❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❋❮❙■❈❆ ●❘❆❉❯❆➬➹❖ ❊▼ ❋❮❙■❈❆
❋❊❘◆❆◆❉❖ ❏❖❙➱ ❉❊ ❆▲▼❊■❉❆
▼➱❚❖❉❖❙ P❆❘❆ ❙■▼❯▲❆➬➹❖ ❉❊ ❙▲❊ ❈❖❘❉❆▲
❋❊❘◆❆◆❉❖ ❏❖❙➱ ❉❊ ❆▲▼❊■❉❆
▼➱❚❖❉❖❙ P❆❘❆ ❙■▼❯▲❆➬➹❖ ❉❊ ❙▲❊ ❈❖❘❉❆▲
▼♦♥♦❣r❛✜❛ ❞❡ ❇❛❝❤❛r❡❧❛❞♦ ❛♣r❡s❡♥t❛❞❛ à ❈♦♦r❞❡♥❛çã♦ ❞❛ ●r❛❞✉❛çã♦ ❞♦ ❈✉rs♦ ❞❡ ❋í✲ s✐❝❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥❞ré ❆✉t♦ ▼♦r❡✐r❛
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❆●❘❆❉❊❈■▼❊◆❚❖❙
❆♦s ▼❡✉s P❛✐s✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ ✐r❡str✐t♦s✳
❆♦ ♣r♦❢❡ss♦r ❏♦sé ❙♦❛r❡s ❞❡ ❆♥❞r❛❞❡ ❏✉♥✐♦r✱ ♣❡❧❛s ♦r✐❡♥t❛çõ❡s✱ ❡ ♣♦r ♠❡ ❞❛r ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ✐♥✐❝✐❛r ♥❛ ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛✳
❆♦ ♣r♦❢❡ss♦r ❆♥❞ré ❆✉t♦ ▼♦r❡✐r❛✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥❝❡♥t✐✈♦ ❞✉✲ r❛♥t❡ ♦s ❞♦✐s ❛♥♦s ❝♦♠♦ s❡✉ ❛❧✉♥♦✱ q✉❡ t♦r♥❛r❛♠ ♣♦ssí✈❡❧ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦ ♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❢ís✐❝❛ ❞❛ ❯❋❈✱ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❢♦r♥❡❝✐❞♦ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞❛ ❛ ❣r❛❞✉❛çã♦❀ ❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋❈ q✉❡ t❛♠❜é♠ t✐✈❡r❛♠ ♣❛rt✐❝✐♣❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ♠❡✉ ❛♣r❡♥❞✐③❛❞♦✳
❆ ❡st❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ♣♦r t♦❞♦s ♦s r❡❝✉rs♦s ❡♠♣r❡❣❛❞♦s ❡ q✉❡ ❛❝♦♠♣❛♥❤❛♠ t♦❞❛ ❛ ❢♦r♠❛çã♦ ❞❡ s❡✉s ❛❧✉♥♦s✱ ❡♠ s✉❛ r♦t✐♥❛ ♥♦ ❝❛♠♣✉s✳
❆♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❆❞❛✐❛s ❙♦✉③❛✱ ❊♠❛♥✉❡❧ ❋♦♥t❡❧❧❡s✱ ❋✐❧✐♣❡ ▼❛rt✐♥s✱ ❏♦ã♦ P❡❞r♦✱ ❏♦sé ❇❡♥t✐✈✐✱ ❑❡♥ ❆✐❦❛✇❛✱ ▼✐❝❤❡❧ ❘♦❞r✐❣✉❡s✱ ◆❛t❤❛♥❛❡❧❧ ❙♦✉s❛✱ P❛❜❧♦ ❘❛♠♦♥✱ P❡❞r♦ ❍❡♥r✐q✉❡✱ ❘❛✉❧ P❡✐①♦t♦✱ ❙♦✜❛ ▼❛❣❛❧❤ã❡s ❡ ❲❡♥❞❡❧ ❖❧✐✈❡✐r❛✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛♣♦✐♦ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳
❘❊❙❯▼❖
▼♦❞❡❧♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s q✉❡ s✐♠✉❧❛♠ ♦ ❢❡♥ô♠❡♥♦ ❞❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡s sã♦ ❝♦♠✉♥s ❡♠ ❢ís✐❝❛ ❡st❛tíst✐❝❛✳ ❆❝r❡❞✐t❛✲s❡ q✉❡✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s✱ ❛ ❡✈♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❛♣r❡s❡♥t❛ ✐♥✈❛r✐â♥❝✐❛ ❝♦♥❢♦r♠❡ ♥❛ r❡❣✐ã♦ ❝rít✐❝❛✳ ❈❛s♦ ❡ss❛ s✉♣♦s✐çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✱ ♥♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦✱ q✉❛♥❞♦ ♦ ❡s♣❛ç❛♠❡♥t♦ ❞❛ r❡❞❡ t❡♥❞❡ ❛ ③❡r♦✱ ❛s ❝✉r✈❛s ❛❧❡❛tór✐❛s q✉❡ ❢♦r♠❛♠ ❛ ❢r♦♥t❡✐r❛ ❞♦s ❛❣❧♦♠❡r❛❞♦s ♥❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ♣♦❞❡r✐❛♠ s❡r ❞❡s❝r✐t❛s ♣♦r ✉♠ ♣r♦❝❡ss♦ ❞✐♥â♠✐❝♦ ❞❡♥♦♠✐♥❛❞♦ SLE✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞❡✜♥✐♠♦s SLE ❝♦r❞❛❧✱ ❡ ❢❛❧❛♠♦s s♦❜r❡ s✉❛
❝♦♥❡①ã♦ ❝♦♠ ♦s ♠♦❞❡❧♦s ❞❡ r❡❞❡✳ ▼♦str❛♠♦s t❛♠❜é♠ ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✉çã♦ ♥✉♠ér✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❙▲❊ ❞✐s❝r❡t♦✳ ❆ ♣❛rt✐r ❞❡❧❡ é ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ♠♦❞❡❧♦s ❞❡ r❡❞❡ ❡♠ SLEκ✱ ♦ q✉❡ ❧❡✈❛ ❛ ✉♠❛ ♠❛✐♦r ❝♦♠♣r❡❡♥sã♦
❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s s✐st❡♠❛s ♥❛ ❢❛s❡ ❝rít✐❝❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥❛ ♦❜t❡♥çã♦ ❞❡ s❡✉s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ❛♦ ♠ét♦❞♦ ❞✐s❝r❡t♦ é ♠♦str❛❞❛✱ q✉❡ ♣r♦❞✉③ ✉♠ ❛❧❣♦r✐t♠♦ ♠❛✐s ❡✜❝✐❡♥t❡ ♥❛ ♠❛✐♦r✐❛ ❞❛s s✐t✉❛çõ❡s✳
❆❇❙❚❘❆❈❚
❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ ♠♦❞❡❧s t❤❛t s✐♠✉❧❛t❡ t❤❡ ♣❤❡♥♦♠❡♥♦♥ ♦❢ ♣❤❛s❡ tr❛♥s✐t✐♦♥s ❛r❡ ❝♦♠♠♦♥ ✐♥ st❛t✐st✐❝❛❧ ♣❤②s✐❝s✳ ■t✬s ❜❡❧✐❡✈❡❞ t❤❛t s♦♠❡ ♠♦❞❡❧s ❛r❡ ❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t ✐♥ t❤❡ s②st❡♠✬s ❝r✐t✐❝❛❧ r❡❣✐♦♥✳ ■❢ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s tr✉❡✱ ✐♥ t❤❡ ❝♦♥t✐♥✉✉♠ ❧✐♠✐t ✇❤❡♥ t❤❡ s♣❛❝✐♥❣ ♦❢ t❤❡ ♠❡s❤ ✐s ③❡r♦✱ r❛♥❞♦♠ tr❛❝❡s t❤❛t ❢♦r♠ t❤❡ ❜♦✉♥❞❛r② ♦❢ ❝❧✉st❡rs ❛t t❤❡ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❝♦✉❧❞ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ❞②♥❛♠✐❝ ♣r♦❝❡ss ❝❛❧❧❡❞ SLE✳ ■♥ t❤✐s ✇♦r❦
✇❡ ❞❡✜♥❡ ❝❤♦r❞❛❧ ❙▲❊✱ ❛♥❞ t❛❧❦ ❛❜♦✉t ✐ts ❝♦♥♥❡❝t✐♦♥ t♦ ❧❛tt✐❝❡ ♠♦❞❡❧s✳ ❲❡ ❛❧s♦ s❤♦✇ ❛ ♠❡t❤♦❞ ❢♦r ♥✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ▲♦❡✇♥❡r ❡q✉❛t✐♦♥✱ ✉s✐♥❣ ❛ ♠❡t❤♦❞ ❦♥♦✇ ❛s ❞✐s❝r❡t❡ ❙▲❊❀ ❢r♦♠ ✇❤✐❝❤ ✐s ♣♦ss✐❜❧❡ t♦ ✈❡r✐❢② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❧❛tt✐❝❡ ♠♦❞❡❧s ✇❤✐❝❤ ❧❡❛❞s t♦ ❛ ❣r❡❛t❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ s✉❝❤ s②st❡♠s✱ ❧✐❦❡ ❣❡tt✐♥❣ t❤❡✐r ❝r✐t✐❝❛❧✲♣♦✐♥t ❡①♣♦♥❡♥ts✳ ❋✐♥❛❧❧②✱ ✇❡ s❤♦✇♥ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❙▲❊✱ ♣r♦❞✉❝✐♥❣ ❛ ♠♦r❡ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠ ✐♥ ♠♦st s✐t✉❛t✐♦♥s✳
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
❋✐❣✉r❛ ✶ ✕ ❉✐❛❣r❛♠❛ ♣r❡ssã♦✲t❡♠♣❡r❛t✉r❛ ❞❛ á❣✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❋✐❣✉r❛ ✷ ✕ ❆♣❧✐❝❛çã♦ ❞❡ gt ❡♠ ✉♠ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ s✐♠♣❧❡s ♣❛r❛ ❛❧❣✉♥s ✐♥s✲
t❛♥t❡s ❞❡ t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ❋✐❣✉r❛ ✸ ✕ ■❧✉str❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❡①♣❧♦r❛çã♦ ❡♠ ✉♠❛ ♣❛rt❡ ❞❛ r❡❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❋✐❣✉r❛ ✹ ✕ Pr♦❝❡ss♦ ❞❡ ❡①♣❧♦r❛çã♦ ❝♦r❞❛❧ ♣❛r❛ n= 104 ♣♦♥t♦s ❞❡ γt✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
❋✐❣✉r❛ ✺ ✕ ❋✉♥çã♦Ut ♦❜t✐❞❛ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞✐s❝r❡t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
❋✐❣✉r❛ ✻ ✕ ●rá✜❝♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡Ut ❝♦♠ ♦ t❡♠♣♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
❙❯▼➪❘■❖
✶ ■◆❚❘❖❉❯➬➹❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ❊◗❯❆➬Õ❊❙ ❉❊ ▲❖❊❲◆❊❘ ❊ ❉❊❋■◆■➬➹❖ ❉❊ ❙▲❊ ✳ ✳ ✳ ✳ ✶✸
✷✳✶ ◆♦çõ❡s ❡♠ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✶ ❘❡❣✐õ❡s ❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✷ ❉♦♠í♥✐♦s s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✸ ▼❛♣❛s ❝♦♥❢♦r♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✸✳✶ Pr❡s❡r✈❛çã♦ ❞♦s â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✸✳✷ ■♥✈❛r✐â♥❝✐❛ ❞❡ ❡s❝❛❧❛ ♣❛r❛ ♣♦♥t♦s ♣ró①✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❊q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r ♥❛ ✈❡rsã♦ ❝♦r❞❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸ ❙▲❊ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ ❙▲❊ ❉■❘❊❚❖ ❊ ❆P▲■❈❆➬➹❖ ❊▼ ❆▲●❯◆❙ ▼❖❉❊▲❖❙ ❉❊
❘❊❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶ ▼ét♦❞♦s ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ♦ ❙▲❊ ❞✐s❝r❡t♦✿ ❆❧❣♦r✐t♠♦ ❩✐♣♣❡r ✶✽ ✸✳✶✳✶ ❖❜t❡♥çã♦ ❞❡ ❯✭t✮ ❝♦♥❤❡❝❡♥❞♦✲s❡ γt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✶✳✷ ❖❜t❡♥çã♦ ❞❡ γt ❝♦♥❤❡❝❡♥❞♦✲s❡ Ut ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✷ ❊st✐♠❛t✐✈❛ ❞❡ ❦ ♣❛r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❞❡ r❡❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷✳✶ P❡r❝♦❧❛çã♦ ❝rít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✹ P❊❘❙P❊❈❚■❱❆❙ P❆❘❆ ❆ ❙❖▲❯➬➹❖ ❉❆ ❊◗❯❆➬➹❖ ❉❊
▲❖❊❲◆❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✶ ❙♦❧✉çã♦ ♣❛r❛ ft ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✶✵
✶ ■◆❚❘❖❉❯➬➹❖
❋❡♥ô♠❡♥♦s ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡♠ ❢ís✐❝❛ sã♦ ❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ❬✷❪✱ ♣♦✐s ♦❝♦rr❡♠ ❛♠♣❧❛♠❡♥t❡ ♥❛ ♥❛t✉r❡③❛✳ ◗✉❛♥❞♦ ❛❧❣✉♠ ♣❛râ♠❡tr♦ ✭t❡♠♣❡r❛t✉r❛✱ ♣♦r ❡①❡♠♣❧♦✮ é ❧❡✈❛❞♦ ❛ ✉♠ ✈❛❧♦r ❝rít✐❝♦✱ ♦ s✐st❡♠❛ s♦❢r❡ ❞rást✐❝❛s ❛❧t❡r❛çõ❡s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s✱ ❞❡✈✐❞♦ ❛ ✉♠❛ ♠✉❞❛♥ç❛ ❣❧♦❜❛❧ ♥❛s ❝❛r❛❝t❡ríst✐❝❛s ♠✐❝r♦s❝ó♣✐❝❛s ❞♦ s✐st❡♠❛✳ ❯♠❛ ❛♥á❧✐s❡ q✉❛♥t✐t❛t✐✈❛ ❞♦ q✉❡ r❡❛❧♠❡♥t❡ ♦❝♦rr❡ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ❡♠ ♠✉✐t♦s ❝❛s♦s ✐♥✈✐á✈❡❧✱ ♣♦✐s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈❛r✐á✈❡✐s ❡ s✉❛s r❡❧❛çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♣♦❞❡♠ s❡r tã♦ ❣r❛♥❞❡s q✉❛♥t♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥st✐t✉✐♥t❡s ❜ás✐❝♦s ❞♦ s✐st❡♠❛✳
❯♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ sã♦ ❛s ♠✉❞❛♥ç❛s ❞❡ ❡st❛❞♦ ❞❛ ♠❛✲ tér✐❛✱ ♦♥❞❡ ❛❧t❡r❛çõ❡s s✐❣♥✐✜❝❛t✐✈❛s ♥❛ ♦r❣❛♥✐③❛çã♦ ♠✐s❝r♦s❝ó♣✐❝❛ ❞♦s ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ♠❛t❡r✐❛❧ sã♦ ♦❜s❡r✈❛❞❛s q✉❛♥❞♦ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❝❛r❛❝t❡ríst✐❝❛ ❛♦ ♠❡s♠♦ é ❛❧❝❛♥ç❛❞❛ ✭♠❛♥t❡♥❞♦ ❛ ♣r❡ssã♦ ♦✉ ♦ ✈♦❧✉♠❡ ❝♦♥st❛♥t❡✮✳ ❆s ❛❧t❡r❛çõ❡s ♣r♦✈♦❝❛♠ ✉♠❛ ♠✉❞❛♥ç❛ q✉❛❧✐t❛t✐✈❛ ♥♦ ♠❛t❡r✐❛❧✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥❛s ♠✉❞❛♥ç❛s ❞❡ ❡st❛❞♦ ❞❛ á❣✉❛ à ✈♦❧✉♠❡ ❝♦♥st❛♥t❡ ✭❋■●❯❘❆ ✶✮✳ ◆❛ ❋✐❣✉r❛ ✶ ♦ ♣♦♥t♦ (Tc, Pc) é ♦ ♣♦♥t♦ ♦♥❞❡ ❛s ❢❛s❡s ❧íq✉✐❞❛ ❡ ❣❛s♦s❛ t♦r♥❛♠✲s❡ ✐♥❞✐st✐♥❣✉í✈❡✐s✿ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ T > Tc ❡ P > Pc✱ ♥ã♦ é
♣♦ssí✈❡❧ r❡❛❧✐③❛r ❛ tr❛♥s✐çã♦ ❧íq✉✐❞♦✲❣❛s♦s♦ ✭♦✉ ✈✐❝❡✲✈❡rs❛✮ ✈❛r✐❛♥❞♦✲s❡ s♦♠❡♥t❡ ♣r❡ssã♦ ♦✉ t❡♠♣❡r❛t✉r❛ ❞♦ ♠❛t❡r✐❛❧✳
❋✐❣✉r❛ ✶ ✕ ❉✐❛❣r❛♠❛ ♣r❡ssã♦✲t❡♠♣❡r❛t✉r❛ ❞❛ á❣✉❛
❋♦♥t❡✿ ❊❧❛❜♦r❛❞♦ ♣❡❧♦ ❛✉t♦r✳ ❆ ❝✉r✈❛ ❞❡ ♣r❡ssã♦ ❞❡ ✈❛♣♦r ❛❝❛❜❛ ❡♠(Tc, Pc)✱ ❡ ❛ ♣❛rt✐r ❞❡st❡ ♣♦♥t♦ ❛s ❢❛s❡s t♦r♥❛♠✲s❡ ✐♥❞✐st✐♥❣✉í✈❡✐s✳
❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ♣r❡s❡♥t❡ ♥❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ✉♠❛ ✈❛r✐á✈❡❧ ❞❡♥♦♠✐♥❛❞❛ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✿ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♦❜s❡r✈á✈❡❧ q✉❡ ❞❡❝r❡s❝❡ ❝♦♠ |Tc −T|β ❡♠ ✉♠❛
✶✶
◆♦ ❡①❡♠♣❧♦ ❞❛❞♦ ❛❝✐♠❛✱ ♥❛ tr❛♥s✐çã♦ ❧íq✉✐❞♦✲❣ás ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ é ❛ ❞✐❢❡r❡♥ç❛
ρl −ρg ❡♥tr❡ ❛s ❞❡♥s✐❞❛❞❡s ❞❛ á❣✉❛ ♥♦ ❡st❛❞♦ ❧íq✉✐❞♦ ❡ ❣❛s♦s♦✱ q✉❡ ✈❛✐ ❛ ③❡r♦ ❛❝✐♠❛
❞❛ r❡❣✐ã♦ ❝rít✐❝❛✳ ❖✉tr❛s q✉❛♥t✐❞❛❞❡s ✭❝♦♠♦ ❝❛♣❛❝✐❞❛❞❡ tér♠✐❝❛ ♣♦r ❡①❡♠♣❧♦✮ ♣♦❞❡♠ ♦❜❡❞❡❝❡r t❛♠❜é♠ ❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛s ❞❛ ❢♦r♠❛|Tc−T|−ν✱ ♦♥❞❡
ν é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛❀
♦✉ s❡❥❛✱ ♣❛râ♠❡tr♦s q✉❡ ✈ã♦ ♣r❛ ✐♥✜♥✐t♦ ❛❝✐♠❛ ❞❛ r❡❣✐ã♦ ❝rít✐❝❛✳ ❖s ❡①♣♦❡♥t❡s β✱ ν ❡
♦✉tr♦s q✉❡ ♦ s✐st❡♠❛ ♣♦❞❡ t❡r✱ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛❧❣✉♠ ♣❛râ♠❡tr♦ q✉❡ ✈❛r✐❛ ♥❛ r❡❣✐ã♦ ❝rít✐❝❛ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♣❡❧❛ ✈❛r✐❛çã♦ ❞❛ t❡♠♣❡r❛t✉r❛❀ sã♦ ❞❡♥♦♠✐♥❛❞♦s ❡①♣♦❡♥t❡s ❞❡ ♣♦♥t♦ ❝rít✐❝♦✳
❆♦ ❧♦♥❣♦ ❞♦s ❡①♣❡r✐♠❡♥t♦s ♠✉✐t♦s s✐st❡♠❛s ❞❡ ❞✐❢❡r❡♥t❡s ♠❛t❡r✐❛✐s ❛♣r❡s❡♥✲ t❛r❛♠ ♦s ♠❡s♠♦s ❡①♣♦❡♥t❡s ❞❡ ♣♦♥t♦ ❝rít✐❝♦✳ ❊ss❡ ♣❛❞rã♦ r❡✈❡❧❛✈❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ✉♥✐✈❡rs❛❧✐❞❛❞❡ ❬✸❪ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ♠❛t❡r✐❛✐s ♥❛ ❢❛s❡ ❝rít✐❝❛ ❞♦ s✐st❡♠❛✳ P♦st❡r✐♦r✲ ♠❡♥t❡ ♦✉tr♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ❢♦r❛♠ ❞❡s❝♦❜❡rt♦s ❡ ♦s s✐st❡♠❛s ♣❛ss❛r❛♠ ❛ s❡r ❛❣r✉♣❛❞♦s ❡♠ ❝❧❛ss❡s ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡✳ ❉♦✐s s✐st❡♠❛s ❝♦♠ ♦s ♠❡s♠♦s ✈❛❧♦r❡s ❞❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♣❡rt❡♥❝❡♠ à ♠❡s♠❛ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡✳
P♦❞❡✲s❡ ✐♥❢❡r✐r ❛ ♣❛rt✐r ❞❛ ✐❞❡✐❛ ❞❛ ✉♥✐✈❡rs❛❧✐❞❛❞❡ q✉❡ s♦♠❡♥t❡ ✉♠ r❡❞✉③✐❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛râ♠❡tr♦s ♠✐❝r♦s❝ó♣✐❝♦s sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡s❝r❡✈❡r ✉♠ s✐st❡♠❛ q✉❛♥❞♦ ♣ró①✐♠♦ ❛ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✳ P❛rt✐❝✉❧❛r✐❞❛❞❡s ♠✐❝r♦s❝ó♣✐❝❛s ♣♦❞❡♠ s❡r ❝♦♠♣❧❡t❛✲ ♠❡♥t❡ ✐❣♥♦r❛❞❛s✳ ▼♦❞❡❧♦s s✐♠♣❧❡s ♣♦❞❡♠ s❡r ❝r✐❛❞♦s ♣❛r❛ ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ❞❡ t❛✐s s✐st❡♠❛s✳ ▼✉✐t♦s ❞❡❧❡s ♣♦❞❡♠ s❡r ❞❡s❝r✐t♦s ✉t✐❧✐③❛♥❞♦✲s❡ ♦s ♠♦❞❡❧♦s ❞❡ r❡❞❡ ❬✾❪✭❧❛tt✐❝❡ ♠♦❞❡❧s✮✳ ▼♦❞❡❧♦ ❞❡ r❡❞❡ é ✉♠❛ ✈❡rsã♦ ❞✐s❝r❡t❛ ❞♦ ♣❧❛♥♦ ❡♠ q✉❡ ♦s ✈ért✐✲ ❝❡s ✭❛r❡st❛s✮ r❡♣r❡s❡♥t❛♠ ♣♦♥t♦s ❞❡ss❡ ♣❧❛♥♦❀ ❡ ❛s ❛r❡st❛s ✭✈ért✐❝❡s✮ sã♦ ❛s ❝♦♥❡①õ❡s ❡♥tr❡ ♣♦♥t♦s ✈✐③✐♥❤♦s ❞♦ s✐st❡♠❛✳ ❆s ❝♦♥❡①õ❡s sã♦ ❢❡✐t❛s ❡♠ ❢✉♥çã♦ ❞❡ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✐♥t❡r❡ss❡ ♥❛ ❛♥á❧✐s❡✱ ❡ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞♦ s✐t❡♠❛ tê♠ ❡♠ ❝♦♠✉♠ ❝❛s♦ ❡st❡❥❛♠ ❝♦♥❡❝✲ t❛❞♦s✳ ▼♦❞❡❧♦s ❞❡ ♣❡r❝♦❧❛çã♦❀ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣❀ ❡ ♠♦❞❡❧♦ ❞❡ P♦tts ❞❡ q ❡st❛❞♦s ✭q✲st❛t❡ P♦tts ♠♦❞❡❧✮ sã♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♠♦❞❡❧♦s ❞❡ r❡❞❡✳
❆❝r❡✐t❛✲s❡ q✉❡ ❛❧❣✉♥s s✐st❡♠❛s ❢ís✐❝♦s ✭♦s ❞❡s❝r✐t♦s ♣❡❧♦s ♠♦❞❡❧♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❡stã♦ ♥❡ss❛ ❧✐st❛✮❛♣r❡s❡♥t❛♠ ✐♥✈❛r✐â♥❝✐❛ ❝♦♥❢♦r♠❡ ♥❛ r❡❣✐ã♦ ❝rít✐❝❛✱ ♥♦ ❧✐♠✐t❡ ❡♠ q✉❡ ♦ ❡s♣❛ç❛♠❡♥t♦ ❞❛ r❡❞❡ t❡♥❞❡ ❛ ③❡r♦ ❬✶✶❪✳ ◆♦ s❡♥t✐❞♦ ❞❡ ❡❧✉❝✐❞❛r ❡ss❛ q✉❡stã♦✱ ❙❝❤r❛♠♠ ❬✶✷❪ ❝♦♠❜✐♥♦✉ ✉♠ ♣r♦❝❡ss♦ ❞✐♥â♠✐❝♦ ❝❤❛♠❛❞♦ ❡✈♦❧✉çã♦ ❞❡ ▲♦❡✇♥❡r ❬✶✸❪ ❝♦♠ ♦ ❝á❧❝✉❧♦ ❡s✲ t♦❝ást✐❝♦✳ ❆ss✐♠✱ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ❛❧❡❛tór✐❛s ❞❡♥♦♠✐♥❛❞❛s ❙▲❊ ✭❙❝❤r❛♠♠✲▲♦❡✇♥❡r ❡✈♦❧✉t✐♦♥ ♦✉ ❙t♦❝❤❛st✐❝ ▲♦❡✇♥❡r ❡✈♦❧✉t✐♦♥✮ ❝♦♠ ♣❛râ♠❡tr♦ κ✱ ♦✉ SLEκ ❢♦r❛♠ ❞❡✜♥✐❞❛s✳
✶✷
r❛♥❞♦♠ ✇❛❧❦✮✱ ❢♦✐ ♠♦str❛❞♦ q✉❡ ❛ ❝✉r✈❛ r❡s✉❧t❛♥t❡ é SLE2 ❬✶✺❪❀ q✉❛♥❞♦ s✐♠♣❧❡s♠❡♥t❡
♣r♦✐❜✐♠♦s q✉❡ ♦ ♣❡r❝✉rs♦ t❡♥❤❛ ❝✐❝❧♦s ✭s❡❧❢✲❛✈♦✐❞✐♥❣ ✇❛❧❦✮✱ ❛ ❝✉r✈❛ ❣❡r❛❞❛ ♣❛r❡❝❡ s❡r
SLE8/3 ❬✶✻❪✳ ❆❧❣✉♥s ♦✉tr♦s ♠♦❞❡❧♦s q✉❡ ❝♦♠♣r♦✈❛❞❛♠❡♥t❡ ❣❡r❛♠ ❝✉r✈❛s ❙▲❊ ♥♦ ❧✐♠✐t❡
❝♦♥tí♥✉♦ sã♦✿ ♣❡r❝♦❧❛çã♦ ❝rít✐❝❛ ❬✶✼❪ ❡ ár✈♦r❡s ❞❡ ❡①t❡♥sã♦ ✉♥✐❢♦r♠❡ ✭✉♥✐❢♦r♠ s♣❛♥♥✐♥❣ tr❡❡s✮❬✶✺❪✳
❖ ❝♦r♣♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❞✐✈✐❞✐❞♦ ❡♠ ✹ ❝❛♣ít✉❧♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡✲ ✜♥✐r❡♠♦s ❙▲❊ ♣❛rt✐♥❞♦ ❞❛ ❡✈♦❧✉çã♦ ❞❡ ▲♦❡✇♥❡r✳ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❙▲❊ s❡rã♦ ❞✐s❝✉t✐❞❛s✳ ◆ã♦ ❢♦❝❛r❡♠♦s ♥♦s ❞❡t❛❧❤❡s ❞❡ ❝♦♠♦ sã♦ ♦❜t✐❞❛s ❛s ❡q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r✱ ♦♥❞❡ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ ❡♠ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛ ❡ ❝á❧❝✉❧♦ ❡st♦❝ást✐❝♦ sã♦ ♥❡❝❡s✲ sár✐♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❛❧❣✉♥s ♠♦❞❡❧♦s ❞❡ r❡❞❡ ❡ s✉❛s ❝♦♥❡①õ❡s ❝♦♠ ❙▲❊✳ ◆♦ ❝❛♣ít✉❧♦ ✸ r❡s♦❧✈❡r❡♠♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ❞❡ ▲♦❡✇♥❡r ♣♦r ✉♠ ♠ét♦❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❙▲❊ ❞✐s❝r❡t♦✱ ♦♥❞❡ ✉s❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❝♦♥str✉✐r ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ q✉❡ ♦❜té♠ ❛ ❢✉♥çã♦ Ut✳ ❯s❛r❡♠♦s ❡ss❡ ♠ét♦❞♦ ♣❛r❛ ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ κ ♣❛r❛ ♦ ♠♦✲
❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❝rít✐❝❛ ❡ ✈❡r✐✜❝❛r s✉❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦SLE6✳ ◆♦ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦
✶✸
✷ ❊◗❯❆➬Õ❊❙ ❉❊ ▲❖❊❲◆❊❘ ❊ ❉❊❋■◆■➬➹❖ ❉❊ ❙▲❊
✷✳✶ ◆♦çõ❡s ❡♠ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s
❆❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❞❡✜♥✐çõ❡s s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛s s❡çõ❡s s❡❣✉✐♥t❡s✱ ❡♥tã♦ r❡s✉♠✐♠♦s ❛s q✉❡stõ❡s ♠❛✐s ♣❡rt✐♥❡♥t❡s✳ ❯s❛♠♦s ❬✶❪ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❡ss❛ s❡çã♦✳ ✷✳✶✳✶ ❘❡❣✐õ❡s ❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦
❉❛❞♦ ✉♠ ♣♦♥t♦ z0 ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ ❞❡✜♥❡✲s❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ǫ ❞❡z0 ❝♦♠♦
t♦❞♦s ♦s ♣♦♥t♦s ③ ✐♥t❡r♥♦s ❛♦ ❝ír❝✉❧♦ ❝❡♥tr❛❞♦ ❡♠z0 ❡ ❞❡ r❛✐♦ ǫ✳ ❖✉ s❡❥❛✱ t♦❞♦s ♦s ♣♦♥t♦s
③ t❛❧ q✉❡
|z−z0|< ǫ. ✭✷✳✶✮
◗✉❛♥❞♦ ǫ é ✐rr❡❧❡✈❛♥t❡ ♣❛r❛ ❛ ❞✐s❝✉ssã♦✱ ❝❤❛♠❛♠♦s ✷✳✶ ❛♣❡♥❛s ❞❡ ✈✐③✐♥❤❛♥ç❛✳
❯♠ ♣♦♥t♦ z0 é ❞✐t♦ ✐♠ ♣♦♥t♦ ✐♥t❡r✐♦r ❛ ✉♠ ❝♦♥❥✉♥t♦ S ⊂ C s❡ ❡①✐st❡ ❛❧❣✉♠❛
✈✐③✐♥❤❛♥ç❛ ❞❡ z0 s♦♠❡♥t❡ ❝♦♠ ♣♦♥t♦s z ∈C✳ P♦r ♦✉tr♦ ❧❛❞♦✱ z0 é ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r ❛ ❙
s❡ ❡①✐st❡ ❛❧❣✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ z0 s♦♠❡♥t❡ ❝♦♠ ♣♦♥t♦s ③ ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❙✳ ❙❡z0 ♥ã♦
é ✐♥t❡r✐♦r ♥❡♠ ❡①t❡r✐♦r ❛ ❙✱ ❡♥tã♦ z0 é ✉♠ ♣♦♥t♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❙✳ ❚♦❞♦s ♦s ♣♦♥t♦s ❞❡
❢r♦♥t❡✐r❛ ❞❡ ❙ ❥✉♥t♦s ❢♦r♠❛♠ ❛ ❢r♦♥t❡✐r❛ ❞❡ ❙✳
❯♠ ❝♦♥❥✉♥t♦ é ❞✐t♦ ❛❜❡rt♦ s❡ ♥ã♦ ❝♦♥té♠ s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❝♦♥❥✉♥t♦ é ❞✐t♦ ❢❡❝❤❛❞♦ s❡ ♥ã♦ ❝♦♥té♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❙ é ❞✐t♦ ❝♦♥❡❝t❛❞♦ s❡ ♣♦❞❡♠♦s ❧✐❣❛r q✉❛✐sq✉❡r ❞♦✐s ♣♦♥t♦s z1, z2 ∈ S
♣♦r ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ s❡❣♠❡♥t♦s ❞❡ ❧✐♥❤❛ q✉❡ ✜❝❛♠ ✐♥t❡✐r❛♠❡♥t❡ ❡♠ ❙✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡ ❝♦♥❡❝t❛❞♦ é ❝❤❛♠❛❞♦ ❞♦♠í♥✐♦ ✭q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ é✱ ♣♦rt❛♥t♦✱ ✉♠ ❞♦♠í♥✐♦✮✳ ❯♠ ❞♦♠í♥✐♦ ❝♦♠ ❛❧❣✉♠ ❞❡ s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛ é ❞❡♥♦♠✐♥❛❞♦ r❡❣✐ã♦✳
✷✳✶✳✷ ❉♦♠í♥✐♦s s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ❯♠❛ ❢✉♥çã♦
z(t) =x(t) +iy(t) (a ≤t ≤b), ✭✷✳✷✮
♦♥❞❡ ①✱② sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞♦ ♣❛râ♠❡tr♦ r❡❛❧ t❀ ❢♦r♠❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❝❤❛♠❛❞♦ ❝✉r✈❛✳ ❆ ❝✉r✈❛ é ❞✐t❛ s✐♠♣❧❡s s❡ ♥ã♦ t♦❝❛ ❡♠ s✐ ♠❡s♠❛ ❡♠ ♥❡♥❤✉♠ ♣♦♥t♦ ✭z(t1)6=z(t2)
q✉❛♥❞♦t1 6=t2✮✳ ❯♠ ❛r❝♦ ❞❡ ❏♦r❞❛♥ ♦✉ ❝✉r✈❛ ❢❡❝❤❛❞❛ s✐♠♣❧❡s é ✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s ❡①❝❡t♦
♣❛r❛z(a) = z(b)✳ ❙❡z(t)t❡♠ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❝♦♥tí♥✉❛✱ ❛ ❝✉r✈❛ é ❞✐t❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❙❡ ✉♠❛ ❝✉r✈❛ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡z′
✶✹
♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ✈❡t♦r ♥❛ ❢♦r♠❛z =a+ib = (a, b)✮
❚= 1 |z|(Re(z
′
), Im(z′
)) ✭✷✳✸✮
é ❜❡♠ ❞❡✜♥✐❞♦ ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛✱ ❝♦♠ ✐♥❝❧✐♥❛çã♦argz′
(t)❀ ♥❡ss❡ ❝❛s♦ ❛ ❝✉r✈❛ é ❞✐t❛ s✉❛✈❡✳
❯♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❝✉r✈❛s s✉❛✈❡s γ1[a1, b1], γ2[a2, b2], ..., γn[an, bn] ❥✉st❛♣♦s✲
t❛s ❞❡ ❢♦r♠❛ q✉❡ γ1(b1) = γ2(a2), γ2(b2) = γ3(a3)..., γn−1(bn−1) = γn(an) é ❞❡♥♦♠✐♥❛❞♦ ❝❛♠✐♥❤♦✳ ❙❡γ(a1) =γ(bn)✱ ❡♥tã♦ ♦ ❝❛♠✐♥❤♦ é ❢❡❝❤❛❞♦ s✐♠♣❧❡s✳
❯♠ ❞♦♠í♥✐♦ ❉ é ❞✐t♦ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦ s❡ q✉❛❧q✉❡r ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ s✐♠♣❧❡s ❞❡♥tr♦ ❞❡❧❡ só ❝♦♥t❡♥❤❛ ♣♦♥t♦s ❞❡ ❉✳ ■♥❢♦r♠❛❧♠❡♥t❡ ♣♦❞❡✲s❡ ❞✐③❡r q✉❡ é ✉♠ ❞♦♠í♥✐♦ s❡♠ ✧❜✉r❛❝♦s✧✳
✷✳✶✳✸ ▼❛♣❛s ❝♦♥❢♦r♠❡s ✷✳✶✳✸✳✶ Pr❡s❡r✈❛çã♦ ❞♦s â♥❣✉❧♦s
❯♠ ♠❛♣❛ ❝♦♥❢♦r♠❡ é ✉♠❛ ❢✉♥çã♦ ❢ q✉❡ ♠❛♣❡✐❛ ♣♦♥t♦ ❛ ♣♦♥t♦ ✉♠ ❞♦♠í♥✐♦ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦ D ⊂ C ❡♠ ♦✉tr♦ ❞♦♠í♥✐♦ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦ D′
⊂ C❀ ♣r❡s❡r✈❛♥❞♦ ♦s â♥❣✉❧♦s r❡❧❛t✐✈♦s ❡♥tr❡ q✉❛✐sq✉❡r ❝✉r✈❛s q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ❛❧❣✉♠ ♣♦♥t♦ ❞♦ ❞♦♠í♥✐♦✳ ❊ss❛ é ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❞✐③❡r q✉❡ ❛ ❢✉♥çã♦ ❢ ❞❡✈❡ s❡r ❛♥❛❧ít✐❝❛ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡ ❉ ❡ f′
(z) 6= 0, ∀z ∈ D✳ ❱❡❥❛♠♦s ❝♦♠♦ ✐ss♦ é ✈❡r❞❛❞❡ ❛♣❧✐❝❛♥❞♦ ❢
❡♠ ❞✉❛s ❝✉r✈❛s s✉❛✈❡s C1 ✭z1 =z1(t), a1 ≤ t ≤ b1✮ ❡ C2 ✭z2 = z2(t), a2 ≤ t ≤ b2✮ q✉❡ s❡
✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ♣♦♥t♦ z0 = z1(t0) = z2(t0)✳ ❚❡♠♦s q✉❡ ♦ â♥❣✉❧♦ ❞♦ ✈❡t♦r t❛♥❣❡♥t❡
à ❝✉r✈❛ ✐♠❛❣❡♠ ❞❡ C1 ♥♦ ♣♦♥t♦ ω0 = f(z0) é φ1 = argω1 = argf′[z1(t0)] +argz1′(t0) =
argf′
[z0] +θ1❀ ❡ ♣❛r❛ ❛ ❝✉r✈❛ ✐♠❛❣❡♠ ❞❡C2 é φ2 =argω1 =argf′[z2(t0)] +argz2′(t0) =
argf′
[z0] +θ2✳ ◆♦s ❞♦✐s ❝❛s♦s ❛ ❧✐♥❤❛ t❛♥❣❡♥t❡ às ❝✉r✈❛s C1 ❡ C2 ♥♦ ♣♦♥t♦z0 s♦❢r❡♠ ✉♠❛
r♦t❛çã♦ ♣♦r ✉♠ â♥❣✉❧♦argf′
(z0)✳ ❏✉♥t❛♥❞♦ ♦s ❞♦✐s r❡s✉❧t❛❞♦s✱
φ2−φ1 =θ2 −θ1, ✭✷✳✹✮
❡ ❛s ❞✐❢❡r❡♥ç❛s ❞❡ â♥❣✉❧♦s sã♦ ♣r❡s❡r✈❛❞❛s✳ ❈♦♠♦ ❢ é ❛♥❛❧ít✐❝❛✱ ❡ss❡ t✐♣♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡ é ✈❡r✐✜❝❛❞❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ z0✱ ♦♥❞❡ f′(z)≈f′(z0)✳ ❆ ❢✉♥çã♦
❢ ♣♦rt❛♥t♦ ♣r❡s❡r✈❛ ♦s ❛♥❣✉❧♦s r❡❧❛t✐✈♦s s❡♠♣r❡ q✉❡ ❢♦r ❛♥❛❧ít✐❝❛ ❡ f′
(z) 6= 0, ∀z ∈ D✱
❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳
✷✳✶✳✸✳✷ ■♥✈❛r✐â♥❝✐❛ ❞❡ ❡s❝❛❧❛ ♣❛r❛ ♣♦♥t♦s ♣ró①✐♠♦s
❉❛❞♦ ✉♠ ♣♦♥t♦ z0 ♥♦ ❞♦♠í♥✐♦ ❉✱ s✉❛ ❞✐stâ♥❝✐❛ ❛ q✉❛❧q✉❡r ♣♦♥t♦ z t❛♠❜é♠
✶✺
|f(z)−f(z0)|✳ P♦rt❛♥t♦ ♦ ❢❛t♦r ❞❡ ❡s❝❛❧❛ é ❛ ❢r❛çã♦
|f(z)−f(z0)|
|z−z0|
. ✭✷✳✺✮
❉❡✈✐❞♦ ❛♥❛❧✐t✐❝✐❞❛❞❡ ❞❛ ❢✉♥çã♦f✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡z0 ♦♥❞❡f′(z0)é ✉♠❛ ❜♦❛ ❛♣r♦✲
①✐♠❛çã♦ ♣❛r❛ ✷✳✺✳ ❊ss❡ r❡s✉❧t❛❞♦ ❣❛r❛♥t❡ q✉❡ ♠❛♣❛s ❝♦♥❢♦r♠❛✐s ❛♣r❡s❡♥t❛♠ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ❡s❝❛❧❛ ❧♦❝❛❧♠❡♥t❡✳
❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ❡ ♦ ❞❛ s❡çã♦ ✭✷✳✶✳✸✳✶✮ ♥♦s ♠♦str❛ ❛ ❧✐❣❛çã♦ ❡♥tr❡ ✐♥✈❛r✐â♥❝✐❛ ❝♦♥❢♦r♠❡ ❡ ♣r❡s❡r✈❛çã♦ ❞❛ s✐♠❡tr✐❛ ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞❡ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ❞♦♠í♥✐♦✳ ▼❡s♠♦ q✉❡ ❛ ❢✉♥çã♦ f tr❛♥s❢♦r♠❡ ♦ ❞♦♠í♥✐♦ ❉ ❞rást✐❝❛♠❡♥t❡ ❡♠ ❧❛r❣❛ ❡s❝❛❧❛✱ r❡❣✐õ❡s
♣❡q✉❡♥❛s ❞♦ ❞♦♠í♥✐♦ tê♠ s✉❛s ❢♦r♠❛s ♣r❡s❡r✈❛❞❛s q✉❛♥❞♦ ♦❜s❡r✈❛❞❛s ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❛ ♠❡s♠❛✳
❙❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ q✉❡ ❛♣r❡s❡♥t❛ ✐♥✈❛r✐â♥❝✐❛ ❝♦♥❢♦r♠❡ ♥❛ r❡❣✐ã♦ ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱
✷✳✷ ❊q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r ♥❛ ✈❡rsã♦ ❝♦r❞❛❧
❚♦❞❛s ❛s ❛✜r♠❛çõ❡s ❛❞✐❛♥t❡ ♥ã♦ s❡rã♦ ♣r♦✈❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦❀ ❢♦r❛♠ ❛♣❡♥❛s ❡①tr❛í❞❛s ❞❡ ♠❛t❡r✐❛✐s s♦❜r❡ ♦ ❛ss✉♥t♦ ✭❬✶✸❪ ❡ ❬✽❪✮✱ ❞❡ ♠♦❞♦ q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r ♣✉❞❡ss❡♠ s❡r ♦❜t✐❞❛s✱ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ft ❡ gt q✉❡ s❛t✐s❢❛③❡♠ ❡ss❛s
❡q✉❛çõ❡s ♣✉❞❡ss❡♠ s❡r ❡①♣❧♦r❛❞❛s ♠❛✐s ❛❞✐❛♥t❡✱ ♦♥❞❡ r❡s♦❧✈❡♠♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ♦SLEκ✳
❙❡❥❛γ(t) ✭t ≥0✮ ✉♠❛ ❝✉r✈❛ ♥♦ s❡♠✐♣❧❛♥♦ s✉♣❡r✐♦r ❝♦♠♣❧❡①♦ ✭H✮✱ ♣❛rt✐♥❞♦ ❞❛ ♦r✐❣❡♠ ❡ q✉❡ ♥ã♦ t♦❝❛ ❛ s✐ ♠❡s♠❛ ✭❡ss❡ t✐♣♦ ❞❡ ❝✉r✈❛ é ♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ♦ t❡r♠♦ ❝♦r❞❛❧ ✉s❛❞♦ ❞✉r❛♥t❡ ♦ tr❛❜❛❧❤♦✮✳ ❙♦❜ ❡ss❛s ❝♦♥❞✐çõ❡s✱ H\γ[0, t] é ✉♠ ❞♦♠í♥✐♦ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡❝t❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ γ(t) → ∞ q✉❛♥❞♦ t → ∞❀ ❡①✐st❡ ✉♠ ♠❛♣❛ ❝♦♥❢♦r♠❡ gt q✉❡
tr❛♥s❢♦r♠❛ H\γ[0, t] ❡♠ H t❛❧ q✉❡ gt(z) → z q✉❛♥❞♦ z → ∞✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❡①♣❛♥❞✐r gt ❡♠ ✉♠❛ sér✐❡ ❞❡ ▲❛✉r❡♥t✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥✉❧♦s ♣❛r❛ ♦s t❡r♠♦s ❞❡ ♦r❞❡♠
♠❛✐♦r q✉❡z2✱
gt(z) =z+C(t)
z +O(
1
|z|2), z → ∞. ✭✷✳✻✮
❈✭t✮ é ❝❤❛♠❛❞❛ ❞❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ γ[0, t] ♥♦ s❡♠✐♣❧❛♥♦ ✭❤❛❧❢✲♣❧❛♥❡ ❝❛♣❛❝✐t②✮✱ ❡ é ❡str✐t❛✲ ♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ♣♦❞❡♠♦s r❡♣❛r❛♠❡tr✐③❛r ❛ ❝✉r✈❛ γ(t) ♣❛r❛ q✉❡ ❈✭t✮❂✷t✳ ❊♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦✱g(t) s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ▲♦❡✇♥❡r✿
∂gt
∂t =
2
gt(z)−Ut; g0(z) =z. ✭✷✳✼✮
❖♥❞❡ Ut ∈R é ❛ ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❞✐r❡t♦r❛ ✭❞r✐✈✐♥❣ ❢✉♥❝t✐♦♥✮✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ Ut k =
✶✻
❣✱ ❡ s✐♠ ❞❛ ❢r♦♥t❡✐r❛ ❞♦ s✐st❡♠❛✮✳ ■♥✈❡rs❛♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❢✉♥çã♦ ❞✐r❡t♦r❛ ❝♦♥❤❡❝✐❞❛✱ ♣♦❞❡♠♦s ♦❜t❡r ♦s ♣♦♥t♦s q✉❡ ❢♦r♠❛♠ γ(t) ❡♠ ✉♠ ♠❛♣❛ q✉❡ tr❛♥s❢♦r♠❛ ♦ ❞♦♠í♥✐♦H ♥♦ ❝♦♥tr❛❞♦♠í♥✐♦H\γ[0, t]✿
∂f ∂t =
2
Ut−z ∂f
∂z; f0(z) =z. ✭✷✳✽✮
❊ss❛s sã♦ ❛s ❞✉❛s ✈❡rsõ❡s ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ▲♦❡✇♥❡r✳
◆❛ ❋✐❣✉r❛ ✷ ✉s❛♠♦s ♦ ♠ét♦❞♦ ❞✐s❝r❡t♦ ✭❞❡s❝r✐t♦ ♠❛✐s ❛❞✐❛♥t❡✮ ♣❛r❛ ✐❧✉str❛r ❝♦♠♦ ❛ ❢✉♥çã♦ gt r❡t✐r❛ ❛ ❝✉r✈❛ γt ❞❛ ❢r♦♥t❡✐r❛ ❞♦ ❞♦♠í♥✐♦ H\γ[0, t]✳ ❆ ❝✉r✈❛ ✉s❛❞❛ é ✉♠ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ ❣❡r❛❞♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❛r❛ ❛ s✐♠✉❧❛çã♦✳ ❖ ♠ét♦❞♦ ❞❡ ❣❡r❛çã♦ ✉s❛❞♦ é ♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❛❧❡❛tór✐♦ ❞♦ ♣♦♥t♦ ♠é❞✐♦ ❬✶✹❪✱ q✉❡ ❣❛r❛♥t❡ q✉❡ ❛ ❝✉r✈❛ ♥ã♦ t♦❝❛ ❡♠ s✐ ♠❡s♠❛✳ ❖ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ ❢♦✐ ❣❡r❛❞♦ ❝♦♠ ✶✵✷✺ ♣♦♥t♦s s❡♣❛r❛❞♦s ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛∆x= 2−10
❡♥tr❡ s✐✳ ❆ ❝✉r✈❛ é ♠♦str❛❞❛ ♣❛r❛ ❛❧❣✉♥s ✐♥st❛♥t❡s ❡s♣❡❝í✜❝♦s✱ ❛♣ós ♦ ♠❛♣❛ gt ❛❧❝❛♥ç❛r ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s✳
❋✐❣✉r❛ ✷ ✕ ❆♣❧✐❝❛çã♦ ❞❡gt ❡♠ ✉♠ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ s✐♠♣❧❡s ♣❛r❛ ❛❧❣✉♥s ✐♥st❛♥t❡s
❞❡ t❡♠♣♦
✶✼
✷✳✸ ❙▲❊ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s
❙▲❊ é ✉♠ ♣r♦❝❡ss♦ q✉❡ ✉t✐❧✐③❛ ❛s ❡q✉❛çõ❡s ❞❡ ▲♦❡✇♥❡r t♦♠❛♥❞♦ Ut = √kBt✱
♦♥❞❡ ❦ é ✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦ ❡Bté ✉♠ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ ❝♦♠hBti= 0❡ ✈❛r✐â♥❝✐❛ h|Bt2−Bt1|
2i=|t
2−t1|✳ ❊ss❡ ♠♦❞❡❧♦ é ❞❡♥♦♠✐♥❛❞♦SLEκ ❝♦r❞❛❧✳ ❆ ❡q✉❛çã♦ ❞❡ ▲♦❡✇♥❡r
♣❛r❛ ❡ss❡ ♠♦❞❡❧♦ ❡♥tã♦ ✜❝❛
∂gt
∂t =
2
gt(z)−
√
kBt
; g0(z) =z. ✭✷✳✾✮
❆ ❝✉r✈❛ γt ❛ss♦❝✐❛✲s❡ ❡♥tã♦ ❝♦♠ ♦ ♣❛râ♠❡tr♦ ❦ ♣❡❧❛ r❡❧❛çã♦
√
kBt=gtk(γ(tk)). ✭✷✳✶✵✮
❖ ✈❛❧♦r ❞❡ ❦ ✐♥✢✉✐ ♥❛t✉r❛❧♠❡♥t❡ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡γt✳ ❖❜s❡r✈❛✲s❡ três ❢❛s❡s ❞✐st✐♥t❛s
❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ss❛ ❝✉r✈❛✳ P❛r❛k ∈[0,4]✱ γ é ✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s ♥❛ ❛♠♣❧❛ ♠❛✐♦r✐❛
❞❛s s✐t✉❛çõ❡s✳ ❚❛♠❜é♠ é ✐♠♣r♦✈á✈❡❧ q✉❡ ❛ ♠❡s♠❛ t♦q✉❡ ♦ ❡✐①♦ ❞♦s r❡❛✐s ❡♠t >0✳ P❛r❛ k ∈(4,8]✱ γ ♥ã♦ é ♠❛✐s s✐♠♣❧❡s ❡ ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦ ❞♦s r❡❛✐s✱ ♠❛s ♥ã♦ ♣r❡❡♥❝❤❡ ♦ ❡s♣❛ç♦
✭df <2✮✳ P❛r❛k >8✱ ❛ ❝✉r✈❛ ❛❧é♠ ❞❡ ♥ã♦ s❡r s✐♠♣❧❡s ❡ t♦❝❛r ♦ ❡✐①♦ ❞♦s r❡❛✐s s❡ ❝♦♠♣♦rt❛ ❞❡ ❢♦r♠❛ ❛ ♣r❡❡♥❝❤❡r ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ✭df = 2✮✳ ❋♦✐ ♣r♦✈❛❞❛ ✉♠❛ r❡❧❛çã♦ ❬✶✵❪ ❡♥tr❡ ❛
❞✐♠❡♥sã♦ ❢r❛❝t❛❧ ❞❛ ❝✉r✈❛γ ❡ ♦ ♣❛râ♠❡tr♦ ❦✱ q✉❡ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛
❝✉r✈❛ SLEκ ❞❡st❛❝❛❞♦ ❛❝✐♠❛✿
df =min
2,1 + k 8
. ✭✷✳✶✶✮
P♦❞❡♠♦s ❝♦♠♣❛r❛r ❛s ❝✉r✈❛s ❣❡r❛❞❛s ♣❡❧♦s ♠♦❞❡❧♦s ❞❡ r❡❞❡ ♥❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ♦♥❞❡ s❛❜❡✲s❡ ♦ ✈❛❧♦r ❞❛ ❞✐♠❡♥sã♦ ❢r❛❝t❛❧✱ ❝♦♠ ❛ ❝✉r✈❛SLEκ❡①tr❛í❞❛ ❞❛ r❡❧❛çã♦ ✷✳✶✶✳
❍á ❛❧❣✉♠❛s ❢♦r♠❛s ❞❡ ❢❛③❡r ❡ss❡ ❡st✉❞♦✳ ▼♦str❛r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✷ ♦ ♠ét♦❞♦ ❞✐r❡t♦✱ ♦♥❞❡ r❡s♦❧✈❡r❡♠♦s ❞✐r❡t❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ❞❡ ▲♦❡✇♥❡r ✉s❛♥❞♦ ✉♠ ♠ét♦❞♦ ❞❡ ❞✐s❝r❡t✐③❛çã♦ ❞❛ ❙▲❊✱ ❞❡ ♠♦❞♦ ❛ ♦❜t❡r ❛ ❞r✐✈✐♥❣ ❢✉♥❝t✐♦♥Ut❀ ❡ s❛❜❡♥❞♦ q✉❡
h|Ut2 −Ut1|
2
i=k|t2−t1|, ✭✷✳✶✷✮
♣♦❞❡♠♦s ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ ❦ ❛♥❛❧✐s❛♥❞♦ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡U ❡♠ r❡❧❛çã♦
✶✽
✸ ❙▲❊ ❉■❘❊❚❖ ❊ ❆P▲■❈❆➬➹❖ ❊▼ ❆▲●❯◆❙ ▼❖❉❊▲❖❙ ❉❊ ❘❊❉❊
✸✳✶ ▼ét♦❞♦s ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ♦ ❙▲❊ ❞✐s❝r❡t♦✿ ❆❧❣♦r✐t♠♦ ❩✐♣♣❡r ✸✳✶✳✶ ❖❜t❡♥çã♦ ❞❡ ❯✭t✮ ❝♦♥❤❡❝❡♥❞♦✲s❡ γt
❙❡❥❛gt ♦ ♠❛♣❡❛♠❡♥t♦ ❥á ♠❡♥❝✐♦♥❛❞♦ ✭H\γ[0, t]→H✮ ❝♦♠ ❈✭t✮❂✷t✿
gt(z) =z+
2t z +O(
1
z2), ✭✸✳✶✮
❡ ❛ ❡q✉❛çã♦ ❞❡ ▲♦❡✇♥❡r ✷✳✼✳ ❉❛❞♦ ✉♠ ✐♥st❛♥t❡ tk✱ t❡♠♦s q✉❡ gtk(γtk) ♠❛♣❡✐❛ γtk ♣❛r❛ ✉♠ ♣♦♥t♦ ❞♦ ❡✐①♦ r❡❛❧✱ ✈✐st♦ q✉❡ Utk é r❡❛❧ ❡ Utk = gtk(γ(tk))✳ ❈♦♠ ✐ss♦✱ ❛ ❝✉r✈❛ ❛♣ós
gtk(γ(tk)) ♥ã♦ ♣❛rt❡ ♠❛✐s ❞❛ ♦r✐❣❡♠✱ ♠❛s ❞♦ ♣♦♥t♦Ut✱ ❡ ♦ r❡st❛♥t❡ γ[tk, t] é ♠❛♣❡❛❞♦ ❞❡ ❢♦r♠❛ ❛ ♦❜t❡r♠♦s ✉♠❛ ♥♦✈❛ ❝✉r✈❛ ❝♦♥tí♥✉❛✳ P❛r❛ q✉❡ ❛ ❝✉r✈❛ s❡❥❛ ♠❛♣❡❛❞❛ ♣❛r❛ ❝♦♠❡ç❛r ♥❛ ♦r✐❣❡♠✱ ♥♦ ❧✉❣❛r ❞❡gt ♣♦❞❡♠♦s ✉s❛r ♦ ♠❛♣❡❛♠❡♥t♦✿
ht(z) =gt(z)−Ut. ✭✸✳✷✮
❙❡❥❛z1, z2, ..., zk✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛γ(t)✳ ❙❡ ❞✐✈✐❞✐r♠♦s t ❡♠ ✐♥t❡r✈❛❧♦s
0 = t0 < t1 < ... < tk = t✱ ♦♥❞❡ γ(tk) = zk❀ ♣♦❞❡♠♦s ❛♣❧✐❝❛r s✉❝❡ss✐✈♦s ♠❛♣❡❛♠❡♥t♦s
h1, h2, ..., hk ✐♥t❡r♠❡❞✐ár✐♦s ❛ ❝❛❞❛ ✐♥st❛♥t❡tk✱ q✉❡ ✈ã♦ ♣❛r❝✐❛❧♠❡♥t❡ r❡t✐r❛♥❞♦ ❛ ❝✉r✈❛ ❞♦
s❡♠✐♣❧❛♥♦ s✉♣❡r✐♦r✱ ❛té q✉❡ t♦❞❛ ❡❧❛ s❡❥❛ r❡t✐r❛❞❛✳ ❆ss✐♠✱ ♥♦ ✐♥st❛♥t❡t1 ♦ ♠❛♣❛ h1 ❧❡✈❛ H\γ[t0 = 0, t1] ♣❛r❛ H✱ ❝♦♠ ❛ ♥♦✈❛ ❝✉r✈❛ s❡♥❞♦ ❞❡s❧♦❝❛❞❛ ♣❛r❛ γ′
[0, tk−t1]✱ ❡ ♦ ♣♦♥t♦
z1 é ❧❡✈❛❞♦ ♣❛r❛ ❛ ♦r✐❣❡♠❀ ♥♦ ✐♥st❛♥t❡ t2 ♦ ♠❛♣❛ h2 ❧❡✈❛ H\γ[0, tk−t1] ♣❛r❛ H✱ ❝♦♠ ❛
♥♦✈❛ ❝✉r✈❛ s❡♥❞♦ ❞❡s❧♦❝❛❞❛ ♣❛r❛γ′′
[0, tk−t2]❡ ♦ ♣♦♥t♦z2 é ❧❡✈❛❞♦ ♣❛r❛ ❛ ♦r✐❣❡♠❀ ❡ ❛ss✐♠
s✉❝❡ss✐✈❛♠❡♥t❡ ❛té q✉❡ ♥♦ ✐♥st❛♥t❡ tk ♦ ♠❛♣❛ hk ❧❡✈❡ H\γ(k)[0, tk−tk−1] ♣❛r❛ H✱ ❝♦♠ t♦❞❛ ❛ ❝✉r✈❛ γ[0, tk] s❡♥❞♦ r❡t✐r❛❞❛✳ ❖ r❡s✉❧t❛❞♦ ❣❡r❛❧ é q✉❡ hk ◦hk−1 ◦...◦h1 ♠❛♣❡✐❛ H\γ[0, tk]→H✱ ❝♦♠♦ ❞❡s❡❥❛❞♦✳ ❈❛❞❛ ♠❛♣❛ hk ✐♥t❡r♠❡❞✐ár✐♦ t❡♠ s✉❛ ♣r♦♣r✐❛ ❡①♣❛♥sã♦ ♥♦ ✐♥✜♥✐t♦✿
hk(z) =z−∆Uk+ 2∆tk
z +O(
1
z2), z → ∞, ✭✸✳✸✮
♦♥❞❡∆Uk =Uk−Uk−1 ❡∆tk =tk−tk−1✳ ❊♥tã♦✱
h1(z) = z−∆U1+
∆t1
z +O(
1
z2), ✭✸✳✹✮
h2◦h1(z) = z−U2+
2t2
z−∆Uk+ 2∆tk
z +O(
1
z2)
+O(1
✶✾
q✉❡ ♣♦❞❡ s❡r s✐♠♣❧✐✜❝❛❞♦ ✉s❛♥❞♦ z → ∞✿
h2◦h1(z) =z−U2+
2t2
z +O(
1
z2). ✭✸✳✻✮
●❡♥❡r❛❧✐③❛♥❞♦✱
hk◦...◦h2◦h1(z) = z−Ut+
2tk z +O(
1
z2), ✭✸✳✼✮
♦♥❞❡Ut =
k P
i=1
∆Ui ❡ t=
k P
i=1
∆ti✳
❖ t✐♣♦ ❞❡ ♠❛♣❡❛♠❡♥t♦ q✉❡ ♠❡♥❝✐♦♥❛r❡♠♦s é ♦ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✭✈❡rt✐❝❛❧ s❧✐t✮✱ ♦♥❞❡ ❝❛❞❛ hi ♠❛♣❡✐❛ H ♠❡♥♦s ❛ ❧✐♥❤❛ ✈❡rt✐❝❛❧ [Re(ωi), ωi] ♣❛r❛ H✱ ❝♦♠ωi =hi−1◦hi−2◦...◦h1(zi)❀ ❡♠ ♣❛rt✐❝✉❧❛r✱hk(ωk) = 0 ✳ ❈♦♠❡ç❛♠♦s ♦ ♣r♦❣r❛♠❛ ❡♥tã♦ ❛♣❧✐❝❛♥❞♦ h1 ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ γ(t)✱ ❣❛r❛♥t✐♥❞♦ ❛♣❡♥❛s q✉❡ ❛ ❧✐♥❤❛ ✈❡rt✐❝❛❧
q✉❡ ❝♦♠❡ç❛ ❡♠ Re(ω1) ❡ t❡r♠✐♥❛ ❡♠ ω1 s❡❥❛ ♠❛♣❡❛❞❛ ♣❛r❛ ♦ ❡✐①♦ r❡❛❧❀ ❡♠ ♣❛rt✐❝✉❧❛r✱
h1(ω1) =h1(z1) = 0✳ ❖s ♣♦♥t♦s r❡st❛♥t❡s ❞❡ h1(zk) ❢♦r♠❛♠ ✉♠❛ ♥♦✈❛ ❝✉r✈❛ q✉❡ ❝♦♠❡ç❛
❡♠ h1(z2) = ω2✳ ❖ ♣❛ss♦ s❡❣✉✐♥t❡ é ❛♣❧✐❝❛r ♦ ♠❛♣❛ h2 ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❛ ❝✉r✈❛
r❡s✉❧t❛♥t❡ ❞♦ ♣❛ss♦ ❛♥t❡r✐♦r✳ ❆ ❧✐♥❤❛ ✈❡rt✐❝❛❧ q✉❡ ❝♦♠❡ç❛ ❡♠ Re(ω2) ❡ t❡r♠✐♥❛ ❡♠ ω2 é
♠❛♣❡❛❞❛ ♣❛r❛ ♦ ❡✐①♦ r❡❛❧✳ ❖ ♣♦♥t♦ h2(ω2) = h2 ◦h1(z1) é ♠❛♣❡❛❞♦ ♣❛r❛ ❛ ♦r✐❣❡♠✳ ❖s
♣♦♥t♦s r❡st❛♥t❡s ❞❡h2◦h1(zk)❢♦r♠❛♠ ✉♠❛ ♥♦✈❛ ❝✉r✈❛ q✉❡ ❝♦♠❡ç❛ ❡♠h2◦h1(z2) = ω2✳ ❖
♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ❛té q✉❡ ✜♥❛❧♠❡♥t❡ ❛♣❧✐❝❛♠♦s ♦ ♠❛♣❛hk✱ ❡ ❛♣ós ✐ss♦ ❛ ❝✉r✈❛ γt ✐♥✐❝✐❛❧
❢♦✐ t♦❞❛ ♠❛♣❡❛❞❛ ♣❛r❛ ♦ ❡✐①♦ r❡❛❧✳ ◆♦t❡✲s❡ q✉❡ ❛ ❝✉r✈❛ ♠❛♣❡❛❞❛ ❡♠ ❝❛❞❛ ❡t❛♣❛ ✐ ❝♦♠❡ç❛ ❡♠ ωi✱ ❡ ♥ã♦ ♥❛ ♦r✐❣❡♠✱ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ✉♠❛ ❙▲❊ ❝♦r❞❛❧❀ ❛ ❧✐♥❤❛ ♠❛♣❡❛❞❛ ❡stá
❛ ✉♠❛ ❞✐stâ♥❝✐❛Re(ωi)❞♦ ♣♦♥t♦ ③❡r♦✳ ◆♦ ❡♥t❛♥t♦✱ q✉❛♥t♦ ♠❛✐s ♣ró①✐♠♦ ♦ ♣♦♥t♦ω1 =z1
❞❡ ③❡r♦✱ ♦s ❝♦♥s❡q✉❡♥t❡sωk ♥❛ s✐♠✉❧❛çã♦ t❛♠❜é♠ ♦ s❡rã♦ ✭♣♦✐s ♦s ♠❛♣❛s hi ❛♣r♦①✐♠❛♠
❛ ❝✉r✈❛ ❝❛❞❛ ✈❡③ ♠❛✐s ♣❛r❛ ♦ ❡✐①♦ r❡❛❧ ❡♠ ❝❛❞❛ ❡t❛♣❛✮✳ ❖ ♣r♦❣r❛♠❛ ❢♦r♥❡❝❡ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ s❡ ❡s❝♦❧❤❡r♠♦sz1 ♦ s✉✜❝✐❡♥t❡ ♣ró①✐♠♦ ❞❡ ③❡r♦✳
P♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ♣❛r❛ ♦ ♠❛♣❛ hk q✉❡ r❡s♦❧✈❛ ♥♦ss♦ ♣r♦✲
❜❧❡♠❛ ♣❛r❛ ❛s ❝♦♥❞✐çõ❡s ❛♣r❡s❡♥t❛❞❛s✳ P❛r❛hk✱ ❛ r❡t❛ ✈❡rt✐❝❛❧ ♥♦ ✐♥t❡r✈❛❧♦ [Re(ωk), ωk]é ❢♦r♠❛❞❛ ♣♦r ♣♦♥t♦s t❛✐s q✉❡z =Re(ωk) +iIm(z)❡Im(z)≤Im(ωk)✳ ❚❡♠♦s ❛s r❡str✐çõ❡s
q✉❡hk(z)∈H❡hk(ωk) = 0✳ ❊♥tã♦hk
teste =−(z−Re(ωk))
2 ❥á r❡s♦❧✈❡ ❛ ♣r✐♠❡✐r❛ r❡str✐çã♦✳
❆ s❡❣✉♥❞❛ r❡str✐çã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ❢❛③❡♥❞♦✲s❡ hkteste +Im(ωk)
2 ✭♣❛r❛ ♠❛♥t❡r ❛ ♣r✐✲
♠❡✐r❛ r❡str✐çã♦hk(z)∈H ✈á❧✐❞❛✮✳ ❆ss✐♠✱ ✜❝❛♠♦s ❝♦♠h′
kteste = (z−Re(ωk))
2+Im(ωk)2✱
q✉❡ r❡s♦❧✈❡ ♦ ♣r♦❜❧❡♠❛✳ P♦ré♠✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ✸✳✸✱ ♦ ♠❛♣❛ hk ♥ã♦ t❡♠ t❡r♠♦s O(z2)✳
❉❡✈❡♠♦s ❡♥tã♦ ❞✐♠✐♥✉✐r ❛ ♦r❞❡♠ ❞♦ r❡s✉❧t❛❞♦ ❞❡h′
kteste✱ s❡♠ ❛❧t❡r❛r ❛s r❡str✐çõ❡s ♠❡♥❝✐✲ ♦♥❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳ ❆ s♦❧✉çã♦ é ♠♦str❛❞❛ ❛❜❛✐①♦✱ q✉❡ é ❛ ✈❡rsã♦ ✜♥❛❧hk✿
hk(z) = ip−(z−Re(ωk))2−Im(ωk)2. ✭✸✳✽✮
✷✵
h2
k ≃z2−2z∆Uk+ 4∆tk+ ∆Uk2 =z2−2zRe(ωk)2+Re(ωk)2+Im(ωk)2✱
∆Uk=Re(ωk), ∆tk= Im(ωk)
2
4 . ✭✸✳✾✮
P♦❞❡♠♦s ❡♥tã♦ ❡s❝r❡✈❡r ✸✳✶✵ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
hk(z) = i p
−(z−∆Uk)2−4∆tk. ✭✸✳✶✵✮
❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ é ❛ ❜❛s❡ ♣❛r❛ ♦ ❛❧❣♦r✐t♠♦ ❩✐♣♣❡r ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦✳ ❊♠ ▲✐st✐♥❣ ✶✱ ♠♦str❛♠♦s ✉♠❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❡♠ ❈ ♣❛r❛ ♦ ❛❧❣♦r✐t♠♦ ❞❡s❝r✐t♦ ❛❝✐♠❛✳ ❖ t❡♠♣♦ ❣❛st♦ ❝♦♠ ♦ ♣r♦❝❡ss❛♠❡♥t♦ é ❞❛ ♦r❞❡♠ ❞❡O(n2)✱ s❡♥❞♦ ♥ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❞❡ γ ❢♦r♥❡❝✐❞♦s✳
■ss♦ ♣♦rq✉ê é ♣r❡❝✐s♦ ❝♦♠♣✉t❛r✱ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❣r❛♠❛✱ ✐ ♠❛♣❛s ♣❛r❛ ❝❛❞❛ ♣♦♥t♦
zi ✭hi ◦hi−1 ◦...◦h1✮✱ ❝❛❞❛ ✉♠ ❣❛st❛♥❞♦ t❡♠♣♦ ❧✐♥❡❛r ✭❛♣❡♥❛s ❛ ❛♣❧✐❝❛çã♦ ❞❛ ❢ór♠✉❧❛ ❢✉♥❝✐♦♥❛❧ ❞❡ hi ✸✳✶✵✮✳ ▲♦❣♦✱ ❛ ❝♦♠♣✉t❛çã♦ ❞♦s ♣r✐♠❡✐r♦s ✐ ♣♦♥t♦s ❣❛st❛ t❡♠♣♦ O(i2)✳
❈♦♠♦ ♦ ú❧t✐♠♦ ♣♦♥t♦ é ♦ ♥✲és✐♠♦✱ ♦ ❛❧❣♦r✐t♠♦ t❡♠ ❝♦♠♣❧❡①✐❞❛❞❡ O(n2)✱ ♦ q✉❡ ♥ã♦ é
♠✉✐t♦ ❡✜❝✐❡♥t❡ q✉❛♥❞♦ s❡ é ♣r❡❝✐s♦ ♣r♦❝❡ss❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝♦♥s✐❞❡rá✈❡❧ ❞❡ ♣♦♥t♦s✳ ✶ ✈♦✐❞ ❣❡t❉r✐✈❋r♠●❛♠♠❛ ✭✐♥t ❙✐③❡ ✱ ❝♦♠♣❧❡① ❞♦✉❜❧❡ ✯❣❛♠♠❛ ✱ ❞♦✉❜❧❡✯ ❯✱ ❞♦✉❜❧❡✯
t✮
✷ ④
✸ ✐♥t ✐✱❥❀
✹ ❯ ❬✵❪ ❂ t ❬✵❪ ❂ ✵❀
✺ ❢♦r ✭✐ ❂ ✶❀ ✐ ❁ ❙✐③❡ ❀ ✐ ✰✰✮ ④
✻ ❯❬✐❪ ❂ ❯❬✐ ✲ ✶❪ ✰ ❝r❡❛❧ ✭ ❣❛♠♠❛ ❬✐❪✮❀
✼ t❬✐❪ ❂ t❬✐ ✲ ✶❪ ✰ ♣♦✇ ✭ ❝✐♠❛❣ ✭ ❣❛♠♠❛ ❬✐❪✮ ✱✷✮ ✴✹❀
✽ ❢♦r ✭❥ ❂ ✐ ✰ ✶❀ ❥ ❁ ❙✐③❡ ❀ ❥ ✰✰✮
✾ ❣❛♠♠❛ ❬❥❪ ❂ ■✯ ❝sqrt ✭✲ ❝♣♦✇ ✭ ❣❛♠♠❛ ❬❥❪ ✲ ❝r❡❛❧ ✭ ❣❛♠♠❛ ❬✐❪✮ ✱ ✷✮ ✲ ♣♦✇
✭ ❝✐♠❛❣ ✭ ❣❛♠♠❛ ❬✐❪✮ ✱✷✮✮❀
✶✵ ⑥
✶✶ ⑥
✷✶
✸✳✶✳✷ ❖❜t❡♥çã♦ ❞❡ γt ❝♦♥❤❡❝❡♥❞♦✲s❡ Ut
◆❡ss❡ ❝❛s♦ ♦ ♠❛♣❛ ❝♦♥❢♦r♠❡ ✉t✐❧✐③❛❞♦ ❢❛③ ♦ ✐♥✈❡rs♦ ❞♦ ❞❡s❝r✐t♦ ♥❛ s❡çã♦ ❛♥✲ t❡r✐♦r✳ ❱✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ❝❛❞❛ ♠❛♣❛ hi ♣❛rt❡ ❞❡ H \[Re(ωi), ωi] ♣❛r❛ H✱ ❝♦♠
ωi =hi−1 ◦hi−2◦...◦h1(zi)✳ ❊♠ ♣❛rt✐❝✉❧❛r hk(ωk) = 0✱ ♦✉ s❡❥❛✿
hk◦hk−1◦...◦h1(zk) = 0. ✭✸✳✶✶✮ ❙❡❥❛ ❛ ❢✉♥çã♦ lk =h
−1
k ✳ P♦❞❡♠♦s ❡♥tã♦ ❡s❝r❡✈❡r ✸✳✶✶ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ l−1
k ◦l
−1
k−1◦...◦l −1
1 (zk) = 0✳
❙❡ ❛♣❧✐❝❛r♠♦s ✭l1◦l2◦...◦lk✮ à ❡sq✉❡r❞❛ ❞❡ ❝❛❞❛ ❧❛❞♦ ❞❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s
zk =l1◦l2◦...◦lk(0). ✭✸✳✶✷✮
P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ❞❡ ❧✱ ♦❜t❡♠♦s ❛ ✐♥✈❡rs❛ ❞♦ ♠❛♣❛hk ♦❜t✐❞❛ ♥❛ s❡çã♦
❛♥t❡r✐♦r✳ ❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❡♠ ✸✳✾ ❡♠ ✸✳✶✵✿
hk(z) = ip−(z−Re(ωk))2−Im(ωk)2. ✭✸✳✶✸✮
❈❛❧❝✉❧❛♥❞♦ ❛ ✐♥✈❡rs❛h−1
k =fk ✉t✐❧✐③❛♥❞♦ ✸✳✶✵✿
lk(z) =pz2−Im(ωk)2+Re(ωk). ✭✸✳✶✹✮
◆❡ss❡ ❝❛s♦✱ ❞❡❝❧❛r❛♠♦s ❦ ♣♦♥t♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ♦ ✈❛❧♦r ③❡r♦✳ ❈❤❛♠❡♠♦s t❛✐s ♣♦♥t♦s ❞❡υ1✱υ2✱✳✳✳✱υk✳ ◆♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ❛♣❧✐❝❛✲s❡li ❛ ❝❛❞❛ ♣♦♥t♦υi ♣❛r❛ i∈[1, k];i∈Z✳ ❈♦♠♦
✐♥✐❝✐❛❧♠❡♥t❡ t♦❞♦s sã♦ ③❡r♦✱ ♦❜t❡♠♦s
li(0) =p−Im(ωi)2+Re(ωi) = ωi, i∈[1, ..., k]. ✭✸✳✶✺✮
❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠♦s υ1 =ω1 =z1✱ ❡♥❝♦♥tr❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ♣♦♥t♦ ❞❛ ❝✉r✈❛ γt✳ ◆♦ ♣❛ss♦
s❡❣✉✐♥t❡✱❛♣❧✐❝❛✲s❡ li−1 ❛ ❝❛❞❛ ♣♦♥t♦ υi = ωi ♣❛r❛ i ∈ [2, k];i ∈ Z✳ ◆❡ss❡ ❝❛s♦✱ ♦❜t❡♠♦s
li−1(ωi) = li−1 ◦l −1
i−1 ◦...◦l −1
1 (zk) = l
−1
i−2◦...◦l −1
1 (zk)✳ P❛rt✐❝✉❧❛♠❡♥t❡✱ ♣❛r❛ ✐ ❂ ✷ t❡♠♦s
l1(ω2) = li−1 ◦l −1
1 (z2) = z2✱ ❡♥❝♦♥tr❛♥❞♦ ♦ s❡❣✉♥❞♦ ♣♦♥t♦ ❞❛ ❝✉r✈❛ γt✳ ❋✐❝❛ ❝❧❛r♦ q✉❡
❛ ❝❛❞❛ ♣❛ss♦ ✐ ✉♠ ♣♦♥t♦ zi ❞❛ ❝✉r✈❛ γt é ❡♥❝♦♥tr❛❞♦✳ ❖ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ❛té ♦ ❦✲
és✐♠♦ ♣❛ss♦✱ ♦♥❞❡ ❡♥❝♦♥tr❛✲s❡ ♦ ♣♦♥t♦ zk✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦✱ sã♦ ♦❜t✐❞♦s ♦s
♣♦♥t♦s z1✱ z2✱✳✳✳✱zk ❞❛ ❝✉r✈❛ γt✳ ➱ ❡✈✐❞❡♥t❡✱ ❞❛❞❛ ❛ ❛♥❛❧✐s❡ ❛❝✐♠❛✱ q✉❡ ♦ ♣r♦❣r❛♠❛ ❢❛③ ♦
♠❛♣❡❛♠❡♥t♦ ✐♥✈❡rs♦ ❞♦ ❞❡s❝r✐t♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ✐st♦ é✱ ♣❛rt❡ ❞♦ ❞♦♠í♥✐♦H❡ ❝❤❡❣❛ ❡♠ H\γt✳ P♦❞❡♠♦s s✉❜st✐t✉✐r ✸✳✾ ❡♠ ✸✳✶✹ ♣❛r❛ ♦❜t❡r
lk(z) = p
z2−4∆t
✷✷
❊♠ ▲✐st✐♥❣ ✷ ❝♦❧♦❝❛♠♦s ✉♠❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❡♠ ❈ ❞♦ ❛❧❣♦r✐t♠♦ q✉❡ ❞❡s❝r❡✈❡♠♦s ❛❝✐♠❛✳ ❆ ❝♦♠♣❧❡①✐❞❛❞❡ t❡♠♣♦r❛❧ é O(n2)✱ ✈✐st♦ q✉❡ ❝❛❞❛ ♣♦♥t♦ zi é ♦❜t✐❞♦ ❛♣ós ❛♣❧✐❝❛r l
1 ◦l2◦
...◦li(0)✱ t♦t❛❧✐③❛♥❞♦ ✐ ♠❛♣❛s ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♥st❛♥t❡ ✭❛♣❡♥❛s ❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ❞❡fi
✸✳✶✻✮✳ ❙❡ t✐✈❡r♠♦s ♥ ♣♦♥t♦s ❛ s❡r❡♠ ♦❜t✐❞♦s✱ ♦ ♣r♦❣r❛♠❛ ❣❛st❛ t❡♠♣♦ O(n2) ❡♠ t♦❞♦ ♦
♣r♦❝❡ss♦✳
✶ ✈♦✐❞ ❣❡t●❛♠♠❛❋r♠❉r✐✈ ✭✐♥t ❙✐③❡ ✱ ❝♦♠♣❧❡① ❞♦✉❜❧❡ ✯❣❛♠♠❛ ✱ ❞♦✉❜❧❡✯ ❯✱ ❞♦✉❜❧❡✯
t✮
✷ ④
✸ ✐♥t ✐✱❥❀
✹ ❞♦✉❜❧❡ ❞❯❬ ❙✐③❡ ❪✱ ❞t❬ ❙✐③❡ ❪❀
✺ ❞❯ ❬✵❪❂ ❞t ❬✵❪❂✵❀
✻ ❣❛♠♠❛ ❬✵❪❂✵❀
✼ ❢♦r ✭✐ ❂ ✶❀ ✐ ❁ ❙✐③❡ ❀ ✐ ✰✰✮ ④
✽ ❣❛♠♠❛ ❬✐❪ ❂ ✵❀
✾ ❞❯❬✐❪ ❂ ❯❬✐❪✲❯❬✐ ✲✶❪❀
✶✵ ❞t❬✐❪ ❂ t❬✐❪✲t❬✐ ✲✶❪❀
✶✶ ⑥
✶✷ ❢♦r ✭✐ ❂ ✶❀ ✐ ❁ ❙✐③❡ ❀ ✐ ✰✰✮
✶✸ ❢♦r ✭❥ ❂ ✐❀ ❥ ❁ ❙✐③❡ ❀ ❥ ✰✰✮
✶✹ ❣❛♠♠❛ ❬❥❪ ❂ ❘❛✐③❈♦♠♣❧❡①❛ ✭ ❝♣♦✇ ✭ ❣❛♠♠❛ ❬❥❪ ✱✷✮ ✲✹✯ ❞t❬❥✲✐ ✰✶❪✮✰❞❯❬❥✲✐
✰✶❪❀
✶✺ ⑥
✷✸
✸✳✷ ❊st✐♠❛t✐✈❛ ❞❡ ❦ ♣❛r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❞❡ r❡❞❡
❯s❛r❡♠♦s ♦ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ▲♦❡✇♥❡r ♣❛r❛ ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ ❦ ♣❛r❛ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❝rít✐❝❛✳
✸✳✷✳✶ P❡r❝♦❧❛çã♦ ❝rít✐❝❛
❉✐✜♥✐r❡♠♦s ❛ ♣❡r❝♦❧❛çã♦ ♣❛r❛ sít✐♦s ❡♠ ✉♠❛ r❡❞❡ tr✐â♥❣✉❧❛r✳ ❚♦❞♦s ♦s ✈ért✐❝❡s ❞❛ r❡❞❡ sã♦ ❞❡✜♥✐❞♦s ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❝♦♠♦ ❛❜❡rt♦✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❀ ♦✉ ❢❡❝❤❛❞♦✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✶✲♣✳ ♣ é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ r❡❞❡✱ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡✜♥✐❞❛ ✐♥✐❝✐✲ ❛❧♠❡♥t❡✳ ❙❛❜❡✲s❡ q✉❡ ♣❛r❛ p > 1/2✱ é q✉❛s❡ ❝❡rt♦ q✉❡ ❡①✐st❛ ✉♠ ❝❛♠✐♥❤♦ ❞❡ ❛r❡st❛s ❝♦♥❡❝t❛❞❛s ✭❧♦❣♦ ❝♦♠ ♦s ✈ért✐❝❡s ❛❜❡rt♦s✮ q✉❡ ❛tr❛✈❡ss❛ t♦❞❛ ❛ r❡❞❡✱ é ❡ ❞✐t♦ q✉❡ ♦❝♦r✲ r❡✉ ❛ ♣❡r❝♦❧❛çã♦ ♥♦ s✐st❡♠❛✳❖ ✈❛❧♦rp = 1/2 é ❝❤❛♠❛❞♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝rít✐❝❛ ❞❛ r❡❞❡ tr✐â♥❣✉❧❛r✳
❯♠ ♠ét♦❞♦ ❡q✉✐✈❛❧❡♥t❡ ❡ ❛✐♥❞❛ ♠❛✐s s✐♠♣❧❡s ❞❡ s✐♠✉❧❛r ❝♦♥s✐st❡ ❡♠ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❝♦❧♦r✐r ♦s ❤❡①á❣♦♥♦s ❞❛ r❡❞❡ ❞✉❛❧✳ ❇❛s✐❝❛♠❡♥t❡✱ ✈✐s✐t❛✲s❡ ❝❛❞❛ ❤❡①á❣♦♥♦ ❡ s♦rt❡✐❛✲s❡ ✉♠ ✈❛❧♦rp′
∈[0 : 1]✳ ❙❡p′
≤p✱ ♦♥❞❡ ♣ é ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡ ❡s❝♦❧❤✐❞♦ ✐♥✐❝✐❛❧♠❡♥t❡❀
❝♦❧♦r❡✲s❡ ♦ ❤❡①á❣♦♥♦ ❝♦♠ ❛ ❝♦r ❛③✉❧✳ ❈❛s♦ p′
> p✱ ❡s❝♦❧❤❡✲s❡ ❛ ❝♦r ❛ ❛♠❛r❡❧❛✳ ➱ s❛❜✐❞♦
q✉❡ ♣❛r❛ ✉♠ ✈❛❧♦r ❞❡ p≤ 1/2✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❤❛✈❡r ✉♠ ❛❣❧♦♠❡r❛❞♦ ❞♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡ ✭❡ q✉❛♥❞♦ ❛ ♦r❞❡♠ ❞❛ r❡❞❡ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✱ ♦ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ❡ss❡ ❛❣❧♦♠❡r❛❞♦✮ é q✉❛s❡ ♥✉❧❛✳ P❛r❛ p >1/2 ❡①✐st✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣ró①✐♠❛ ❞❡ ✶✱ ✉♠ ❛❣❧♦♠❡r❛❞♦ ❛③✉❧ ❞♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡✳ ❆ ❢r♦♥t❡✐r❛ q✉❡ s❡♣❛r❛ ♦ ❛❣❧♦♠❡r❛❞♦ ❛③✉❧ ❞❡ t♦❞♦ ♦ r❡st♦ é ❛ ♠❡s♠❛ ❝✉r✈❛ ❣❡r❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✳
❈♦♠♦ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦✱ ♦s ❤❡①á❣♦♥♦s q✉❡ t♦❝❛♠ ✉♠❛ ❞❡ s✉❛s ❛r❡st❛s ♥♦ ❡✐①♦ r❡❛❧ ♣♦s✐t✐✈♦ sã♦ ❝♦❧♦r✐❞♦s ❞❡ ❛③✉❧✱ ❡ ♦s q✉❡ t♦❝❛♠ ❛ ❧✐♥❤❛ r❡❛❧ ♥❡❣❛t✐✈❛ sã♦ ❝♦❧♦r✐❞♦s ❞❡ ❛♠❛r❡❧♦✳ ◆❛ ❋✐❣✉r❛ ✶ t❡♠♦s ✉♠❛ ✐❧✉str❛çã♦ ❡♠ ♣❡q✉❡♥❛ ❡s❝❛❧❛ ❞❛ r❡❞❡ ♥❛ r❡❣✐ã♦ ❝rít✐❝❛✳ ❊ss❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❣❛r❛♥t❡♠ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠✐♥❤♦ ❞❡ ❛r❡st❛s ❞♦s ❤❡①á❣♦♥♦s✱ ❝♦♠❡ç❛♥❞♦ ❞❛ ♦r✐❣❡♠✱ q✉❡ s❡♣❛r❛ ♦ ❛❣❧♦♠❡r❛❞♦ ❞❡ ❤❡①á❣♦♥♦s ❛③✉✐s ❧✐❣❛❞♦s ❛ s❡♠✐r❡t❛ ♣♦s✐t✐✈❛ ❞♦s r❡❛✐s❀ ❞♦s ❛❣❧♦♠❡r❛❞♦s ❞❡ ❤❡①á❣♦♥♦s ❛♠❛r❡❧♦s ❧✐❣❛❞♦s à s❡♠✐r❡t❛ ♥❡❣❛t✐✈❛ ❞♦s r❡❛✐s✳ ❚❛❧ ❝❛♠✐♥❤♦ é ❝❤❛♠❛❞♦ ♣r♦❝❡ss♦ ❞❡ ❡①♣❧♦r❛çã♦ ❝♦r❞❛❧ ❞❡ ✵ ❛ ∞ ♥♦ s❡♠✐♣❧❛♥♦✳
❋✐❣✉r❛ ✸ ✕ ■❧✉str❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❡①♣❧♦r❛çã♦ ❡♠ ✉♠❛ ♣❛rt❡ ❞❛ r❡❞❡✳
❋♦♥t❡✿ ❬✾❪✳
✷✹
♦ ❛❧❣♦r✐t♠♦ ✵✶ ♣❛r❛ ♦❜t❡r ❛ ❢✉♥çã♦ ❞✐r❡t♦r❛Ut✳ ❙❛❜❡♠♦s q✉❡ ❡ss❛ ❝✉r✈❛ é ✉♠ ♠♦✈✐♠❡♥t♦
❜r♦✇♥✐❛♥♦ ❝♦♠ ✈❛r✐â♥❝✐❛ ✷✳✶✷✳ ❖ ♠ét♦❞♦ q✉❡ ✉t✐❧✐③❛♠♦s é ❞❡s❝r✐t♦ ❡♠ ❬✷✵❪✳ ❆ ❝✉r✈❛ é ❞✐✈✐❞✐❞❛ ❡♠ ♥ ♣❛rt❡s ❞❡ ✐♥t❡r✈❛❧♦s ❞❡∆t ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛✐s ❡ ♦ s♦♠❛tór✐♦
h|Uti −Uti−∆t|2i∆t =
N X
i=0
(Uti−Uti−∆t)2/(N + 1), ✭✸✳✶✼✮ é ❡①❡❝✉t❛❞♦ ♣❛r❛ ✈❛❧♦r❡s ❛r❜✐trár✐♦s ❞❡ ♥✳ ◆ é ♦ ♥ú♠❡r♦ ❞❡ ✧♣❡❞❛ç♦s✧❞❛ ❝✉r✈❛ q✉❡ ❝♦♠✲ ♣õ❡♠ ♦ s♦♠❛tór✐♦✱ ❡ ❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❞❡ ♥✳ ◆❛ ❋✐❣✉r❛ ✵✺ t❡♠♦s ❛ ❝✉r✈❛ ❞❡s❡❥❛❞❛✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ ❛♣r♦①✐♠❛❞❛ é ❡①❛t❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❡ ❦ ✱ q✉❡ ♥♦ ❡①♣❡r✐♠❡♥t♦ r❡s✉❧t♦✉ ❡♠ k ≈ 6.19153±0.03521✳ ❖ ✈❛❧♦r q✉❡ ♦❜t✐❞♦ é ♣ró①✐♠♦ ❛♦ ❦❂✻ ♦❜t✐❞♦ ♣♦r
❙♠✐r♥♦✈ ❬✶✼❪✱ ❡♠❜♦r❛ ♦ ♠ét♦❞♦ ✉s❛❞♦ ♣❛r❛ ♦❜t❡r ❡ss❡ ❝♦❡✜❝✐❡♥t❡ ❛✐♥❞❛ ♣r❡❝✐s❡ s❡r ♠❡❧❤♦✲ r❛❞♦✱ ❞❛❞❛ ❛s ✢✉t✉❛çõ❡s ♦❜s❡r✈❛❞❛s ♥♦ ❣rá✜❝♦✳ ❯s❛♥❞♦ ✷✳✶✶♦❜t❡♠♦sdf = 1,77✱ ♣ró①✐♠♦ ❞♦ ✈❛❧♦r ✶✱✼✺ ❡s♣❡r❛❞♦✳
❋✐❣✉r❛ ✹ ✕ Pr♦❝❡ss♦ ❞❡ ❡①♣❧♦r❛çã♦ ❝♦r❞❛❧ ♣❛r❛ n = 104 ♣♦♥t♦s ❞❡ γt✳
✷✺
❋✐❣✉r❛ ✺ ✕ ❋✉♥çã♦ Ut ♦❜t✐❞❛ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞✐s❝r❡t♦✳
❋♦♥t❡✿ ❊❧❛❜♦r❛❞♦ ♣❡❧♦ ❛✉t♦r✳
❋✐❣✉r❛ ✻ ✕ ●rá✜❝♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡ Ut ❝♦♠ ♦ t❡♠♣♦✳
✷✻
✹ P❊❘❙P❊❈❚■❱❆❙ P❆❘❆ ❆ ❙❖▲❯➬➹❖ ❉❆ ❊◗❯❆➬➹❖ ❉❊ ▲❖❊❲◆❊❘
✹✳✶ ❙♦❧✉çã♦ ♣❛r❛ ft
❈♦♠❡ç❛♠♦s ❛❞♠✐t✐♥❞♦ q✉❡ ❢✭③✱t✮ ♣♦ss✉✐ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ▲❛✉r❡♥t✿
f(z) = ∞
X
i=−∞
ai(t)zi. ✭✹✳✶✮
❚❡♠♦s q✉❡ ❢✭③✮→③ q✉❛♥❞♦z → ∞✳ P♦rt❛♥t♦✱ t♦❞♦s ♦s t❡r♠♦s ❞❡ ♦r❞❡♠O(z2)♦✉ s✉♣❡r✐♦r
tê♠ ❝♦❡✜❝✐❡♥t❡s ♥✉❧♦s❀ ❛❧é♠ ❞✐ss♦✱ a0 ❂ ✵ ❡ a1 = 1✳ P♦rt❛♥t♦✱
f(z) =z+ ∞
X
i=0
ai(t)
zi . ✭✹✳✷✮
❱❛♠♦s ✉t✐❧✐③❛r ✹✳✷ ♣❛r❛ r❡s♦❧✈❡r ❝❛❞❛ t❡r♠♦ ❞❛ ❡q✉❛çã♦ ✷✳✽✳ P❛r❛ ∂f ∂t✿ ∂f ∂t = ∞ X i=0 1 zi ∂ai
∂t . ✭✹✳✸✮
❏á ∂f
∂z ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿
∂f
∂z = 1−
∞
X
i=0
iai
zi+1. ✭✹✳✹✮
P❛r❛ ♦ t❡r♠♦ r❡st❛♥t❡ ♥❛ ❡q✉❛çã♦ ✷✳✽✱ ♣♦❞❡♠♦s ❛❥✉st❛✲❧♦ tr❛♥s❢♦r♠❛♥❞♦✲♦ ❡♠ ✉♠❛ sér✐❡ ❣❡♦♠étr✐❝❛✿
2
Ut−z
= −2/z 1−Ut/z
=−2
z ∞ X i=0 (Ut z )
i. ✭✹✳✺✮
❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❛❝✐♠❛ ♥❛ ❡q✉❛çã♦ ✷✳✽✿
∂a0
∂t = 0; ∂an
∂t =−2U n−1
t + 2
n−2
X
i=0
iaiUn−i−2
t . ✭✹✳✻✮
❈♦♠❡ç❛♠♦s ❛ s✐♠✉❧❛çã♦ ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s(t1;Ut1)✱ (t2;Ut2)✱✳✳✳✱(tk;Utk) ❝♦♥❤❡✲ ❝✐❞♦s ❞❡ Ut✱ ❛❧é♠ ❞♦ ♣♦♥t♦ t0 = 0;Ut0 = 0)✳ ❆ ❝❛❞❛ ♣❛ss♦ ❞♦ ♣r♦❣r❛♠❛✱ ❡♥❝♦♥tr❛♠♦s
❝❛❞❛ t❡r♠♦ ∂an
∂t ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛ ✹✳✻✳ ❆♣ós ✐ss♦✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ s❡❣✉✐♥t❡
❛♣r♦①✐♠❛çã♦✿
an(ti) = ∂an
∂t ∆ti+an(ti−1), ✭✹✳✼✮
♦♥❞❡∆ti =ti−ti−1✳ ■♥✐❝✐❛❧♠❡♥t❡✱ t❡♠♦san(t0) = 0✳ P♦rt❛♥t♦✱ t❡♠♦s t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ♣r❛ r❡s♦❧✈❡r ♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ❡ ❡♥❝♦♥tr❛ran(t1) ✭❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ft1✮✳ ❙❡❣✉✐♥❞♦ ❡st❡
