˜gt(z)−1)

dWt

gt(1)−Wt

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

u(t) =

Z t

0

ds (gs(1)−Ws)2

♦♥❡ ❣❡ts t❤❡ ❛✉t♦♥♦♠♦✉s ❙❉❊✿

dwu= (wu−1)

κ−_w2

u(1 +wu)

du+ (wu−1)dW˜u

✇❤❡r❡ ( ˜Wu/√κ)u≥0 ✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳

▲❡t ✉s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ t✐♠❡ ❝❤❛♥❣❡✳ ▲❡tYt=gt(1)−Wt❀ t❤❡♥✱dYt=−dWt+ 2dt/Yt✱

s♦ t❤❛t(Yt/√κ)t≥0✐s ❛ ❇❡ss❡❧ ♣r♦❝❡ss ♦❢ ❞✐♠❡♥s✐♦♥(1 + 4/κ)✳ ❋♦rκ≤4✱ t❤✐s ❞✐♠❡♥s✐♦♥ ✐s ♥♦t

s♠❛❧❧❡r t❤❛♥ 2✱ s♦ t❤❛tY ❛❧♠♦st s✉r❡❧② ♥❡✈❡r ✈❛♥✐s❤❡s ✭s❡❡ ❡✳❣✳ ❬❘❡✈❨♦r✾✹❪✮❀ ♠♦r❡♦✈❡r✱ ❛✳s✳✱

Z ∞ 0 dt Y2 t =∞

■♥❞❡❡❞✱ ❧❡tTn= inf{t >0 : Yt= 2n}✳ ❚❤❡♥✱ t❤❡ ♣♦s✐t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s(R Tn+1

Tn dt/Y

2

t, n≥

1) ❛r❡ ✐✳✐✳❞✳ ✭✉s✐♥❣ t❤❡ ▼❛r❦♦✈ ❛♥❞ s❝❛❧✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❇❡ss❡❧ ♣r♦❝❡ss❡s✮✳ ❍❡♥❝❡✿

Z ∞ 0 dt Y2 t ≥ ∞ X n=1

Z Tn+1

Tn

dt Y2

t

=∞ ❛✳s✳

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s ✸✶

❲❤❡♥κ >4✱ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ssY ✐s s♠❛❧❧❡r t❤❛♥2✱ s♦ t❤❛tτ1<∞❛❧♠♦st

s✉r❡❧②✳ ■♥ t❤✐s ❝❛s❡✱ ✉s✐♥❣ ❛ s✐♠✐❧❛r ❛r❣✉♠❡♥t ✇✐t❤ t❤❡ st♦♣♣✐♥❣ t✐♠❡s Tn ❢♦rn <0✱ ♦♥❡ s❡❡s

t❤❛t

Z τ1

0

dt Y2

t

=∞

❍❡♥❝❡✱ ✐❢κ >4✱ t❤❡ t✐♠❡ ❝❤❛♥❣❡ ✐s ❛✳s✳ ❛ ❜✐❥❡❝t✐♦♥[0, τ1)→R+✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❢♦r ❛❧❧ κ >0✱ t❤❡ st♦❝❤❛st✐❝ ✢♦✇(˜gu)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✿

dF(wu) =

F′(wu)(κ− 2

wu

(1 +wu)) +κ

2F ′′_(w

u)(wu−1)

(wu−1)du+F′(wu)(wu−1)dW˜u

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

F′_(w)

κ− 2

w(1 +w)

+F′′_(w)κ

2(w−1) = 0

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉❝❤ t❤❛t

F′_(w)∝wα₋1_(w

−1)β−1_,

✇❤❡r❡

α = 1−4κ

β = 8_κ−1 ❋♦rκ= 4✱F(w) = log(w)✐s ❛ s♦❧✉t✐♦♥✳

✹ 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✐❢②F❛s t❤❡ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈✲ ❛❧❡♥❝❡ ♦❢ (H_,0,1,∞) ♦♥t♦ ❛♥ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ _{(Tκ, a, b, c)} ✇✐t❤ ❛♥❣❧❡s _{aˆ = ˆc = απ =}

(1− 4

κ)π ❛♥❞ ˆb = βπ = (

8

κ −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(˜gt

∧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✱limt_րτz(gt(z)−Wt) =

0✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ (gt(z)−Wt) ✐s ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ ③❡r♦ ✐❢ t st❛②s ❜♦✉♥❞❡❞

❛✇❛② ❢r♦♠τz✳ ❘❡❝❛❧❧ t❤❛t

˜ gt(z) =

gt(z)−Wt

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

❙♦✱ ❛stրτ1,z✱ ˜gt(z)→ ∞✐❢τ1< τz ❛♥❞ ˜gt(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❤❛tg˜t(z) t❡♥❞s t♦ 1✱ ❛s ✐s ❡❛s✐❧② s❡❡♥ ❜②

♠❛♣♣✐♥❣Ht♦ str✐♣s✳

◆♦✇ ♦♥❡ ❝❛♥ ❛♣♣❧② t❤❡ ♦♣t✐♦♥❛❧ st♦♣♣✐♥❣ t❤❡♦r❡♠ t♦ t❤❡ ♠❛rt✐♥❣❛❧❡sF(˜gt∧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✐♦♥ ✶✳

❚❤❡ ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ♦❢w =F(z)✐♥ t❤❡ tr✐❛♥❣❧❡ Tκ ❛r❡ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t❤❡

❡✈❡♥tsτz< τ1✱τz=τ1✱τz> τ1✳

❉❡✜♥❡T0_={w∈T

κ : τz< τ1}✱T1={w∈Tκ : τz=τ1}✱T∞={w∈Tκ : τz> τ1}✱

✇❤✐❝❤ ✐s ❛ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ ♦❢ Tκ✳ ❚❤❡s❡ t❤r❡❡ s❡ts ❛r❡ ❛✳s✳ ❜♦r❡❧✐❛♥❀ ✐♥❞❡❡❞✱ T∞ =

F(H_\Kτ

1)✐s ❛✳s✳ ♦♣❡♥✱ ❛♥❞ T

0 ₌S

t<τ1Kt ✐s ❛✳s✳ ❛♥ Fσ ❜♦r❡❧✐❛♥✳ ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡

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

❈♦r♦❧❧❛r② ✶✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❤♦❧❞s✿

E_(A(T0_{)) =}E_(A(T1_{)) =}E_(A(T∞_{)) =} A(Tκ)

3 ✇❤❡r❡A❞❡s✐❣♥❛t❡s t❤❡ ❛r❡❛✳

απ

απ

T

1

T

∞

βπ

✶

F

(

γ

τ₁

)

T

0

∞

✵

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s ✸✸

❈♦r♦❧❧❛r② ✷ ✭❈❛r❞②✬s ❋♦r♠✉❧❛✮✳

▲❡t γ ❜❡ t❤❡ tr❛❝❡ ♦❢ ❛ ❝❤♦r❞❛❧ SLEκ ❣♦✐♥❣ ❢r♦♠ a t♦ c ✐♥ t❤❡ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡ Tκ✱

4 < κ <8✳ ▲❡t τ ❜❡ t❤❡ ✜rst t✐♠❡ γ ❤✐ts (b, c)✳ ❚❤❡♥ γτ ❤❛s ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦♥

(b, c)✳

❖♥❡ ❝❛♥ tr❛♥s❧❛t❡ t❤✐s r❡s✉❧t ♦♥ t❤❡ ✉s✉❛❧ ❤❛❧❢✲♣❧❛♥❡ s❡t✉♣✳

❈♦r♦❧❧❛r② ✸✳ ▲❡t γ❜❡ t❤❡ tr❛❝❡ ♦❢ ❛ ❝❤♦r❞❛❧ SLEκ ❣♦✐♥❣ ❢r♦♠0 t♦∞✐♥ t❤❡ ❤❛❧❢✲♣❧❛♥❡✱

❛♥❞γτ1 ❜❡ t❤❡ ✜rst ❤✐t ♦❢ t❤❡ ❤❛❧❢✲❧✐♥❡[1,∞)❜②γ✳ ❚❤❡♥✱ ✐❢ 4< κ <8✱ t❤❡ ❧❛✇ ♦❢1/γτ1

✐s t❤❛t ♦❢ t❤❡ ❜❡t❛ ❞✐str✐❜✉t✐♦♥B(1−4

κ,

8

κ−1)✳

■t ✐s ❡❛s② t♦ s❡❡ t❤❛t✱ t❤❡ ❧❛✇ ♦❢ γτ1 ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ δ1 ✇❤❡♥ κր 8✳ ❚❤✐s ✐s ♥♦t

s✉r♣r✐s✐♥❣ ❛s ❢♦r κ≥8✱ t❤❡ ❙▲❊ tr❛❝❡γ✐s ❛✳s✳ ❛ P❡❛♥♦ ❝✉r✈❡✱ ❛♥❞ γτ1 = 1❛✳s✳

• ❈❛s❡ κ= 4

■♥ t❤✐s ❝❛s❡✱ F(w) = log(w) ✐s ❛ s♦❧✉t✐♦♥✳ ❖♥❡ ❝❛♥ ❝❤♦♦s❡ ❛ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ❧♦❣❛r✐t❤♠ s✉❝❤ t❤❛t ℑ(log(H)) = (0, π)✳ ❚❤❡♥ _{ℑ(log(˜gt(z))) = arg(˜gt(z))}✐s ❛ ❜♦✉♥❞❡❞

❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡✳ ▲❡t H_r ✭r❡s♣✳ H_l✮ ❜❡ t❤❡ ♣♦✐♥ts ✐♥ H❧❡❢t ♦♥ t❤❡ r✐❣❤t ✭r❡s♣✳ ♦♥ t❤❡

❧❡❢t✮ ❜② t❤❡ ❙▲❊ tr❛❝❡ ✭❛ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ✐s t♦ ❜❡ ❢♦✉♥❞ ✐♥ ❬❙❝❤✵✶❪✮✳ ■❢ z ∈ H_l✱ t❤❡

❤❛r♠♦♥✐❝ ♠❡❛s✉r❡ ♦❢ gt−1((Wt,∞)) s❡❡♥ ❢r♦♠z ✐♥ H\γ[0,t] t❡♥❞s t♦ 0 ❛s t → ∞=τz✳

❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❛r❣✉♠❡♥t ♦❢g˜t(z)t❡♥❞s t♦π✳ ❋♦rz∈Hr✱ ❛♥ ❛r❣✉♠❡♥t s✐♠✐❧❛r t♦

t❤❡ ❝❛s❡4< κ <8s❤♦✇s t❤❛t˜gt(z)→1✳ ❍❡♥❝❡✱ ❛♣♣❧②✐♥❣ t❤❡ ♦♣t✐♦♥❛❧ st♦♣♣✐♥❣ t❤❡♦r❡♠

t♦ t❤❡ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡arg(˜gt(z))✱ ♦♥❡ ❣❡ts✿

arg(z) = 0×P_(z∈H_{r) +π}P_(z∈H_l)

♦r P_(z∈H_l) = arg(z)/π✱ ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ❬❙❝❤✵✶❪✳

✶

✵

∞

❋✐❣✉r❡ ✷✿ F(H)✱ ❝❛s❡_κ= 4✿ s❧✐t

• ❈❛s❡ κ= 8 ▲❡tF(z) =R

w−1

2(w−1)−1dw❀ F ♠❛♣s(H,0,1,∞) ♦♥t♦ ❛ ❤❛❧❢✲str✐♣(D, a,∞, b)✳ ❖♥❡

♠❛② ❝❤♦♦s❡ F s♦ t❤❛t F(H_{) = {z : 0 < ℜz < 1,ℑz > 0}}✳ ❚❤❡♥ _{F(∞) = 0} ❛♥❞

F(0) = 1✳ ▼♦r❡♦✈❡r✱ℜF(˜gt(z))✐s ❛ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡✳ ■♥ t❤❡ ❝❛s❡κ≥8✱ ✐t ✐s ❦♥♦✇♥

t❤❛t τ1 <∞✱τz <∞✱ ❛♥❞τ16=τz ❛✳s✳ ✐❢z6= 1✭s❡❡ ❬❘♦❤❙❝❤✵✶❪✮✳ ❍❡♥❝❡✱ ✐❢τ =τ1∧τz✱

˜

gτ(z) ❡q✉❛❧s 0 ♦r ∞✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡rτz < τ1 ♦r τz > τ1✳ ❆♣♣❧②✐♥❣ t❤❡ ♦♣t✐♦♥❛❧

st♦♣♣✐♥❣ t❤❡♦r❡♠ t♦ t❤❡ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡ℜF(˜gt(z))✱ ♦♥❡ ❣❡ts✿

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

✵

✶

∞

❋✐❣✉r❡ ✸✿ F(H)✱ ❝❛s❡_κ= 8✿ ❤❛❧❢✲str✐♣

• ❈❛s❡ κ >8

■♥ t❤✐s ❝❛s❡✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡F s♦ t❤❛t ✐t ♠❛♣s(H_,0,1,∞)♦♥t♦_(D,1,∞,0)✇❤❡r❡

D=

z : ℑz >0,0<arg(z)<

1−4

κ

π,4

κπ <arg(z−1)< π

❚❤❡♥ F(H₎ ✐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❡F s♦ t❤❛t ✐t ♠❛♣s(H_,0,1,∞)♦♥t♦_(D,∞,0,∞)✱ ✇❤❡r❡

D=

z : ℑz <1,−

4 κ−1

π <arg(z)< 4 κπ

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s ✸✺

✵

∞

✶

(8

/κ

−

1)

π

❋✐❣✉r❡ ✺✿ F(H)✱ ❝❛s❡ 8

3 ≤κ≤4

✺ ❘❛❞✐❛❧ ❙▲❊

▲❡tD ❜❡ t❤❡ ✉♥✐t ❞✐s❦✳ ❘❛❞✐❛❧ ❙▲❊ ✐♥ D st❛rt✐♥❣ ❢r♦♠ ✶ ✐s ❞❡✜♥❡❞ ❜②g0(z) =z✱ z∈D ❛♥❞

t❤❡ ❖❉❊s✿

∂tgt(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❡ ✭❬❘♦❤❙❝❤✵✶❪✮✳ ❉❡✜♥❡˜gt(z) =gt(z)ξt−1✱ s♦ t❤❛t

˜

gt(0) = 0✱g(γ˜ t) = 1✱ ✇❤❡r❡(γt)✐s t❤❡ ❙▲❊ tr❛❝❡✳ ❖♥❡ ♠❛② ❝♦♠♣✉t❡✿

d˜gt(z) =−g˜t(z)˜gt(z) + 1

˜ gt(z)−1

dt+ ˜gt(z)(−idWt−1

2κdt)

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

F′_(z)

−zz+ 1 z−1 −

κ 2z

−κ

2F

′′_(z)z2_{= 0}

✐✳❡✳✱

F′′_(z) F′_(z) =

₂

κ−1

₁

z− 4 κ

1 z−1.

▼❡r♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❞❡✜♥❡❞ ♦♥ D ❡①✐st ❢♦r κ= 2/n✱ n ∈N∗_{✳ ❋♦rκ= 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

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

♣r♦❝❡ss ❢♦r ❝r✐t✐❝❛❧ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r q ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ t❤❡ tr❛❝❡ ♦❢ SLEκ

❢♦rq∈(0,4)✱ ✇❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❤♦❧❞s✿

κ= 4π

cos−1₍₋√_q/2)

❚❤❡♥ t❤❡ ❛ss♦❝✐❛t❡❞ ✐s♦s❝❡❧❡s tr✐❛♥❣❧❡Tκ ❤❛s ❛♥❣❧❡saˆ= ˆc= cos−1(√q/2)✱ˆb=π−2ˆa✳ ▲❡tΓn

❜❡ ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ tr✐❛♥❣❧❡Tκ ♦♥ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡ ✇✐t❤ ♠❡s❤ 1_n❀ ❛❧❧ ✈❡rt✐❝❡s

♦♥ t❤❡ ❡❞❣❡s(a, b]❛♥❞[b, c)❛r❡ ✐❞❡♥t✐✜❡❞✳ ▲❡tΓ†

n❜❡ t❤❡ ❞✉❛❧ ❣r❛♣❤✳ ❚❤❡ ❞✐s❝r❡t❡ ❡①♣❧♦r❛t✐♦♥

♣r♦❝❡ssβ r✉♥s ❜❡t✇❡❡♥ t❤❡ ♦♣❡♥❡❞ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢(a, b]∪[b, c)✐♥Γn ❛♥❞ t❤❡ ❝❧♦s❡❞

❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢(a, c)✐♥ Γ†

n✳

❈♦♥❥❡❝t✉r❡ ✶✳ ❈❛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)✳

❑❡♥②♦♥ ❬❑❡♥✵✷❪ ❤❛s ♣r♦♣♦s❡❞ ❛♥ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧ ❢♦r ❛♥② ✐s♦r❛❞✐❛❧ ❧❛tt✐❝❡✱ ✐♥ ♣❛rt✐❝✉❧❛r ❢♦r ❛♥② r❡❝t❛♥❣✉❧❛r ❧❛tt✐❝❡✳ ▲❡t κ✱q ❛♥❞ α❜❡ ❛s ❛❜♦✈❡✱ ✐✳❡✳ 4 < κ <8✱ 4_κπ = cos−1₍₋√_q/2)

❛♥❞α= 1−κ4✳ ❈♦♥s✐❞❡r t❤❡ r❡❝t❛♥❣✉❧❛r ❧❛tt✐❝❡Zcosαπ+iZsinαπ✳ ❚❤❡♥ ✐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❣❡✮Tκtr✐❛♥❣❧❡✱ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❜♦✉♥❞❛r②

❝♦♥❞✐t✐♦♥s✳ ❆ ❝♦♥✜❣✉r❛t✐♦♥ω∈ {0,1}E ♦❢ ♦♣❡♥ ❡❞❣❡s ❤❛s ♣r♦❜❛❜✐❧✐t②✿

pΓ(ω)∝qk(ω)νheh(ω)νvev(ω)

✇❤❡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=√q sin(2α

2_π)

sin(α(1−2α)π)

νh=

q νv

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✱ κ= 16

3✱ ♦♥❡ r❡tr✐❡✈❡s t❤❡ ✉s✉❛❧ ❝r✐t✐❝❛❧ ❋❑ ♣❡r❝♦❧❛t✐♦♥ ♦♥ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡✳

❙▲❊ ❛♥❞ 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❡rq= 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✐❛♥❣❧❡ T24

5 ✱ ✇❤✐❝❤ ❤❛s ❛♥❣❧❡saˆ = ˆc =

π

6✱

ˆ b= 2π

3 ✳ ❚❤❡ ♣♦ss✐❜❧❡ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ t❤❡q= 3P♦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❤✳ ▲❡tRn,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 ❛sL ❣♦❡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✱ ❛♥❞ ❧❡tb ❜❡ t❤❡ ♦t❤❡r tr✐♣❧❡ ♣♦✐♥t ♦❢ t❤❡

♠✐♥✐♠❛❧ s✉❜tr❡❡ ❝♦♥t❛✐♥✐♥❣(0,0)✱(n,0)✱wn❛♥❞(0, nL)✳ ❖♥❡ ❝❛♥ ❢♦r♠✉❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❛s②

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

▲❡♠♠❛ ✶✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐ts ❤♦❧❞✿ lim

n_→∞Llim_→∞

P_R

n,L(b ❜❡❧♦♥❣s t♦ t❤❡ ♦r✐❡♥t❡❞ ❛r❝[0, a]∪[a, wn]✐♥T) =ℜw

lim

L→∞nlim_→∞

P_R

n,L(b ❜❡❧♦♥❣s t♦ t❤❡ ♦r✐❡♥t❡❞ ❛r❝[0, a]∪[a, wn]✐♥T) =ℜw

▲❡t ✉s ❝❧❛r✐❢② t❤❡ ❛❧t❡r♥❛t✐✈❡ ✭✉♣ t♦ ❡✈❡♥ts ♦❢ ♥❡❣❧✐❣✐❜❧❡ ♣r♦❜❛❜✐❧✐t②✮✿ ❡✐t❤❡rb ❜❡❧♦♥❣s t♦ t❤❡ ✭♦r✐❡♥t❡❞✮ ❛r❝ [0, a]∪[a, wn]✱ ♦r t♦ t❤❡ ✭♦r✐❡♥t❡❞✮ ❛r❝ [wn;a]∪[a,1]✳ ❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡

❝♦♠♣✉t❡❞ P_(τF−1_(w) > τ₁) = ℜw ❢♦r ❛ ❝❤♦r❞❛❧ SLE₈ ❣♦✐♥❣ ❢r♦♠ 0 t♦ 1 ✐♥ t❤❡ ❤❛❧❢✲str✐♣ ✭✐♥

❛❝❝♦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❝ ❥♦✐♥✐♥❣at♦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✳

❛

✵

♥

✭✵✱▲♥✮

w

n

❜

❛

✵

♥

w

n

❜

✭✵✱▲♥✮

❋✐❣✉r❡ ✽✿ ❚❤❡ ❛❧t❡r♥❛t✐✈❡

■♥ ❢❛❝t✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ t❤❡ ❧❡♠♠❛ ✇✐t❤♦✉t ✉s✐♥❣ t❤❡ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ❢♦r ❯❙❚✳ ■♥❞❡❡❞✱ ❧❡tw†

n

❜❡ ❛ ♣♦✐♥t ♦♥ t❤❡ ❞✉❛❧ ❣r✐❞ st❛♥❞✐♥❣ ❛t ❞✐st❛♥❝❡ √2

2 ❢r♦♠wn✳ ❚❤❡♥✱ ❛s nt❡♥❞s t♦ ✐♥✜♥✐t②✱

P_R

n,L(b❜❡❧♦♥❣s t♦ t❤❡ ❛r❝[0, a]∪[a, wn]✐♥ T)

−P

R†

n,L(w

†

n ✐s ❝♦♥♥❡❝t❡❞ t♦ t❤❡ r✐❣❤t✲❤❛♥❞ ❜♦✉♥❞❛r② ✐♥ t❤❡ ❞✉❛❧ tr❡❡)→0

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

n

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♦♠w†

❙▲❊ ❛♥❞ 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❡❡ ❬❑❡♥✵✵❪✮✳ ▲❡tγ ❜❡ 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

L→∞nlim→∞

P_R

n,L(wn ❧✐❡s ❛❜♦✈❡γ) =ℑz

Pr♦♦❢✳ ❲❡ ✉s❡ ❛ s✐♠✐❧❛r ❛r❣✉♠❡♥t t♦ t❤❡ ♦♥❡ ❣✐✈❡♥ ✐♥ ❬❑❡♥✾✼❪✱ ✹✳✼✳ ▲❡t R1✱ R2 ❜❡ t❤❡ t✇♦

♣♦❧②♦♠✐♥♦s✱ ❛♥❞h1✱h2t❤❡ ❤❡✐❣❤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❤❛th=h1−h2= 0♦♥ t❤❡ ❜♦tt♦♠ s✐❞❡✱ ❛♥❞ h= 4♦♥ t❤❡ t❤r❡❡ ♦t❤❡r s✐❞❡s✳ ▲❡tx

❜❡ ❛♥ ✐♥♥❡r ❧❛tt✐❝❡ ♣♦✐♥t✳ ❚❤❡♥✿

E_{(h(x)) = 4}P_(x ❧✐❡s ❛❜♦✈❡_γ)

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

❢♦r±4 ✇✐t❤ ❡q✉❛❧ ♣r♦❜❛❜✐❧✐t② ✐♥h(x)✳ ▼♦r❡♦✈❡r✱ ❝r♦ss✐♥❣γ❢r♦♠ ❜❡❧♦✇ ✐♥❝r❡❛s❡sh❜②4✳ ❚❤✐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❛♥❣❧❡RLt♦ t❤❡ ✇❤♦❧❡ s❧✐t✱ ✜①✐♥❣ ❛ ❣✐✈❡♥ ♣♦✐♥tx❀ 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♦❝✐❛t❡❞ ✇✐t❤ ❛ ❝❡rt❛✐♥ SLE ♣r♦❝❡ss✱ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ✐s♦s❝❡❧❡ tr✐❛♥❣❧❡s ✇❡r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ SLEκ♣r♦❝❡ss❡s✳

✼✳✶

SLE(

κ, ρ

)

♣r♦❝❡ss❡s

▲❡t ✉s ❜r✐❡✢② ❞❡s❝r✐❜❡ SLE(κ, ρ) ♣r♦❝❡ss❡s✱ ❞❡✜♥❡❞ ✐♥ ❬▲❙❲✵✷❜❪✳ ▲❡t(Wt, Ot)t_≥0 ❜❡ ❛ t✇♦✲

❞✐♠❡♥s✐♦♥❛❧ s❡♠✐♠❛rt✐♥❣❛❧❡ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❉❊s✿

_dW

t=√κdBt+_Wt−ρ_Otdt

dOt= _Ot−2_Wtdt ✭✶✮

✇❤❡r❡B✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ❛s ✇❡❧❧ ❛s t❤❡ ✐♥❡q✉❛❧✐t②Wt≤Ot✈❛❧✐❞ ❢♦r ❛❧❧ ♣♦s✐t✐✈❡

t✐♠❡s ✭t❤❡ ❝♦♥✈❡♥t✐♦♥ ❤❡r❡ ❞✐✛❡rs ❢r♦♠ t❤❡ ♦♥❡ ✐♥ ❬▲❙❲✵✷❜❪✮✳ ❚❤✐s ♣r♦❝❡ss ✐s ✇❡❧❧ ❞❡✜♥❡❞ ❢♦r κ >0✱ρ >−2✳ ■♥❞❡❡❞✱ ♦♥❡ ♠❛② ❝♦♥s✐❞❡rZt=Ot−Wt✳ ❚❤❡ ♣r♦❝❡ss(Zt/√κ)t_≥0 ✐s ❛ ❇❡ss❡❧

♣r♦❝❡ss ✐♥ ❞✐♠❡♥s✐♦♥ d= 1 + 2ρ+2_κ ✳ ❙✉❝❤ ♣r♦❝❡ss❡s ❛r❡ ✇❡❧❧ ❞❡✜♥❡❞ s❡♠✐♠❛rt✐♥❣❛❧❡s ✐❢d >1✱ ♦r ρ >−2✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬❘❡✈❨♦r✾✹❪✮✳ ❚❤❡♥Ot= 2R

t

0

du

Zu ❛♥❞Wt=Ot−Zt✳

❍❡♥❝❡ ♦♥❡ ♠❛② ❞❡✜♥❡ ❛SLE(κ, ρ)❛s ❛ st♦❝❤❛st✐❝ ▲♦❡✇♥❡r ❝❤❛✐♥ t❤❡ ❞r✐✈✐♥❣ ♣r♦❝❡ss ♦❢ ✇❤✐❝❤ ❤❛s t❤❡ ❧❛✇ ♦❢ t❤❡ ♣r♦❝❡ss (Wt) ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✭♦r r❛t❤❡r st❛t❡✮ ♦❢ t❤❡

♣r♦❝❡ss ✐s ❛ ❝♦✉♣❧❡ (w, o) ✇✐t❤ w≤ o✱ ✉s✉❛❧❧② s❡t t♦ (0,0+_{)✳ ❚❤❡♥O}

t r❡♣r❡s❡♥ts t❤❡ ✐♠❛❣❡

✉♥❞❡r t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣gt♦❢ t❤❡ r✐❣❤t♠♦st ♣♦✐♥t ♦❢∂Kt∪O0✳ ❖❜✈✐♦✉s❧②✱ ❢♦rρ= 0✱ ♦♥❡

r❡❝♦✈❡rs ❛ st❛♥❞❛r❞SLE(κ)♣r♦❝❡ss✳

Pr♦♣♦s✐t✐♦♥ ✸✳ ▲❡t(Wt, Ot)❜❡ t❤❡ ❞r✐✈✐♥❣ ♣r♦❝❡ss ♦❢ ❛SLE(κ, ρ)♣r♦❝❡ss st❛rt✐♥❣ ❢r♦♠(0,1)✱

❛♥❞ (gt) ❜❡ t❤❡ ❛ss♦❝✐❛t❡❞ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡s✳ ▲❡t z ∈ H✳ ❚❤❡♥ ✐❢ F ✐s ❛♥② ❛♥❛❧②t✐❝

❢✉♥❝t✐♦♥ ♦♥ H✱ t❤❡ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ s❡♠✐♠❛rt✐♥❣❛❧❡

t7→F

gt(z)−Wt

Ot−Wt

✐s ❛ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢✿

F′_(z)∝z−4

κ(1−z)2 ρ−κ+4

κ .

❚❤❡ ♣r♦♦❢ ✐s r♦✉t✐♥❡ ❛♥❞ ✐s ♦♠✐tt❡❞✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ F ♠❛② ❜❡ ✐❞❡♥t✐✜❡❞ ✉s✐♥❣ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛✳

✼✳✷ ❆ ♣❛rt✐❝✉❧❛r ❝❛s❡

■♥ ❬❙❝❤✵✶❪✱ ❙❝❤r❛♠♠ ❞❡r✐✈❡s ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❢♦r♠

❙▲❊ ❛♥❞ tr✐❛♥❣❧❡s ✹✶

✇❤❡r❡γ ✐s t❤❡ tr❛❝❡ ♦❢ ❛SLE(κ)♣r♦❝❡ss✱ ❢♦rκ≤4✳ ❚❤❡ ❢✉♥❝t✐♦♥Fκ ✐♥✈♦❧✈❡s ❤②♣❡r❣❡♦♠❡tr✐❝

❢✉♥❝t✐♦♥s✱ ❛♥❞Fκ(x)∝x✐✛κ= 4✭✐♥ t❤✐s ❝❛s❡F◦arg✐s ❛ ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥✮✳ ◆♦✇ ✐t ✐s ❡❛s✐❧②

s❡❡♥ t❤❛t ❢♦r ❛♥② κ > 0✱ ρ > −2✱ ✐❢ δ ❞❡s✐❣♥❛t❡s t❤❡ r✐❣❤t ❜♦✉♥❞❛r② ♦❢ ❛ SLE(κ, ρ) ♣r♦❝❡ss st❛rt✐♥❣ ❢r♦♠ (0,0+_{)✱ t❤❡♥ ❛ s✐♠♣❧❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ s❝❛❧✐♥❣ ✐s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥F}

κ,ρ

s✉❝❤ t❤❛t✿

P_(z∈H❧✐❡s t♦ t❤❡ ❧❡❢t ♦❢_{δ) =Fκ,ρ(argz)}

▼♦r❡♦✈❡r✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ♥♦t ✐❞❡♥t✐❝❛❧❧② ③❡r♦ ✐❢ ρ ≥κ/2−2✳ ❚❤✐s ♠♦t✐✈❛t❡s t❤❡ ❢♦❧❧♦✇✐♥❣

r❡s✉❧t✿

Pr♦♣♦s✐t✐♦♥ ✹✳ ▲❡t κ >4✱ρ= κ

2 −2✳ ❚❤❡♥✿

P_(z∈H❧✐❡s t♦ t❤❡ ❧❡❢t ♦❢_{δ) = argz/π}

Pr♦♦❢✳ ▲②✐♥❣ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ r✐❣❤t ❜♦✉♥❞❛r② ♦❢ t❤❡ ❤✉❧❧ ✐s t❤❡ s❛♠❡ t❤✐♥❣ ❛s ❜❡✐♥❣ ❛❜s♦r❜❡❞ ✐❢ κ > 4✳ ▲❡t (Wt, Ot) ❜❡ t❤❡ ❞r✐✈✐♥❣ ♠❡❝❤❛♥✐s♠ ♦❢ t❤❡ SLE(κ,κ₂ −2)✱ ❛♥❞ ❧❡t zt = gt(z)✳

❙✉♣♣♦s❡ ❢♦r ♥♦✇ t❤❛t t❤❡ st❛rt✐♥❣ st❛t❡ ♦❢ t❤❡ SLE✐s(W0, O0) = (0,1)✳ ▲❡th:H→C❜❡ ❛

❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✳ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦rh(zt−Wt

Ot−Wt)t♦

❜❡ ❛ ✭C✲✈❛❧✉❡❞✮ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡ ✐s t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

h′′_(z) h′_(z) =−

4 κ 1 z −2

ρ−κ+ 4 κ

1 1−z

♦r h(z)∝z−4

κ(1−z)2 ρ−κ+4

κ ✳ ■♥ t❤❡ ❝❛s❡ ρ=κ/2−2✱ ✉s✐♥❣ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛

✭s❡❡ ❬❆❤❧✼✾❪✮✱ ♦♥❡ s❡❡s t❤❛t h✐s ✭✉♣ t♦ ❛ ❝♦♥st❛♥t ❢❛❝t♦r✮ t❤❡ ❝♦♥❢♦r♠❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ (H_,0,1,∞)❛♥❞_(D,0,1,∞)✱ ✇❤❡r❡ _D✐s t❤❡ ❞❡❣❡♥❡r❛t❡ tr✐❛♥❣❧❡ ❞❡✜♥❡❞ ❜②✿

D={z∈H _{: arg(z)≤π(1−4/κ),arg(z−1)≥π(1−4/κ)}}

▲❡tϕ(z) =ℜz−cotan(π(1−4/κ))ℑz✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢D ✉♥❞❡r t❤✐sR✲❧✐♥❡❛r ❢♦r♠ ✐s_[0,1]✳

❍❡♥❝❡ ϕ◦h(zt−Wt

Ot−Wt) ✐s ❛ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛❧❡✳ ▼♦r❡♦✈❡r✱ st❛♥❞❛r❞ ❝♦♥✈❡r❣❡♥❝❡ ❛r❣✉♠❡♥ts

✐♠♣❧② t❤❛t zt−Wt

Ot−Wt ❣♦❡s t♦0✐♥ ✜♥✐t❡ t✐♠❡ ✐❢z✐s ❛❜s♦r❜❡❞ ❛♥❞ t♦ ✶ ✐♥ ✐♥✜♥✐t❡ t✐♠❡ ✐♥ t❤❡ ♦t❤❡r

❝❛s❡✳ ❆ str❛✐❣❤t❢♦r✇❛r❞ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♦♣t✐♦♥❛❧ st♦♣♣✐♥❣ t❤❡♦r❡♠ ②✐❡❧❞s✿

P_(z∈H❧✐❡s t♦ t❤❡ r✐❣❤t ♦❢_{δ) =ϕ}

Z z

0

w−4

κ(1−w)

4

κ−1dw

/B(1−4/κ,4/κ)

❚❛❦✐♥❣ t❤❡ ❛s②♠♣t♦t✐❝s ♦❢ t❤✐s ❢♦r♠✉❧❛ ✇❤❡♥z=rexpiθ _{❣♦❡s t♦ ✐♥✜♥✐t② ✇✐t❤ ❝♦♥st❛♥t ❛r❣✉✲}

♠❡♥t ✭♠❛❦✐♥❣ ✉s❡ ♦❢ B(1−x, x) = π/sin(πx)✮✱ ♦♥❡ ✜♥❞s t❤❛t ❢♦r ❛ SLE(κ,κ

2 −2) st❛rt✐♥❣

❢r♦♠(0,0+_)✿

P_(z∈H❧✐❡s t♦ t❤❡ r✐❣❤t ♦❢_{δ) = 1−argz/π}

■♥ ♦t❤❡r ✇♦r❞s✱ Fκ,κ/2₋2 =F4 ❢♦r ❛❧❧ κ≥4✳ ❚❤✐s r❛✐s❡s s❡✈❡r❛❧ q✉❡st✐♦♥s✱ s✉❝❤ ❛s ✇❤❡t❤❡r

t❤✐s st✐❧❧ ❤♦❧❞s ❢♦rκ <4✱ ♦r ✇❤❡t❤❡r ✐♥ ❢✉❧❧ ❣❡♥❡r❛❧✐t② Fκ,ρ =F2κ/(ρ+2)✱ t❤✐s ❧❛st ❝♦♥❥❡❝t✉r❡

❜❡✐♥❣ ❜❛s❡❞ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss(Ot−Wt)✱ ✇❤❡r❡ (Wt, Ot)❞❡s✐❣♥❛t❡s t❤❡

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

❘❡❢❡r❡♥❝❡s

❬❆❤❧✼✾❪ ▲✳ ❆❤❧❢♦rs✱ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✱ ✸r❞ ❡❞✐t✐♦♥✱ ▼❝●r❛✇✲❍✐❧❧✱ ✶✾✼✾

❬❈❛✾✷❪ ❏✳▲✳ ❈❛r❞②✱ ❈r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥ ✐♥ ✜♥✐t❡ ❣❡♦♠❡tr✐❡s✱ ❏✳ P❤②s✳ ❆✱ ✷✺✱ ▲✷✵✶✲▲✷✵✻✱ ✶✾✾✷ ❬●r✐✾✼❪ ●✳ ❘✳ ●r✐♠♠❡tt✱ P❡r❝♦❧❛t✐♦♥ ❛♥❞ ❞✐s♦r❞❡r❡❞ s②st❡♠s✱ ✐♥ ▲❡❝t✉r❡s ♦♥ Pr♦❜❛❜✐❧✐t② ❚❤❡✲

♦r② ❛♥❞ ❙t❛t✐st✐❝s✱ ❊❝♦❧❡ ❞✬été ❞❡ ♣r♦❜❛❜✐❧✐tés ❞❡ ❙❛✐♥t✲❋❧♦✉r ❳❳❱■✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✶✻✻✺✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✾✼

❬❑❡♥✾✼❪ ❘✳ ❑❡♥②♦♥✱ ▲♦❝❛❧ st❛t✐st✐❝s ♦❢ ❧❛tt✐❝❡ ❞✐♠❡rs✱ ❆♥♥✳ ■♥st✳ ❍✳ P♦✐♥❝❛ré Pr♦❜❛❜✳ ❙t❛t✐st✳✱ ✸✸✱ ♣♣ ✺✾✶✕✻✶✽✱ ✶✾✾✼

❬❑❡♥✵✵❪ ❘✳ ❑❡♥②♦♥✱ ❈♦♥❢♦r♠❛❧ ✐♥✈❛r✐❛♥❝❡ ♦❢ ❞♦♠✐♥♦ t✐❧✐♥❣✱ ❆♥♥✳ Pr♦❜❛❜✳✱ ✷✽✱ ♥♦✳✷✱ ♣♣ ✼✺✾✕ ✼✾✺✱ ✷✵✵✵

❬❑❡♥✵✷❪ ❘✳ ❑❡♥②♦♥✱ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❞✐♠❡r ♠♦❞❡❧✱ ▲❡❝t✉r❡ ♥♦t❡s ❢♦r ❛ s❤♦rt ❝♦✉rs❡ ❛t t❤❡ ■❈❚P✱ ✷✵✵✷

❬▲❛✇✵✶❪ ●✳ ▲❛✇❧❡r✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❙t♦❝❤❛st✐❝ ▲♦❡✇♥❡r ❊✈♦❧✉t✐♦♥✱ ♣r❡♣r✐♥t✱ ✷✵✵✶ ❬▲❛✇❙❝❤❲❡r✵✶❪ ●✳ ▲❛✇❧❡r✱ ❖✳ ❙❝❤r❛♠♠✱ ❲✳ ❲❡r♥❡r✱ ❱❛❧✉❡s ♦❢ ❇r♦✇♥✐❛♥ ✐♥t❡rs❡❝t✐♦♥ ❡①♣♦✲

♥❡♥ts ■✿ ❍❛❧❢✲♣❧❛♥❡ ❡①♣♦♥❡♥ts✱ ❆❝t❛ ▼❛t❤✳ ✶✽✼✱ ✷✸✼✕✷✼✸✱ ✷✵✵✶

❬▲❛✇❙❝❤❲❡r✵✷❪ ●✳ ▲❛✇❧❡r✱ ❖✳ ❙❝❤r❛♠♠✱ ❲✳ ❲❡r♥❡r✱ ❈♦♥❢♦r♠❛❧ ■♥✈❛r✐❛♥❝❡ ♦❢ ♣❧❛♥❛r ❧♦♦♣✲ ❡r❛s❡❞ r❛♥❞♦♠ ✇❛❧❦s ❛♥❞ ✉♥✐❢♦r♠ s♣❛♥♥✐♥❣ tr❡❡s✱ ♣r❡♣r✐♥t✱ ❛r❳✐✈✿♠❛t❤✳P❘✴✵✶✶✷✷✸✹✱ ✷✵✵✷ ❬▲❙❲✵✷❜❪ ●✳ ▲❛✇❧❡r✱ ❖✳ ❙❝❤r❛♠♠✱ ❲✳ ❲❡r♥❡r✱ ❈♦♥❢♦r♠❛❧ r❡str✐❝t✐♦♥✳ ❚❤❡ ❝❤♦r❞❛❧ ❝❛s❡✱

♣r❡♣r✐♥t✱ ❛r❳✐✈✿♠❛t❤✳P❘✴✵✷✵✾✸✹✸✱ ✷✵✵✷

❬P✐t❨♦r✽✵❪ ❏✳ P✐t♠❛♥✱ ▼✳ ❨♦r✱ ❇❡ss❡❧ Pr♦❝❡ss❡s ❛♥❞ ■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❧❛✇s✱ ✐♥ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛❧s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✽✺✶✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ♣♣ ✷✽✺✕✸✼✵

❬❘❛❣❍❡♥❆r♦✾✼❪ ❘✳ ❘❛❣❤❛✈❛♥✱ ❈✳ ▲✳ ❍❡♥❧❡②✱ ❙✳ ▲✳ ❆r♦✉❤✱ ◆❡✇ t✇♦✲❝♦❧♦rs ❞✐♠❡r ♠♦❞❡❧s ✇✐t❤ ❝r✐t✐❝❛❧ ❣r♦✉♥❞ st❛t❡s✱ ❏✳ ❙t❛t✐st✳ P❤②s✳✱ ✽✻✭✸✲✹✮✿✺✶✼✕✺✺✵✱ ✶✾✾✼

❬❘❡✈❨♦r✾✹❪ ❉✳ ❘❡✈✉③✱ ▼✳ ❨♦r✱ ❈♦♥t✐♥✉♦✉s ♠❛rt✐♥❣❛❧❡s ❛♥❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ✷♥❞ ❡❞✐t✐♦♥✱ ●r✉♥❞❧❡❤r❡♥ ❞❡r ♠❛t❤❡♠❛t✐s❝❤❡♥ ✇✐ss❡♥s❝❤❛❢t❡♥ ✷✾✸✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ✶✾✾✹

❬❘♦❤❙❝❤✵✶❪ ❙✳ ❘♦❤❞❡✱ ❖✳ ❙❝❤r❛♠♠✱ ❇❛s✐❝ Pr♦♣❡rt✐❡s ♦❢ ❙▲❊✱ ❛r❳✐✈✿♠❛t❤✳P❘✴✵✶✵✻✵✸✻✱ ✷✵✵✶ ❬❙❝❤✵✵❪ ✱ ❖✳ ❙❝❤r❛♠♠✱ ❙❝❛❧✐♥❣ ❧✐♠✐ts ♦❢ ❧♦♦♣✲❡r❛s❡❞ r❛♥❞♦♠ ✇❛❧❦s ❛♥❞ ✉♥✐❢♦r♠ s♣❛♥♥✐♥❣ tr❡❡s✱

■sr❛❡❧ ❏✳ ▼❛t❤✳✱ ✶✶✽✱ ✷✷✶✕✷✽✽✱ ✷✵✵✵

❬❙❝❤✵✶❪ ❖✳ ❙❝❤r❛♠♠✱ ❆ ♣❡r❝♦❧❛t✐♦♥ ❢♦r♠✉❧❛✱ ❛r❳✐✈✿♠❛t❤✳P❘✴✵✶✵✼✵✾✻✈✷✱ ✷✵✵✶

❬❙♠✐✵✶❪ ❙✳ ❙♠✐r♥♦✈✱ ❈r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥ ✐♥ t❤❡ ♣❧❛♥❡✳ ■✳ ❈♦♥❢♦r♠❛❧ ■♥✈❛r✐❛♥❝❡ ❛♥❞ ❈❛r❞②✬s ❢♦r♠✉❧❛ ■■✳ ❈♦♥t✐♥✉✉♠ s❝❛❧✐♥❣ ❧✐♠✐t✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✱ ✷✵✵✶

❬❲✐❧✼✹❪ ❉✳ ❲✐❧❧✐❛♠s✱ P❛t❤ ❉❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝♦♥t✐♥✉✐t② ♦❢ ❧♦❝❛❧ t✐♠❡ ❢♦r ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❞✐✛✉s✐♦♥s✱ ■✱ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ❙❡r ✸✱ ✷✽✱ ♣♣ ✼✸✽✕✼✻✽✱ ✶✾✼✹