Contribuição ao estudo das redes complexas: extensão do modelo de afinidade

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ❚❡rr❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛ ❡ ❊①♣❡r✐♠❡♥t❛❧

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

❈♦♥tr✐❜✉✐çã♦ ❛♦ ❊st✉❞♦ ❞❛s ❘❡❞❡s ❈♦♠♣❧❡①❛s✿ ❊①t❡♥sã♦ ❞♦ ▼♦❞❡❧♦

❞❡ ❆✜♥✐❞❛❞❡

♣♦r

❛♥t♦♥✐♦ ♠❛rq✉❡s ❞♦s s❛♥t♦s

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ❚❡rr❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛ ❡ ❊①♣❡r✐♠❡♥t❛❧

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛

❈♦♥tr✐❜✉✐çã♦ ❛♦ ❊st✉❞♦ ❞❛s ❘❡❞❡s ❈♦♠♣❧❡①❛s✿ ❊①t❡♥sã♦ ❞♦ ▼♦❞❡❧♦ ❞❡ ❆✜♥✐❞❛❞❡

❆♥t♦♥✐♦ ▼❛rq✉❡s ❞♦s ❙❛♥t♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞♦ ❉❡♣❛rt❛✲ ♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛ ❡ ❊①♣❡r✐♠❡♥t❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡ ❝♦♠♦ ♣❛rt❡ ❞♦s ♣ré✲r❡q✉✐s✐t♦s à ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❊❙❚❘❊ ❡♠ ❋❮❙■❈❆✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉❝✐❛♥♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛

◆❛t❛❧ ❘◆✱ ❇r❛s✐❧ ❏✉❧❤♦ ❞❡ ✷✵✶✵

❆❣r❛❞❡❝✐♠❡♥t♦s

❆q✉✐ ❝♦♠❡ç❛ ✉♠❛ ❞❛s ♣❛rt❡s ♠❛✐s ❞✐❢í❝❡✐s✱ ♠❛s ❛♦ ♠❡s♠♦ t❡♠♣♦ ♣r❛③❡r♦s❛ ❞❡st❡ tr❛❜❛❧❤♦✱ q✉❡ é ❧❡♠❜r❛r ❞❡ t♦❞♦s q✉❡ ❞✐r❡t♦ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ❛ r❡❛❧✐③❛çã♦ ❞❡st❛ ❞✐ss❡r✲ t❛çã♦✳

❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦✿

• ➚ ❉❡✉s ♥♦ss♦ ❝r✐❛❞♦r❀

• ❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞✐r❡t♦ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉✐r❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡

tr❛❜❛❧❤♦❀

• ❆♦ s❡♠♣r❡ ♣r❡st❛t✐✈♦ ♣r♦❢❡ss♦r ▲✉❝✐❛♥♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡

r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❛♣♦✐♦ ❡♠ t♦❞♦ ♦ ♣❡rí♦❞♦ ❡♠ q✉❡ ❡st✐✈❡ ♥♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡❀

• ➚ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡

❛❞q✉✐r✐r ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♥ã♦ ♣♦ss✉í❛❀

• ❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❋❚❊ q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦❀

• ❆ s❡❝r❡tár✐❛ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❈❡❧✐♥❛ P✐♥❤❡✐r♦ ♣♦r t♦❞❛s ❛s ♣❛❧❛✈r❛s ❞❡

❛♣♦✐♦ ❞✉r❛♥t❡ ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s q✉❡ ♣❛ss❡✐ ♥♦ Pr♦❣r❛♠❛❀

• ➚ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

• ❆♦ ♣r♦❢❡ss♦r ❆♥tô♥✐♦ ❏♦sé ❙✐❧✈❛ ❖❧✐✈❡✐r❛ ❡ ❛ ♣r♦❢❡ss♦r❛ ■✈♦♥❡ ▲♦♣❡s ▲✐♠❛ ♣♦r t♦❞♦ ♦

❛♣♦✐♦ q✉❡ ♠❡ ❞❡r❛♠ ♣❛r❛ ❢❛③❡r ♣❛rt❡ ❞♦ ♣r♦❣r❛♠❛❀

• ❆ ♠❡✉s ❛♠✐❣♦s ❞❡ ❣r✉♣♦ ❡♠ ❡s♣❡❝✐❛❧ ▼❛❝❡❞♦✱ ▼❛✉rí❝✐♦ ❡ ●❛❜r✐❡❧ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛✱

♣❛❝✐ê♥❝✐❛✱ ❛t❡♥çã♦ ❡ ❞❡❞✐❞❛çã♦ ❞✉r❛♥t❡ ❛ ❡①❡❝✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛s ❞✐❝❛s q✉❡ ❢♦r❛♠ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❡♥r✐q✉❡❝✐♠❡♥t♦ ❞♦ ♠❡s♠♦❀

• ❆ t♦❞♦s ❞❛ s❛❧❛ ❈és❛r ▲❛tt❡s ❆r♠❛♥❞♦ ❆r❛ú❥♦✱ ❋✐❧✐♣❡ ❡ ❏❡♥♥② P❛♦❧❛ ♣❡❧♦s ♠♦♠❡♥t♦s

❞❡ ❞❡s❝♦♥tr❛çã♦✱ ♥♦✈❛♠❡♥t❡ ❛ ❏❡♥♥② P❛♦❧❛ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❞❛❞♦s ♥♦s ♠♦♠❡♥t♦s q✉❡ ♠❛✐s ♣r❡❝✐s❡✐❀

• ❆ ❛❧❣✉♥s ❛♠✐❣♦s q✉❡ ✜③ ♥♦ ✜♥❛❧ ❞♦ ♠❡str❛❞♦✱ ❋r❛♥❝✐s❝♦ ❏â♥✐♦✱ ▼❛t❡✉s ❇❛r❜♦s❛✱ ▼✐r✐❛♥✱

●✐s❧❛♥❛✱ ❍❡②❞s♦♥ ❡ ❘í③✐❛ ❘♦❞r✐❣✉❡s✱ ♥♦✈❛♠❡♥t❡ ❛ ❘í③✐❛ ♣❡❧♦ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡✉ ♥❛ ❡t❛♣❛ ✜♥❛❧ ❞❛ ❡❧❡❜♦r❛çã♦ ❞♦ tr❛❜❛❧❤♦ ♣❡❧❛ ❛t❡♥çã♦✱ ♣r❡♦❝✉♣❛çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ❝❛r✐♥❤♦✳

• ❆ ❉♦♥❛ ❊✉③é❧✐❛✱ ❙❡✉ ■✈♦✱ ❱❡r✉s❦❛ ❡ ❚❛✐♥♥á ♣❡❧♦s ❞♦✐s ❛♥♦s ❞❡ ❞❡❞✐❝❛çã♦✱ ❝❛r✐♥❤♦ ❡

❛t❡♥çã♦✳

• ❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ❣r❛♥❞❡s ❛♠✐❣♦s ❆♥❛ ❑❛r♦❧✐♥❛ ●♦♠❡s ❆r❛ú❥♦✱

▼❛r❝❡❧♦ ❇r✐t♦ ❞❛ ❙✐❧✈❛✱ ▼❛r✐❛ ❞❛s ●r❛ç❛s ❉✐❛s ❞❛ ❙✐❧✈❛ ❡ ▼❛r✐❛ ▲✐❞✉í♥❛ ❞❛s ❈❤❛❣❛s ♣❡❧♦s ❞♦✐s ❛♥♦s q✉❡ ♣❛ss❛♠♦s ♣♦r ❞✐✜❝✉❧❞❛❞❡s✱ ✐♥s❡❣✉r❛♥ç❛s✱ ❡rr♦s✱ ❛❝❡rt♦s✱ ✈✐tór✐❛s ❡ ❛❧❡❣r✐❛s✳ ❈❤❡❣❛♠♦s ❛♦ ✜♥❛❧ ❝♦♠ ❛ ❝❡rt❡③❛ ❞♦ ❞❡✈❡r ❝✉♠♣r✐❞♦✳ ❉✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❢♦♠♦s ❝♦❧❡❣❛s✱ ❛♠✐❣♦s ❡ ❛té ✐r♠ã♦s✱ ❝❤♦r❛♠♦s ❡ s♦rr✐♠♦s ♠✉✐t❛s ✈❡③❡s ❥✉♥t♦s ❡ ✐ss♦ ♥♦s ❢❡③ ♣❡ss♦❛s ❞✐❢❡r❡♥t❡s✱ ❞✐❢❡r❡♥t❡s ♣♦rq✉❡ ♦ r✐s♦ ❡ ❛ ❧á❣r✐♠❛ t❡♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♥✐r ♣❡ss♦❛s ❡ ❛♦ ♥♦s s❡♣❛r❛r♠♦s✱ ❧❡✈❛♠♦s ✉♠ ♣♦✉❝♦ ✉♠ ❞♦ ♦✉tr♦ ❡ ❞❡✐①❛♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ♥ós✳ ▼✉✐t❛s ❧✉t❛s ♥♦s ❡s♣❡r❛♠✱ ♠❛s t❡♥❤❛♠♦s s❡♠♣r❡ ❡♠ ♥ós ❡ss❛ ❢♦rç❛ q✉❡ ♥♦s tr♦✉①❡ ❛té ❛q✉✐ ❡ q✉❡ ❛❣♦r❛ ♥♦s ❧❡✈❛ ❛ s❡❣✉✐r ❝❛♠✐♥❤♦s ❞✐❢❡r❡♥t❡s ❛ s❛✉❞❛❞❡ ❞❡ t♦❞♦s ❡ ❛ ❡s♣❡r❛♥ç❛ ❞❡ ✉♠ ❜r❡✈❡ r❡❡♥❝♦♥tr♦ ❡st❛rá s❡♠♣r❡ ❡♠ ♥♦ss♦s ❝♦r❛çõ❡s❀

• ❆ t♦❞♦s ♦s ♠❡✉s ✐r♠ã♦s ❲❛❧❜❡r♠❛r❦✱ ❲❛❧❜❡t✐s❡✱ ❇❛rt♦❧♦♠❡✉✱ ❲❛❧❜❡rt♦✱ ❲❛❧❜❡❧✐❝❡✱

❲❛❧❜❡♥✐❝❡ ❡ ❲❛❧❜❡♥✐s❡ ♣♦r ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♥❡♠ ❡✉ ♠❡s♠♦ ❝♦♥✜❛✈❛✱ ♣❡❧❛s ♣❛❧❛✈r❛s q✉❡ ❡✉ ❞❡✈✐❛ ❢❛❧❛r ❡ ♥❛♦ ❢❛❧❛✈❛✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡r❛♠ ♠❡s♠♦ ♥❛ ❞✐stâ♥❝✐❛✱ ♣♦r t♦❞❛s ❛s ♣❛❧❛✈r❛s q✉❡ ❢♦r❛♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❝❤❡❣❛r ♦♥❞❡ ❡✉ ❝❤❡❣✉❡✐✱ ♦❜r✐❣❛❞♦✳

• ❋✐♥❛❧♠❡♥t❡✱ ❛♦s ♠❡✉s ♣❛✐s✱ ❇❛rt♦❧♦♠❡✉ ❋❡rr❡✐r❛ ❞♦s ❙❛♥t♦s ❡ ▼❛r✐❛ ❞❛ ❈♦♥❝❡✐çã♦

▼❛rq✉❡s ❞♦s ❙❛♥t♦s✱ q✉❡ ♠❡ ❞❡r❛♠ ❛ ✈✐❞❛ ❡ ♠❡ ❡♥s✐♥❛r❛♠ ❛ ✈✐✈ê✲❧❛ ❝♦♠ ❞✐❣♥✐❞❛❞❡✱ ♥ã♦ ❜❛st❛r✐❛ ✉♠ ♦❜r✐❣❛❞♦✱ ❛ ✈♦❝ês✱ q✉❡ ✐❧✉♠✐♥❛r❛♠ ♦s ❝❛♠✐♥❤♦s ♦❜s❝✉r♦s ❝♦♠ ❛❢❡t♦ ❡ ❞❡❞✐❝❛çã♦ ♣❛r❛ q✉❡ ❡✉ tr✐❧❤❛ss❡ s❡♠ ♠❡❞♦ ❡ ❝❤❡✐♦ ❞❡ ❡s♣❡r❛♥ç❛✱ ♥ã♦ ❜❛st❛r✐❛ ✉♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦✱ ❛ ✈♦❝ês✱ q✉❡ s❡ ❞♦❛r❛♠ ✐♥t❡✐r♦s ❡ r❡♥✉♥❝✐❛r❛♠ ❛♦s s❡✉s s♦♥❤♦s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ r❡❛❧✐③❛r ♦s ♠❡✉s✱ ❛ ✈♦❝ês ♣♦r ♥❛t✉r❡③❛✱ ♣♦r ♦♣çã♦ ❡ ❛♠♦r✱ ♥ã♦ ❜❛st❛r✐❛ ❞✐③❡r✱ q✉❡ ♥ã♦ t❡♥❤♦ ♣❛❧❛✈r❛s ♣❛r❛ ❛❣r❛❞❡❝❡r t✉❞♦ ✐ss♦✳ ▼❛s é ♦ q✉❡ ❛❝♦♥t❡❝❡ ❛❣♦r❛✱ q✉❛♥❞♦

♣r♦❝✉r♦ ❛r❞✉❛♠❡♥t❡ ✉♠❛ ❢♦r♠❛ ✈❡r❜❛❧ ❞❡ ❡①♣r✐♠✐r ✉♠❛ ❡♠♦çã♦ í♠♣❛r✳ ❯♠❛ ❡♠♦çã♦ q✉❡ ❥❛♠❛✐s s❡r✐❛ tr❛❞✉③✐❞❛ ♣♦r ♣❛❧❛✈r❛s✳ ❆♠♦ ✈♦❝ês✦

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❡❧❛❜♦r❛♠♦s ❡ ❞✐s❝✉t✐♠♦s ✉♠❛ r❡❞❡ ❝♦♠♣❧❡①❛ s❡♠ ❡s❝❛❧❛✱ ♦✉ s❡❥❛✱ ✉♠❛ r❡❞❡ ❝✉❥❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ s❡❣✉❡ ✉♠❛ ❧❡✐ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ♣♦tê♥❝✐❛✳ ◆♦ss♦ tr❛❜❛❧❤♦ ♣♦❞❡ s❡r r❡s✉♠✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ P❛r❛ ❡❢❡✐t♦ ❞❡ ❞✐❞át✐❝❛ ✈❛♠♦s ❝♦♠❡ç❛r ❝♦♠ r❡❞❡s ❛❧❡❛tór✐❛s q✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ s✐t✉❛çõ❡s r❡❛✐s ❡ ❛rt✐✜❝✐❛✐s✱ ❡ ❞❡♣♦✐s ❝♦♠❡♥t❛r ❛s r❡❞❡s ❧✐✈r❡s ❞❡ ❡s❝❛❧❛✱ ❝♦♠♦ ♣r♦♣♦st♦ ♣♦r ❇❛r❛❜ás✐✲❆❧❜❡rt ✭❇❆✮✳ ❉❡♣♦✐s ❞✐ss♦✱ ❞✐s❝✉t✐♠♦s ✉♠❛ ❡①t❡♥sã♦ ❞❡st❡ ♠♦❞❡❧♦✱ ♦♥❞❡ ❇❛r❛❜❛s✐ ❡ ❇✐❛♥❝♦♥✐ ✭❇❇✮ ✐♥❝❧✉❡♠ ❛ q✉❛❧✐❞❛❞❡✳ ❉✐s❝✉✲ t✐♠♦s t❛♠❜é♠ ♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ Πi = _P{1−|ηi−ηj|}ki

i{1−|ηi−ηj|}ki ✭❱❡r ❆❧♠❡✐❞❛ ❡t ❛❧✮✳

❋✐♥❛❧♠❡♥t❡ ✈❛♠♦s ♠♦str❛r ♦ ♥♦ss♦ ♠♦❞❡❧♦✱ ✉♠❛ ❡①t❡♥sã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡ ❞❛❞❛ ♣♦r Πi = _P{1−|ηi−ηj|σ}ki

i

{1−|ηi−ηj|σ}ki ❡ ❛♣r❡s❡♥t❛r ♦s r❡s✉❧t❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ P❛r❛ r❡❛❧✐③❛r t❛❧ t❛r❡❢❛ ♠♦❞✲

✐✜❝❛♠♦s ❛ r❡❣r❛ ❞❡ ❧✐❣❛çã♦ ♣r❡❢❡r❡♥❝✐❛❧ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❇ ❝♦❧♦❝❛♥❞♦ ✉♠ ❢❛t♦r q✉❡ ❛♣r❡s❡♥t❛ ♦ ❣r❛✉ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♥tr❡ ♦s sít✐♦s ❞❛ r❡❞❡✳ ❊st❛ q✉❛♥t✐❞❛❞❡ é ❢❡✐t❛ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ q✉❛❧✐❞❛❞❡ ❞♦ ♥♦✈♦ sít✐♦ ❡ ❛ q✉❛❧✐❞❛❞❡ ❞♦s ❛♥t❡r✐♦r❡s✳ ❊st❡ ♥♦✈♦ ♣❛râ♠❡tr♦σ ♣r♦❞✉③ ♥♦✈♦s r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s✿ ❛ ❞✐str✐❜✉✐çã♦ P(k) q✉❡ s❡❣✉❡ ✉♠❛ ❧❡✐ ❞❡ ❡s♣❡❝✐❛❧ ❞❡ ♣♦tê♥❝✐❛✱ ❡①✲ ♣♦❡♥t❡ ❛♣r♦♣r✐❛❞♦✳ ❆ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦ sít✐♦ ki t❛♠❜é♠ é ❝❛❧❝✉❧❛❞❛ ✭ki ∝(_tt0)

β_{✮✳ ❆❧é♠ ❞✐ss♦✱ ♠♦str❛♠♦s ❛ ❡✈♦❧✉çã♦ ❞❡ γ ❝♦♠ σ✳}

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ❡❧❛❜♦r❛t❡ ❛♥❞ ❞✐s❝✉ss ❛ s❝❛❧❡✲❢r❡❡ ❝♦♠♣❧❡① ♥❡t✇♦r❦✱ t❤❛t ♠❡❛♥s✱ ❛ ♥❡t✇♦r❦ ✇❤♦s❡ ❞✐str✐❜✉✐♦♥ ♦❢ ❝♦♥♥❡❝t✐✈✐t② ❢♦❧❧♦✇s ❛ ♣♦✇❡r✲❧❛✇ ❞✐str✐❜✉t✐♦♥✳ ❖✉r ✇♦r❦ ♠❛② ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦❧❧♦✇s✿ ❇② ❞✐❞❛❝t✐❝❛❧ ♣✉r♣♦s❡ ✇❡ st❛rt ✇✐t❤ r❛♥❞♦♠ ♥❡t✇♦r❦s t❤❛t ❛r❡ r❡❧❛t❡❞ t♦ r❡❛❧ ❛♥❞ ❛rt✐✜❝✐❛❧ s✐t✉❛t✐♦♥s✱ ❛♥❞ t❤❡♥ ✇❡ ❝♦♠♠❡♥t t❤❡ s❝❛❧❡ ❢r❡❡ ♥❡t✇♦r❦s ❛s ♣r♦♣♦s❡❞ ❜② ❇❛r❛❜ás✐✲❆❧❜❡rt ✭❇❆✮✳ ❆❢t❡r t❤❛t ✇❡ ❞✐s❝✉ss ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ♠♦❞❡❧ ✇❤❡r❡ ❇❛r❛❜❛s✐ ❛♥❞ ❇✐❛♥❝♦♥✐ ✭❇❇✮ ✐♥❝❧✉❞❡ ✜t♥❡ss✳ ❲❡ ❞✐s❝✉ss ❛❧s♦ t❤❡ ❆✣♥✐t② ♠♦❞❡❧✱ ✐✳❡✳✱ Πi = _P{1−|ηi−ηj|}ki

i{1−|ηi−ηj|}ki

✭s❡❡ ❆❧♠❡✐❞❛ ❡t ❛❧✮✳ ❋✐♥❛❧❧② ✇❡ s❤♦✇ ♦✉r ♠♦❞❡❧✱ ❛ ❡①t❡♥t✐♦♥ ♦❢ t❤❡ ❆✣♥✐t② ▼♦❞❡❧ ❣✐✈❡♥ ❜② t❤❡ Πi = _P{1−|ηi−ηj|σ}ki

i

{1−|ηi−ηj|σ}ki ❛♥❞ ♣r❡s❡♥t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✉❧ts✳ ❚♦ ♣❡r❢♦r♠ s✉❝❤ t❛s❦ ✇❡

♠♦❞✐❢② t❤❡ ♣r❡❢❡r❡♥t✐❛❧ ❛tt❛❝❤♠❡♥t r✉❧❡ ♦❢ t❤❡ ❇❇ ♠♦❞❡❧ ❜② ♣✉tt✐♥❣ ❛ ❢❛❝t♦r t❤❛t ✐♥tr♦❞✉❝❡s t❤❡ ❞❡❣r❡❡ ♦❢ ❧✐❦❡❧✐❤♦♦❞ ❛♠♦♥❣ t❤❡ s✐t❡s ♦❢ t❤❡ ❧❛tt✐❝❡✳ ❚❤✐s q✉❛♥t✐t② ✐s ❞♦♥❡ ❜② t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ q✉❛❧✐t② ♦❢ t❤❡ ♥❡✇ s✐t❡ ❛♥❞ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡✈✐♦✉s ♦♥❡s✳ ❚❤✐s ♥❡✇ ♣❛r❛♠❡t❡r σ ♣r♦❞✉❝❡s ♥❡✇ ✐♥t❡r❡st✐♥❣ r❡s✉❧ts✿ t❤❡ ❞✐str✐❜✉t✐♦♥ P(k) ✇❤✐❝❤ ❢♦❧❧♦✇s ❛ s♣❡❝✐❛❧ ♣♦✇❡✲❧❛✇✱ ✇✐t❤ ❛ ♣r♦♣❡r ❡①♣♦♥❡♥t✳ ❚❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈✐t② ♦❢ t❤❡ s✐t❡ki ✐s ❛❧s♦ ❝❛❧❝✉❧❛t❡❞ ✭ki ∝(t

t0)

β_{✮✳ ❋✉rt❤❡r♠♦r❡ ✇❡ s❤♦✇ t❤❡ γ ❡✈♦❧✉t✐♦♥ σ✳}

❙❯▼➪❘■❖

❘❡s✉♠♦ ✈✐

❆❜str❛❝t ✈✐✐

❧✐st❛ ❞❡ ✜❣✉r❛s ①✐✈

■♥tr♦❞✉çã♦ ✶

✶ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✸

✶✳✶ ❘❡❞❡s ❘❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✶ ❖ ▼✉♥❞♦ é P❡q✉❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✷ ❘❡❞❡s s♦❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✸ ❘❡❞❡s ❞❡ ✐♥❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✹ ❘❡❞❡s t❡❝♥♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✺ ❘❡❞❡s ❜✐♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✻ ❘❡❞❡s ❝❡❧✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✶✳✼ ❘❡❞❡s ❡❝♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✽ ❘❡❞❡s ❞❡ ❝❤❛♠❛❞❛s t❡❧❡❢ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ❖s Pr✐♠❡✐r♦s ▼♦❞❡❧♦s ❞❡ ❘❡❞❡s ❈♦♠♣❧❡①❛s ✶✹

✷✳✶ ▼♦❞❡❧♦ ❞❡ ❊r❞ös✲❘é♥②✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✶ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ▼♦❞❡❧♦ ❞❡ ❲❛tts ❡ ❙tr♦❣❛t③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸ ❘❡❞❡s ▲✐✈r❡ ❞❡ ❊s❝❛❧❛ ✷✸

✸✳✶ ▼♦❞❡❧♦ ❞❡ ❇❛r❛❜ás✐ ❡ ❆❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ▼♦❞❡❧♦ ❞❡ ❇❛rá❜ás✐✲❆❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷✳✶ ❈♦♠♣r✐♠❡♥t♦ ❞♦ ▼❡♥♦r ❈❛♠✐♥❤♦ ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷✳✷ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❆❣r❡❣❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ▼♦❞❡❧♦ ❞❡ ❇✐❛♥❝♦♥✐✲❇❛r❛❜ás✐ ♦✉ ▼♦❞❡❧♦ ❞❡ ◗✉❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸✳✶ ▼♦❞❡❧♦ ❞❡ ❊s❝❛❧❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✸✳✸✳✷ ❉✐str✐❜✉✐çã♦ ❞❡ q✉❛❧✐❞❛❞❡ ✉♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ▼♦❞❡❧♦ ❞❡ ❆✜♥✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹ ❊①t❡♥sã♦ ❞♦ ▼♦❞❡❧♦ ❞❡ ❆✜♥✐❞❛❞❡ ✹✼

❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s ✺✼

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✶✳✶ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ r❡❞❡✱ ❝♦♠ ✽ ♥ós ✭❡♠ ✈❡r♠❡❧❤♦✮ ❡ ✶✵ ❛r❡st❛s ✭❛③✉✐s✮✳✳ ✳ ✳ ✹ ✶✳✷ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣r❛çã♦ ❞♦ ✈ért✐❝❡ ν✱ é ♦❜t✐❞♦ q✉❛♥❞♦ ❝♦♥❤❡❝❡♠♦s

s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❡ s❛❜❡♠♦s ❝♦♠♦ ❡st❡s ❡stã♦ ❧✐❣❛❞♦s ❡♥tr❡ s✐✳ P❛r❛ ❡st❛ ✜❣✉r❛ t❡♠♦szν = 6 ❡ y= 5 q✉❡ ✐♠♣❧✐❝❛ ❡♠cν = 1/3✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✹❪✳ ✳ ✳ ✻

✶✳✸ ▼❛♣❛ ❞♦s ❊st❛❞♦s ❯♥✐❞♦s✳ ❖s ❡st❛❞♦s ❡♠ ✈❡r♠❡❧❤♦ r❡♣r❡s❡♥t❛♠ ❛ ♦r✐❣❡♠ ❞❛s ❝♦r✲ r❡s♣♦♥❞ê♥❝✐❛s ❡ ♦ ❡st❛❞♦ ❡♠ ✈❡r❞❡ r❡♣r❡s❡♥t❛ ♦ ❞❡st✐♥♦ ✜♥❛❧✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ❍✐st♦❣r❛♠❛ ♠♦str❛♥❞♦ q✉❡ ♥♦ ✜♥❛❧ ❞♦ ❡①♣❡r✐♠❡♥t♦ ❛s ❝❛rt❛s q✉❡ ❝❤❡❣❛r❛♠ ❛♦ ❞❡s✲

t✐♥♦ ♣❛ss❛r❛♠ ♣♦r s❡✐s ♣❡ss♦❛s ❡♠ ♠é❞✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✶ ■❧✉str❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❡✈♦❧✉çã♦ ❝♦♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝✉♣❛çã♦ ❞❛ ❧✐❣❛çã♦✱ ❞❡ ✉♠ ❣r❛❢♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❊r❞ös✲❘é♥②✐✳ ❈♦♠❡ç❛♠♦s ❝♦♠ N = 10 ♥ós ✐s♦❧❛❞♦s

✭✜❣✉r❛ ♠❛✐s ❛❝✐♠❛✱ ♦♥❞❡p= 0✮✱ ❡ ❝♦♥❡❝t❛♠♦s ❝❛❞❛ ♣❛r ❞❡ ♥ós ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡p✳

❆s ❞✉❛s ✜❣✉r❛s ♠❛✐s ❛❜❛✐①♦✱ ♠♦str❛♠ ❞♦✐s ❡stá❣✐♦s ❞✐❢❡r❡♥t❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❣r❛❢♦✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❛ p= 0.1 ❡ p= 0.15✳ ❖❜s❡r✈❛♠♦s ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ár✈♦r❡s

✭✉♠❛ ár✈♦r❡ ❞❡ ♦r❞❡♠ ✸✱ ❞❡s❡♥❤❛❞❛ ♣❡❧❛s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s✮ ❡ ❝✐❝❧♦s ✭✉♠ ❝✐❝❧♦ ❞❡ ♦r❞❡♠ ✸✱ ❞❡s❡♥❤❛❞♦ ♣❡❧❛s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s✮ ♥♦ ❣r❛❢♦✱ ❡ ✉♠ ❛❣❧♦♠❡r❛❞♦ ❝♦♥❡❝t❛♥❞♦ ♠❡t❛❞❡ ❞♦s ♥ós ❡♠p= 0.15 = 1.5/N✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✷✳✷ ❉✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❊r❞ös✲❘é♥②✐✱ ♣❛r❛ r❡❞❡s ❝♦♠N = 10000

❡ p = 0.0006 ✭❝ír❝✉❧♦s✮✱p = 0.001 ✭q✉❛❞r❛❞♦s✮ ❡p = 0.0015 ✭❞✐❛♠❛♥t❡s✮✳ ❖❜s❡r✈❛✲

♠♦s q✉❡ ♦s ♥ós tê♠✱ ❡♠ ♠é❞✐❛✱ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❧✐❣❛çõ❡s✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷✳✸ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ♠é❞✐♦ ❞♦s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s ❞❡ r❡❞❡s r❡❛✐s ✭sí♠❜♦✲ ❧♦s✮ ❝♦♠ ❛ ♣r❡✈✐sã♦ ❞❛ ❊q✉❛çã♦ ✷✳✸ ♣❛r❛ r❡❞❡s ❛❧❡❛tór✐❛s ✭❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛✮✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✹ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❛❣r❡❣❛çã♦ ❞❡ r❡❞❡s r❡❛✐s ❡ ❞❡ ❣r❛❢♦s ❛❧❡❛tór✐♦s✳ ❆ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ❝♦rr❡s♣♦♥❞❡ ❛ ♣r❡✈✐sã♦ ❞❛ ❊q✉❛çã♦ ✷✳✹✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷✳✺ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ r❡❝♦♥❡❝t❛r ♦s ✈ért✐❝❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ❲❛tts✲❙tr♦❣❛t③ q✉❡ tr❛♥s❢♦r♠❛ ✉♠❛ r❡❞❡ ✭❝✐r❝✉❧❛r✮ r❡❣✉❧❛r ❡♠ ✉♠❛ r❡❞❡ ❛❧❡❛tór✐❛ s❡♠ ❢❛③❡r ♠✉❞❛♥ç❛ ♥♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡ ♦✉ ❧✐❣❛çã♦✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ✉♠❛ r❡❞❡ ❞❡ N = 20 ♥ós✱ ❝❛❞❛ ✉♠ ❝♦♥❡❝t❛❞♦

❛♦s s❡✉s4✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s ✭❞♦✐s ❞❡ ❝❛❞❛ ❧❛❞♦ ❞♦ ✈ért✐❝❡ ❡♠ q✉❡stã♦✱ ♦✉ s❡❥❛✱

z=k/2✮✱ ❡♠ s❡❣✉✐❞❛ ❡s❝♦❧❤❡♠♦s ✉♠ ✈ért✐❝❡ ❡ ✉♠❛ ❧✐❣❛çã♦ q✉❡ ❝♦♥❡❝t❛ ✉♠ ❞❡ s❡✉s

✈✐③✐♥❤♦s✳ ❈♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ p r❡❝♦♥❡❝t❛♠♦s ❡ss❛ ❧✐❣❛çã♦ ❛ ✉♠ ♦✉tr♦ ✈ért✐❝❡

❡s❝♦❧❤✐❞♦ ❛❧❡❛t♦r✐❛♠❡♥t❡✳ ❋❛③❡♠♦s ✐ss♦ ❛té q✉❡ t♦❞❛s ❛s ❧✐❣❛çõ❡s✱ ❞❛ r❡❞❡ ♦r✐❣✐♥❛❧✱ t❡♥❤❛♠ s✐❞♦ ❝♦♥s✐❞❡r❛❞❛s✳ ❘❡❛❧✐③❛çõ❡s ❞❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦ sã♦ ♠♦str❛❞♦s ❛q✉✐ ♥❡st❛ ✜❣✉r❛✱ ♣❛r❛ três ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❞❡ p✳ ◆❛ ♣r✐♠❡✐r❛✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱

❢♦✐ ✉s❛❞♦ p = 0✱ ❝♦♠ ✐ss♦ ❛ r❡❞❡ ♦r✐❣✐♥❛❧ ♣❡r♠❛♥❡❝❡ ✐♥❛❧t❡r❛❞❛ ✭r❡❞❡ r❡❣✉❧❛r✮✳ ◆❛

s❡❣✉♥❞❛✱ ❝♦♠ ♦ ✈❛❧♦r ❞❡ p ♥♦ ✐♥t❡r✈❛❧♦ 0 < p < 1 ✭r❡❞❡ ❞❡ ♠✉♥❞♦ ♣❡q✉❡♥♦✮✳ ◆❛

t❡r❝❡✐r❛✱ é ♦❜t✐❞❛ ✉♠❛ r❡❞❡ t♦t❛❧♠❡♥t❡ ❛❧❡❛tór✐❛✱ ❝♦♠ p= 1✱ ♦♥❞❡ t♦❞❛s ❛s ❧✐❣❛çõ❡s

❢♦r❛♠ ❛❧❡❛t♦r✐❛♠❡♥t❡ r❡❝♦♥❡❝t❛❞❛s✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✻ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ C(p) ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ♠é❞✐♦ ❞♦s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s l(p)

❞♦ ♠♦❞❡❧♦ ❞❡ ❲❛tts✲❙tr♦❣❛t③✳ ❖s ❞❛❞♦s ❢♦r❛♠ ♥♦r♠❛❧✐③❛❞♦s ♣❡❧♦s ✈❛❧♦r❡s ❞❡ C(0)

❡ l(0)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦❞❡ s❡r ✈✐st♦ ❛ rá♣✐❞❛ q✉❡❞❛ ❞❡ l(p)✱ ❞❡♠♦♥str❛♥❞♦ q✉❡ ❛

r❡❞❡ ❛♣r❡s❡♥t❛ ♦ ❡❢❡✐t♦ ♠✉♥❞♦ ♣❡q✉❡♥♦✳ ❉✉r❛♥t❡ ❡ss❛ q✉❡❞❛ ❞❡l(p)✱C(p)♣❡r♠❛♥❡❝❡

♣r❛t✐❝❛♠❡♥t❡ ❝♦♥st❛♥t❡✱ ♦ q✉❡ ✐♥❞✐❝❛ q✉❡ ❛ tr❛♥s✐çã♦ ♣❛r❛ ✉♠❛ r❡❞❡ ❞❡ ♠✉♥❞♦ ♣❡✲ q✉❡♥♦ é q✉❛s❡ q✉❡ ✐♠♣❡r❝❡♣tí✈❡❧ ❛ ♥í✈❡❧ ❧♦❝❛❧✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✼❪✳ ✳ ✳ ✳ ✷✶ ✷✳✼ ❈♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❲❛tts✲❙tr♦❣❛t③✳

❘❡❞❡ ❝♦♠ N = 10000 ❡ k = 6 ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ p✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡

❞❡ r❡❢✳ ❬✶✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✸✳✶ ❆ ✜❣✉r❛ ♠♦str❛ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞❡ ❞♦✐s sít✐♦s✱ ❛❞✐❝✐♦♥❛❞♦s ♥❛ r❡❞❡ ❡♠ t1 = 5 ❡ t2 = 95✳ ❆ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ t❡♠ ✐♥❝❧✐♥❛çã♦ 0.5 ❝♦♠♦ ♣r❡✈✐st♦ ♣❡❧❛

❊q✉❛çã♦ ✸✳✺✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✹✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✷ ❉✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❛r❛❜ás✐✲❆❧❜❡rt✱ ❝♦♠N = 300000♦♥❞❡✿

❝ír❝✉❧♦s ✭m0 = m = 1✮❀ q✉❛❞r❛❞♦s ✭m0 = m = 3✮❀ ❞✐❛♠❛♥t❡s ✭m0 = m = 5✮ ❡

tr✐â♥❣✉❧♦s ✭m0 = m = 7✮✳ ❆ ✐♥❝❧✐♥❛çã♦ ❞❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ é γ = 2.9✱ ❝♦♥❝♦r❞❛♥❞♦

❝♦♠ ❛ ♣r❡✈✐sã♦ t❡ór✐❝❛ ❞❛ ❊q✉❛çã♦ ✸✳✶✵✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✸ ●rá✜❝♦ ❞♦ ♠❡♥♦r ❝❛♠✐♥❤♦ ♠é❞✐♦l♣❡❧♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡N✱ ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❛r❛❜ás✐✲

❆❧❜❡rt ❝♦♠hki= 4✱ ❝♦♠♣❛r❛❞♦ ❝♦♠ ✉♠ ❣r❛❢♦ ❛❧❡❛tór✐♦ ❞❡ ✐❣✉❛❧ t❛♠❛♥❤♦ ❡ ♠❡s♠❛

❝♦♥❡❝t✐✈✐❞❛❞❡ ♠é❞✐❛✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸✳✹ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ✈❡rs✉s ♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❛r❛❜ás✐✲❆❧❜❡rt ❝♦♠ hki = 4✱ ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ❞❡ ✉♠ ❣r❛❢♦ ❛❧❡❛tór✐♦✱

Crand≃ hki/N✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✺ ❙✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠m= 1 ❡N = 105 _{♠♦str❛♥❞♦ ❛ ❞❡♣❡♥❞ê♥❝✐❛}

t❡♠♣♦r❛❧ ❞❛ ❝♦♥❡❝t✐✈✐❞❛❞❡kη(t)✱ ♣❛r❛ sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡ η = 0.3✱ 0.6 ❡ 0.9.◆♦t❡ q✉❡ ❡♠ ❝❛❞❛ ❝❛s♦ kη(t) s❡❣✉❡ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❡ ♦ ❡①♣♦❡♥t❡ ❞✐♥â♠✐❝♦β(η)✱ ❞❛❞♦ ♣❡❧❛ ✐♥❝❧✐♥❛çã♦ ❞❡ k(t)✱ ❛✉♠❡♥t❛ ❝♦♠ η✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✻ ❈♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ C ❝♦♠ t → ∞✱ ♥♦t❡ q✉❡ ❤á ❛❝♦r❞♦ ❝♦♠ ❛ ♣r❡✈✐sã♦

❛♥❛❧ít✐❝❛✱ ❧✐♥❤❛ tr❛❝❡❥❛❞❛✱ ✭❊q✉❛çã♦ ✸✳✷✺✮✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✺✶❪✳✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✼ ❉✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♥♦ ♠♦❞❡❧♦ ❞❡ q✉❛❧✐❞❛❞❡✱ ♦❜t✐❞❛ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠

m= 1❡ N = 105_{✳ ❖s ❝ír❝✉❧♦s só❧✐❞♦s r❡♣r❡s❡♥t❛♠ ❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ q✉❡ ❡stá ❞❡}

❛❝♦r❞♦ ❝♦♠ ❛ ♣r❡❞✐çã♦ t❡ór✐❝❛ ❞❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ✸✳✷✻ ✭❧✐♥❤❛ ❛③✉❧✮✱ ❝♦♠γ = 2,255✳

❆ ❧✐♥❤❛ ♣r❡t❛ r❡♣r❡s❡♥t❛ ✉♠ ❛❥✉st❡ ❝♦♠ P(k)∼k−2.255 s❡♠ ❛ ❝♦rr❡çã♦ ❧♦❣❛r✐t♠✐❝❛✳

❆ ❧✐♥❤❛ ✈❡r♠❡❧❤❛ ❝♦rr❡s♣♦♥❞❡ ❛ P(k) ∼ k−3 ❝♦♠♦ ♣r❡✈ê ♦ ♠♦❞❡❧♦ ❞❡ ❡s❝❛❧❛ ❧✐✈r❡✱

♦♥❞❡ t♦❞❛s ❛s q✉❛❧✐❞❛❞❡s sã♦ ✐❣✉❛✐s✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✽ ❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ❡①♣♦❡♥t❡ ❞✐♥â♠✐❝♦ β(η) ❝♦♠ ♦ ♣❛râ♠❡tr♦ ❞❛ q✉❛❧✐❞❛❞❡ η ♣❛r❛ ♦

❝❛s♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✉♥✐❢♦r♠❡✱ ρ(η) = constante. ❖s ❝ír❝✉❧♦s ❢♦r❛♠ ♦❜t✐❞♦s

♣♦r s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✱ ❡♥q✉❛♥t♦ ❛ ❧✐♥❤❛ ❝♦rr❡s♣♦♥❞❡ ❛ ♣r❡❞✐çã♦ ❛♥❛❧ít✐❝❛ β(η) =

η/1,255✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✹✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✸✳✾ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡✱ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠ m = 1 ❡ N = 105✱ ❡♠ ❡s❝❛❧❛ ❧♦❣✲❧♦❣✳ ❋✐❣✉r❛

♣r♦✈❡♥✐❡♥t❡ ❞❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❛✉rí❝✐♦ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✸✳✶✵ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ q✉❛❧✲ ✐❞❛❞❡ ✉♥✐❢♦r♠❡ ✭ r❡❞❡ ❝♦♠ N = 105 _{❡ m = 1✮✳ ❖❜s❡r✈❛♠♦s q✉❡ ❛ ❡✈♦❧✉çã♦ ❞❛}

❝♦♥❡❝t✐✈✐❞❛❞❡ s❡❣✉❡ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ♥♦ t❡♠♣♦ ❡ q✉❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞♦s ❣rá✜❝♦s ✭❡①♣♦❡♥t❡ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✮ ❝r❡s❝❡ ❛té ♦ sít✐♦ ❛♣r❡s❡♥t❛r ✉♠ ♣❛râ♠❡tr♦ ❞❡ q✉❛❧✐❞❛❞❡ ✐❣✉❛❧ ❛0.5✳ ❆ ♣❛rt✐r ❞❛í✱ ♦✉ s❡❥❛✱ ♣❛r❛ sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s s✉♣❡r✐♦r❡s ❛ ✵✳✺✱ t❡♠♦s

♦ ❡①♣♦❡♥t❡ ❞❡❝r❡s❝❡♥❞♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ s❡✉s ✈❛❧♦r❡s sã♦✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ✐❣✉❛✐s ❛♦s ❞♦s sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s ❛❜❛✐①♦ ❞❡ ✵✳✺✳ ❖s ❣rá✜❝♦s ✐♥t❡r♥♦s ♥♦s ❛❥✉❞❛♠ ❛ ♦❜s❡r✈❛r ✐ss♦✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❛✉rí❝✐♦ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✸✳✶✶ ❙✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠m= 1❡N = 105✱ ♠♦str❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡

♦ ❡①♣♦❡♥t❡ ❞✐♥â♠✐❝♦ β✱ ✈❡rs✉s ♦s ♣❛râ♠❡tr♦s ❞❡ q✉❛❧✐❞❛❞❡s✱ η✳ ❖ ❣rá✜❝♦ ❡①♣❧✐❝✐t❛

q✉❡ sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s ❞❡ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ✐♥t❡r♠❞✐ár✐❛ sã♦ ♦s q✉❡ ❝♦♥s❡❣✉❡♠ ♠❛✐s ❧✐❣❛çõ❡s ❛♦ ❧♦♥❣♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ r❡❞❡✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❛✉rí❝✐♦ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✸✳✶✷ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ q✉❛❧✐✲ ❞❛❞❡ ✉♥✐❢♦r♠❡✳ ▼❡♥♦r ❝❛♠✐♥❤♦ ♠é❞✐♦ ✈❡rs✉s ♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡ ♥✉♠ ❣rá✜❝♦ ❧✐♥❡❛r✲ ❧♦❣ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡m❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ❧❡❣❡♥❞❛ ❛❝✐♠❛✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡

❞❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❛✉rí❝✐♦ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✸ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ q✉❛❧✲

✐❞❛❞❡ ✉♥✐❢♦r♠❡✳ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ♠é❞✐♦ ✈❡rs✉s ♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡ ♥✉♠ ❣rá✲ ✜❝♦ ❧♦❣✲❧♦❣ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ m ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ❧❡❣❡♥❞❛ ❛❝✐♠❛✳ s❋✐❣✉r❛

♣r♦✈❡♥✐❡♥t❡ ❞❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❛✉rí❝✐♦ ❬✺✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✹✳✶ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞❛ ❡①t❡♥sã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡✱ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠ m= 1 ❡ N = 105✱ ❡♠ ❡s❝❛❧❛ ❧♦❣✲❧♦❣✱ ❛

♠❡❞✐❞❛ q✉❡ ♦ ❡①♣♦❡♥t❡ σ q✉❡ ✐♥s❡r✐♠♦s ♥❛ ❛✜♥✐❞❛❞❡ ✈❛✐ ❛✉♠❡♥t❛♥❞♦✱ ♦ ❡①♣♦❡♥t❡ γ

❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ✈❛✐ ❞✐♠✐♥✉✐♥❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✹✳✷ ❘❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ❞❛ ❡①t❡♥sã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛✜♥✐❞❛❞❡ ❝♦♠ ❞✐s✲ tr✐❜✉✐çã♦ ❞❡ q✉❛❧✐❞❛❞❡ ✉♥✐❢♦r♠❡ ✭ r❡❞❡ ❝♦♠ N = 105 _{❡ m = 1✮✳ ❖❜s❡r✈❛♠♦s q✉❡ ❛}

❡✈♦❧✉çã♦ ❞❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ s❡❣✉❡ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ♥♦ t❡♠♣♦ ❡ q✉❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞♦s ❣rá✜❝♦s ✭❡①♣♦❡♥t❡ ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✮ ❝r❡s❝❡ ❛té ♦ sít✐♦ ❛♣r❡s❡♥t❛r ✉♠ ♣❛râ♠❡tr♦ ❞❡ q✉❛❧✐❞❛❞❡ ✐❣✉❛❧ ❛ 0.5✳ ❆ ♣❛rt✐r ❞❛í✱ ♦✉ s❡❥❛✱ ♣❛r❛ sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s s✉♣❡r✐♦r❡s ❛

✵✳✺✱ t❡♠♦s ♦ ❡①♣♦❡♥t❡ ❞❡❝r❡s❝❡♥❞♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ s❡✉s ✈❛❧♦r❡s sã♦✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ✐❣✉❛✐s ❛♦s ❞♦s sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s ❛❜❛✐①♦ ❞❡ ✵✳✺✳ ❖s ❣rá✜❝♦s ✐♥t❡r♥♦s ♥♦s ❛❥✉❞❛♠ ❛ ♦❜s❡r✈❛r ✐ss♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✸ ❙✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ♣❛r❛ ✉♠❛ r❡❞❡ ❝♦♠m= 1❡N = 105✱ ♠♦str❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡

♦ ❡①♣♦❡♥t❡ ❞✐♥â♠✐❝♦ β✱ ✈❡rs✉s ♦s ♣❛râ♠❡tr♦s ❞❡ q✉❛❧✐❞❛❞❡s✱ η✳ ❖ ❣rá✜❝♦ ❡①♣❧✐❝✐t❛

q✉❡ sít✐♦s ❝♦♠ q✉❛❧✐❞❛❞❡s ❞❡ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ✐♥t❡r♠❞✐ár✐❛ sã♦ ♦s q✉❡ ❝♦♥s❡❣✉❡♠ ♠❛✐s ❧✐❣❛çõ❡s ❛♦ ❧♦♥❣♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ r❡❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✹ r❡❧❛çã♦ ❡♥tr❡ ♦s ❡①♣♦❡♥t❡s γ ❡ σ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✹✳✺ r❡❧❛çã♦ ❡♥tr❡ |ηi−ηj|❡ |ηi−ηj|σ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

■◆❚❘❖❉❯➬➹❖

❖ ♠❛t❡♠át✐❝♦ ❊✉❧❡r ❢♦✐ ♦ r❡s♣♦♥sá✈❡❧ ♣❡❧♦s ♣r✐♠❡✐r♦s ♣❛ss♦s ❞❛ t❡♦r✐❛ ❞❛s r❡❞❡s ❡ ❢♦✐ ♦ ❝r✐❛❞♦r ❞❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✳ ❊✉❧❡r ❡s❝r❡✈❡✉ ✉♠ ❝✉rt♦ ❛rt✐❣♦ s♦❜r❡ ✉♠ ❞✐✈❡rt✐❞♦ ♣r♦❜❧❡♠❛ q✉❡ t❡✈❡ ❝♦♠♦ ♣❛❧❝♦ ❑♦♥✐❣s❜❡r❣✱ ✉♠❛ ❝✐❞❛❞❡ ♥ã♦ ♠✉✐t♦ ❞✐st❛♥t❡ ❞❛ ❝❛s❛ ❞❡ ❊✉❧❡r ❡♠ ❙t✳ P❡t❡rs❜✉r❣✳ ❉❡✈✐❞♦ ❛ ❡①✐stê♥❝✐❛ ❞♦ r✐♦ Pr❡❣❡❧ ❡ ❞❛ ✈♦❝❛çã♦ ❞❛ ❝✐❞❛❞❡ ♣❛r❛ ♦s ♥❡❣ó❝✐♦s✱ ✉♠❛ ✈❡③ q✉❡ ♣♦ss✉í❛ ✉♠❛ ❢r♦t❛ ♠✉✐t♦ r❡q✉✐s✐t❛❞❛ ❞❡ ♥❛✈✐♦s✱ ♦s ♠❡r❝❛❞♦r❡s ❧♦❝❛✐s ❡ s❡✉s ❢❛♠✐❧✐❛r❡s ♣♦❞✐❛♠ ❞❡s❢r✉t❛r ❞❡ ✉♠❛ ✈✐❞❛ ❝♦♥❢♦rtá✈❡❧✳ ❆s ❜♦❛s ❝♦♥❞✐çõ❡s ❡❝♦♥ô♠✐❝❛s ❞❛ ❝✐❞❛❞❡ ❡r❛♠ t❛✐s q✉❡ ♦s r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ♠❡s♠❛ ❝♦♥str✉ír❛♠ ♥❛❞❛ ♠❡♥♦s ❞♦ q✉❡ s❡t❡ ♣♦♥t❡s ❡♠ ✈ár✐♦s ♣♦♥t♦s ❞❛ ❝✐❞❛❞❡✳ ❆ ♠❛✐♦r✐❛ ❞❛s ♣♦♥t❡s✱ ✺ ❞❡ ✉♠ t♦t❛❧ ❞❡ ✼✱ ❝♦♥❡❝t❛✈❛♠ ❛ ❡❧❡❣❛♥t❡ ✐❧❤❛ ❑♥❡✐♣❤♦❢✱ ❝❡r❝❛❞❛ ♣❡❧♦ r✐♦ Pr❡❣❡❧✱ ❝♦♠ ♦ r❡st❛♥t❡ ❞❛ ❝✐❞❛❞❡✳ ❆s ♦✉tr❛s ❞✉❛s ♣♦♥t❡s ❧✐❣❛✈❛♠ ♦s ♦✉tr♦s ♣♦♥t♦s ❞❛ ❝✐❞❛❞❡✳ ❆ ♣♦♣✉❧❛çã♦ ❞❡ ❑♦♥✐❣s❜❡r❣✱ ❛♣r♦✈❡✐t❛♥❞♦ ♦ ❝❧✐♠❛ ❞❡ ❤❛r♠♦♥✐❛ ❡ ❛ ♣❛③ ❞❛ ❝✐❞❛❞❡✱ ♥♦ t❡♠♣♦ ❧✐✈r❡ s❡ ❞✐✈❡rt✐❛ ❝♦♠ ♦ s❡❣✉✐♥t❡ q✉❡❜r❛✲❝❛❜❡ç❛s✿ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ❝❛♠✐♥❤♦ ❛tr❛✈❡ss❛♥❞♦ ❛s s❡t❡ ♣♦♥t❡s s❡♠ ♥✉♥❝❛ ❛tr❛✈❡ss❛r ✉♠❛ ♠❡s♠❛ ♣♦♥t❡ ❞✉❛s ✈❡③❡s❄✳ ❈♦♥t✉❞♦✱ ♥✐♥❣✉é♠ ❢♦✐ ❝❛♣❛③ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❝❛♠✐♥❤♦ q✉❡ s❛t✐s✜③❡ss❡ ♦ q✉❡❜r❛✲ ❝❛❜❡ç❛s ❛té q✉❡ ✉♠❛ ♥♦✈❛ ♣♦♥t❡ ❢♦ss❡ ❝♦♥tr✉í❞❛ ❡♠ ✶✽✼✺✳ ❊♥tr❡t❛♥t♦✱ q✉❛s❡ ✶✺✵ ❛♥♦s ❛♥t❡s ❞❛ ❝♦♥str✉çã♦ ❞❛ ♥♦✈❛ ♣♦♥t❡✱ ❡♠ ✶✼✸✻✱ ❊✉❧❡r ♦❢❡r❡❝❡✉ ✉♠❛ r✐❣♦r♦s❛ ♣r♦✈❛ ♠❛t❡♠át✐❝❛ ❞✐③❡♥❞♦ q✉❡ ♥ã♦ ❡①✐st✐❛ ✉♠ ❝❛♠✐♥❤♦ q✉❡ ♣❛ss❛ss❡ ♣♦r t♦❞❛s ❛s s❡t❡ ♣♦♥t❡s ✉♠❛ ú♥✐❝❛ ✈❡③✳ ❆❧é♠ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛s ♣♦♥t❡s ❞❡ ❑♦♥✐❣s❜❡r❣✱ ❊✉❧❡r✱ ❞❡ ❢♦r♠❛ ♥ã♦ ✐♥t❡♥❝✐♦♥❛❧✱ ✐♥✐❝✐♦✉ ✉♠❛ ♥♦✈❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✳ ❊ss❛ t❡♦r✐❛ é ❤♦❥❡ ❛ ❜❛s❡ ❞❡ t♦❞♦ ♥♦ss♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ r❡❞❡s✳ ❖ ❣r❛❢♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ós✱ ❝♦♥❡❝t❛❞♦s ♣♦r ❛r❡st❛s q✉❡✱ ❡♠ ❝♦♥❥✉♥t♦✱ ❢♦r♠❛♠ ✉♠❛ r❡❞❡✳ ❆ ♣❛rt✐r ❞❡ ❊✉❧❡r✱ ♦✉tr♦s ❡st✉❞✐♦s♦s s❡ ❞❡❞✐❝❛r❛♠ ❛ ❝♦♠♣r❡❡♥❞❡r ♦s ✈ár✐♦s t✐♣♦s ❞❡ ❣r❛❢♦s ❡ ❝♦♠♦ s❡ ❞❛✈❛ s✉❛ ❝♦♥str✉çã♦✱ ♦✉ s❡❥❛✱ ❝♦♠♦ s❡✉s ♥ós s❡ ❛❣r✉♣❛✈❛♠✳

✷

◆♦s ú❧t✐♠♦s ❛♥♦s✱ ✈ár✐❛s r❡❞❡s r❡❛✐s tê♠ s✐❞♦ ❡st✉❞❛❞❛s ♣❡❧❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✱ s❡♥❞♦ q✉❡ ❛❧❣✉♠❛s ❞❡❧❛s s❡ t♦r♥❛r❛♠ ❢❛♠♦s❛s✳ ▼❛r❦ ◆❡✇♠❛♥ ♣r♦♣ôs ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s r❡❞❡s ❝♦♠♣❧❡①❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ t❡❝♥♦❧ó❣✐❝❛s✱ ❜✐♦❧ó❣✐❝❛s✱ s♦❝✐❛✐s ❡ ❞❡ ✐♥❢♦r♠❛çã♦✳ ❊♠❜✲ ♦r❛ ♥♦ ♣❛ss❛❞♦ ❛s r❡❞❡s ❢♦ss❡♠ ✈✐st❛s ❝♦♠♦ ♦❜❥❡t♦s ❡str✉t✉r❛✐s✱ ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ✜①❛❞❛s ♥♦ t❡♠♣♦✱ ♦s ♥♦✈♦s ❡st✉❞♦s ♠♦str❛♠ q✉❡ ❡❧❛s sã♦ ♥❛ ✈❡r❞❛❞❡ ❡❧❡♠❡♥t♦s ❞✐♥â♠✐❝♦s✳ ❉✐❢❡r❡♥t❡s ❣r❛♥❞❡③❛s ♦✉ ♣❛râ♠❡tr♦s sã♦ ✉s❛❞♦s ♥❛ ❞❡s❝r✐çã♦ ♦✉ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s r❡❞❡s ❝♦♠♣❧❡①❛s✱ t❛❧✈❡③ ❛ ❣r❛♥❞❡③❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ s❡❥❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡✱ q✉❡ é ❞❡✜♥✐❞❛ ❝♦♠♦ s❡♥❞♦ ❛ ❢r❛çã♦ ❞❡ ✈ért✐❝❡s ❞❛ r❡❞❡ q✉❡ tê♠ ❝♦♥❡❝t✐✈✐❞❛❞❡✱ ♦✉ ❛✐♥❞❛✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ✉♠ ✈ért✐❝❡ ❡s❝♦❧❤✐❞♦ ❛♦ ❛❝❛s♦ t❡♥❤❛ ✉♠❛ ❝❡rt❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ✳ ❖✉tr❛ ❣r❛♥❞❡③❛ ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ♥❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s r❡❞❡s é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ q✉❡ é ♦❜t✐❞♦ ❝❛❧❝✉❧❛♥❞♦✲s❡ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ q✉❡ ♦s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❞❡ ✉♠ ❞❡✲ t❡r♠✐♥❛❞♦ ✈ért✐❝❡ t❛♠❜é♠ ❡st❡❥❛♠ ❝♦♥❡❝t❛❞♦s ❡♥tr❡ s✐✳ ❊♠ ❛❧❣✉♠❛s r❡❞❡s r❡❛✐s ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡✱ P(k)✱ ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♥ó ❡s❝♦❧❤✐❞♦ ❛❧❡❛t♦r✐❛♠❡♥t❡✱ t❡♠ ❝♦♥❡❝t✐✈✐❞❛❞❡ k ❜❡♠ ❞❡✜♥✐❞❛ ❞❡s✈✐❛♥❞♦✲s❡ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐s✲ s♦♥✱ ❡①✐❜✐❞❛ ♣❡❧❛s r❡❞❡s ❛❧❡❛tór✐❛s✳ ▼✉✐t❛s ❞❛s r❡❞❡s r❡❛✐s ❛♣r❡s❡♥t❛♠ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ✉♠❛ ❝❛✉❞❛ ❡♠ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✱ ❡①♣r❡ss❛ ♣♦r

P(k)∼k−γ

♦♥❞❡ ♦ ❡①♣♦❡♥t❡γ t❡♠ ✈❛❧♦r✱ ❡♠ ❣❡r❛❧✱ ❡♥tr❡2 ❡ 3✳ ❆❧é♠ ❞✐ss♦ ❛❧❣✉♠❛s r❡❞❡s r❡❛✐s t❛♠❜é♠ sã♦ ❝❛r❛❝t❡r✐③❛❞❛s ♣♦r ❛♣r❡s❡♥t❛r❡♠ ❝❛♠✐♥❤♦s r❡❧❛t✐✈❛♠❡♥t❡ ❝✉rt♦s ❡♥tr❡ ❞♦✐s ♥ós q✉❛✐s✲ q✉❡r ✭♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠✉♥❞♦ ♣❡q✉❡♥♦✮ ❡ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❡ ❝✐r❝✉✐t♦s r❡tr❛t❛❞♦s ♣♦r ♦✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❝❤❛♠❛❞❛ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦✳

❈❆P❮❚❯▲❖ ✶

◆❖▼❊◆❈▲❆❚❯❘❆ ❊ ❈❖◆❈❊■❚❖❙

❇➪❙■❈❖❙

❈♦♠♦ é ❤❛❜✐t✉❛❧ ❞♦ ❤♦♠❡♠✱ ❞❡♣♦✐s ❞❡ ♦❜t❡r♠♦s ❛s r❡s♣♦st❛s ♣❛r❛ ❛❧❣✉♠❛s ♣❡r❣✉♥t❛s ❧♦❣♦ s✉r❣❡♠ ♥♦✈❛s ♣❡r❣✉♥t❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ❛ ❞✐stâ♥❝✐❛ ♠é❞✐❛ ❡♥tr❡ ✷ ♣❡ss♦❛s q✉❛✐sq✉❡r ♥♦ ♠✉♥❞♦ é s❡♠♣r❡ ❞❡ ✻ ♣❛ss♦s✱ ♣♦❞❡♠♦s ♥♦s ♣❡r❣✉♥t❛r q✉❛❧ ❛ ❡str✉t✉r❛ ❞❛ r❡❞❡ ❢♦r♠❛❞❛ ♣♦r ❡ss❛s ♣❡ss♦❛s❄✳ ❆s r❡s♣♦st❛s ❛ ❡ss❛s ♣❡r❣✉♥t❛s ❝♦♠❡ç❛r❛♠ ❛ s✉r❣✐r ❤á ❛❧❣✉♥s ❛♥♦s ❛trás✳ ❍♦❥❡ ❡♠ ❞✐❛ ❛❧é♠ ❞❡ s❛❜❡r♠♦s q✉❡ ❛s r❡❞❡s s♦❝✐❛✐s sã♦ ♠✉♥❞♦s ♣❡q✉❡♥♦s✱ s❛❜❡♠♦s t❛♠❜é♠ ❝♦♠♦ ♣♦❞❡♠♦s ❝❧❛ss✐✜❝á✲❧❛s ❡ ❝❛❧❝✉❧❛r ✈ár✐❛s ❞❡ s✉❛s ❣r❛♥❞❡③❛s✳

• ❖ q✉❡ sã♦ r❡❞❡s ❄

❙ã♦ ❢♦r♠❛s ❞❡ t❡♥t❛r ♠♦❞❡❧❛r ❛ ♥❛t✉r❡③❛✱ ♦♥❞❡ ✉♠ ❞❛❞♦ ❣r✉♣♦ ❞❡ ❡❧❡♠❡♥t♦s ❝♦♥✲ st✐t✉✐♥t❡s ❞❡ ✉♠ s✐st❡♠❛ ♥❛t✉r❛❧ q✉❛❧q✉❡r ❡stã♦ ❧✐❣❛❞♦s✳ ❉❡✈❡♠♦s ❞❡t❡r♠✐♥❛r r❡❣r❛ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❧✐❣❛çã♦ ❡♥tr❡ ❡ss❡s ❡❧❡♠❡♥t♦s q✉❡ ♣♦❞❡♠ s❡r ♣❡ss♦❛s✱ ♣r♦t❡í♥❛s✱ ❛ ✐♥t❡r♥❡t✱ ❛❡r♦♣♦rt♦s✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✳ ❆s ❧✐❣❛çõ❡s ❞❡♣❡♥❞❡♠ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ q✉❡ s❡ q✉❡r ❡st✉❞❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❡ss♦❛s ♣♦❞❡♠ ❡st❛r ❧✐❣❛❞❛s ♣♦r ❝♦♥❡①õ❡s ❞❡ ❛♠✐③❛❞❡ ♦✉ ❞❡✈✐❞♦ ❛♦ ❝♦♠♣❛rt✐❧✲ ❤❛♠❡♥t♦ ❞❡ ❛❧❣✉♠❛ ♦♣✐♥✐ã♦ ❡ ❛❡r♦♣♦rt♦s ❡st❛rã♦ ❧✐❣❛❞♦s s❡ ♣♦ss✉❡♠ r♦t❛s q✉❡ ♦s ❝♦♥❡❝t❛♠✳ ❯♠❛ ✐❧✉str❛çã♦ ❞❡ r❡❞❡ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ❋✐❣✉r❛ ✶✳✶ ♦♥❞❡ t❡♠♦s ✉♠❛ r❡❞❡ ❝♦♠ ✽ ♥ós ❡ ✶✵ ❛r❡st❛s✳

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✹

❋✐❣✉r❛ ✶✳✶✿ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ♣❡q✉❡♥❛ r❡❞❡✱ ❝♦♠ ✽ ♥ós ✭❡♠ ✈❡r♠❡❧❤♦✮ ❡ ✶✵ ❛r❡st❛s ✭❛③✉✐s✮✳

• ●r❛❢♦s

❯♠ ❣r❛❢♦ ❝♦♥s✐st❡ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❡ ♥ã♦ ✈❛③✐♦s✱ ♦♥❞❡ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s é ❢♦r♠❛❞♦s ❞❡ ♣♦♥t♦s ❞❡♥♦♠✐♥❛❞♦s ✈ért✐❝❡s ✭q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ sít✐♦s✮ ❡ ♦✉tr♦ ❞❡ s❡❣♠❡♥t♦s ❞❡♥♦♠✐♥❛❞♦s ❛r❡st❛s ✭q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ❧✐❣❛çã♦✮ q✉❡ ❝♦♥❡❝t❛♠ ❡st❡s ✈ért✐❝❡s✳ ❉❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✱ ✉♠ ❣r❛❢♦ ❝♦♥s✐st❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❝♦♥❡❝t❛❞♦s ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❧✐♥❤❛s✳ P♦r ❡①❡♠♣❧♦✱ ♦s ♣♦♥t♦s ♣♦❞❡♠ r❡♣r❡s❡♥t❛r ❝✐❞❛❞❡s ❡ ♦s s❡❣♠❡♥t♦s ❡str❛❞❛s✳

• ❉✐str✐❜✉✐çã♦ ❞❡ ❈♦♥❡❝t✐✈✐❞❛❞❡

❊❧❛ ♥♦s ❞✐③ ❝♦♠♦ s❡ ❞✐str✐❜✉❡♠ ❛s ❧✐❣❛çõ❡s ♣❡❧♦s ✈ért✐❝❡s✳ ❙ã♦ ❡①❡♠♣❧♦s tí♣✐❝♦s ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♦❜s❡r✈❛❞❛s ♥♦ ❡st✉❞♦ ❞❡ r❡❞❡s ❝♦♠♣❧❡①❛s✿

✭✶✮ ❉✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✿

❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ t✐♣♦ P♦✐ss♦♥ é ❡①♣r❡ss❛ ♣❡❧❛ ❊q✉❛çã♦ ✶✳✶❀

P(k) = e

−hki_hkik

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✺

♦♥❞❡hkié ❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♠é❞✐❛ ❬✷❪✳ ❯♠ ❣r❛❢♦ ❛❧❡❛tór✐♦ t❡♠ s✉❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ s❡❣✉✐♥❞♦ ✉♠❛

❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥ q✉❛♥❞♦ s❡✉ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ ❡ s✉❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♠é❞✐❛✱hki✱ é ✜①❛✳ ❊ss❡ ✈í♥❝✉❧♦ ♣❛r❛ hki ❝♦♥st✐t✉✐ ❛ ❡s❝❛❧❛ tí♣✐❝❛ ❞❛ ❞✐str✐❜✉✐çã♦✳

✭✷✮ ❉✐str✐❜✉✐çã♦ t✐♣♦ ▲❡✐ ❞❡ P♦tê♥❝✐❛✿

❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ t✐♣♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ é ❡①♣r❡ss❛ ♣❡❧❛ ❊q✉❛çã♦ ✶✳✷❀

P(k)∝k−γ ✭✶✳✷✮

❝♦♠ k 6= 0 ❡ γ s❡♥❞♦ ♦ ❡①♣♦❡♥t❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❬✷❪✳ ❆♦ ❝♦♥trár✐♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐s✲ s♦♥✱ r❡❞❡s q✉❡ ♣♦ss✉❡♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡①õ❡s ❝♦♠♣♦rt❛♥❞♦✲s❡ ❝♦♠♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ♥ã♦ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❡s❝❛❧❛ ❝❛r❛❝t❡ríst✐❝❛ ♣❛r❛ s✉❛ ❝♦♥❡❝t✐✈✐❞❛❞❡✳ ❊ss❛ ✏❧✐❜❡r❞❛❞❡ ❞❡ ❝♦♥❡①ã♦✑ ❢❛③ ❝♦♠ q✉❡ t❛✐s r❡❞❡s s❡❥❛♠ ❝❤❛♠❛❞❛s ❞❡ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛✳

• ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦

❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ❞❡ ✉♠ ✈ért✐❝❡ ❞❡s❝r❡✈❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦♥❡①õ❡s ❡♥tr❡ ♦s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❞❡❧❡✳ ❬✷✱ ✶✶❪✳

P♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ r❡❞❡ ❝♦♠ ❛r❡st❛s s❡♠ ❞✐r❡çõ❡s ❡ t♦♠❛♠♦s ✉♠ ✈ért✐❝❡ ✐ q✉❡ ❛♣r❡s❡♥t❛z ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐✲ ♠♦s✳ ❖❜s❡r✈❡♠♦s ❞❡♥tr❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✈✐③✐♥❤♦s ❞♦ ✈ért✐❝❡ ✐ q✉❛♥t♦s ❡stã♦ ❧✐❣❛❞♦s ❡♥tr❡ s✐✳ ❆ ❛❣r❡❣❛çã♦ ♠á①✐♠❛ ❞♦ ✈ért✐❝❡ ✐ é ♦❜t✐❞❛ q✉❛♥❞♦ t♦❞❛s ❛s ③(③−1)/2 ❛r❡st❛s ♣♦ssí✈❡✐s ❡stã♦ ♣r❡s❡♥t❡s ❝♦♥❡❝t❛♥❞♦ t♦❞♦s ♦s s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳

❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ❞❡ ✉♠ ✈ért✐❝❡ ✐ q✉❛❧q✉❡r ❞❛ r❡❞❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ❛ r❛③ã♦ ❡♥tr❡ ❛ q✉❛♥t✐❞❛❞❡ t♦t❛❧ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❡❝t❛♠ s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ♣♦ssí✈❡❧ ❞❡ ❛r❡st❛s ❡♥tr❡ t♦❞♦s ❡st❡s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s✱ ❝♦♥❢♦r♠❡ ❛ ❊q✉❛çã♦ ✶✳✸✿

Ci =

2yi

zi(zi−1) ✭✶✳✸✮

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✻

❋✐❣✉r❛ ✶✳✷✿ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣r❛çã♦ ❞♦ ✈ért✐❝❡ν✱ é ♦❜t✐❞♦ q✉❛♥❞♦ ❝♦♥❤❡❝❡♠♦s s❡✉s

♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❡ s❛❜❡♠♦s ❝♦♠♦ ❡st❡s ❡stã♦ ❧✐❣❛❞♦s ❡♥tr❡ s✐✳ P❛r❛ ❡st❛ ✜❣✉r❛ t❡♠♦s zν = 6 ❡ y= 5 q✉❡ ✐♠♣❧✐❝❛ ❡♠cν = 1/3✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✹❪✳

❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ❞❡ t♦❞❛ ❛ r❡❞❡✱ ♦✉ s❡❥❛✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r❡❣❛çã♦ ♠é❞✐♦✱ é ♦❜t✐❞♦ ❢❛③❡♥❞♦ ❛ ♠é❞✐❛ ❞♦s Ci′_s ♣❡❧♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✈ért✐❝❡s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊q✉❛çã♦ ✶✳✹

C = 1 N

X

i

Ci = 1 N

X

i

2yi

zi(zi−1) ✭✶✳✹✮

♦♥❞❡ C ❡①♣r❡ss❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ❛r❡st❛s ❡♥tr❡ ♦s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❞❡ ✉♠ ❞❛❞♦ ✈ért✐❝❡ ❡s❝♦❧❤✐❞♦ ❛❧❡❛t♦r✐❛♠❡♥t❡✳

• ❱ért✐❝❡

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✼

• ▲✐❣❛çã♦

➱ ❛ ❧✐♥❤❛ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s✳

• ❈♦♥❡❝t✐✈✐❞❛❞❡

◆ú♠❡r♦ ❞❡ ❧✐❣❛çõ❡s ♣r❡s❡♥t❡s ♥✉♠ ✈ért✐❝❡✳ ❙❡ ❛ r❡❞❡ é ❞✐r✐❣✐❞❛ ❢❛❧❛✲s❡ ❞❡ ❝♦♥❡❝t✐✈✐✲ ❞❛❞❡ ❞❡ ❡♥tr❛❞❛ ❡ ❞❡ s❛í❞❛✳

• ❉✐â♠❡tr♦

➱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s q✉❛✐sq✉❡r✳

• ▼❡♥♦r ❈❛♠✐♥❤♦ ▼é❞✐♦

◆✉♠ ❣r❛❢♦✱ ❝♦♥❡①õ❡s q✉❡ ❧✐❣❛♠ ♦s ✈ért✐❝❡s ❢♦r♠❛♠ ✉♠ ❝❛♠✐♥❤♦✳ ❉❛❞♦s ❞♦✐s ✈ért✐❝❡s ♣♦❞❡ ❤❛✈❡r ❝❛♠✐♥❤♦ ❡♥tr❡ ❡❧❡s ♠❡s♠♦ q✉❡ ❡st❡s ♥ã♦ ❡st❡❥❛♠ ❞✐r❡t❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♠❡♥♦r ❝❛♠✐♥❤♦✱ ♦✉ ❝❛♠✐♥❤♦ ❣❡♦❞és✐❝♦✱ é ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s q✉❛✐sq✉❡r✳ ◆♦ ❝❛s♦ ❞❡ ✉♠ ❣r❛❢♦ ❞❡s❝♦♥❡❝t❛❞♦ ❝♦♠♣♦st♦ ♣♦r ♠❛✐s ❞❡ ✉♠❛ ❝♦♠♣♦✲ ♥❡♥t❡✱ ♦ ❞✐â♠❡tr♦ é ✐♥✜♥✐t♦ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠ ❝❛♠✐♥❤♦ ❡♥tr❡ ❛s ❞✐✈❡rs❛s ❝♦♠♣♦♥❡♥t❡s✳ ❈♦♥t✉❞♦✱ s❡♠♣r❡ é ♣♦ssí✈❡❧ r❡❞❡✜♥✐r ♦ ❞✐â♠❡tr♦ ❞♦ ❣r❛❢♦ ❝♦♠♦ s❡♥❞♦ ♦ ❞✐â♠❡tr♦ ♠á①✐♠♦ ❞❡ s❡✉s ❛❣r✉♣❛♠❡♥t♦s✳

❆ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ♠é❞✐❛ ❡♥tr❡ ♣❛r❡s ❞❡ ✈ért✐❝❡s ❞❡ ✉♠❛ r❡❞❡ ♥ã♦ ❞✐r❡❝✐♦♥❛❞❛ é ❞❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ✶✳✺✿

l = 2

N(N −1)

X

i>j

di,j ✭✶✳✺✮

♦♥❞❡di,j é ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛✶ ❡♥tr❡ ♦s ✈ért✐❝❡s i❡ j✳ ❆ ♠é❞✐❛ ❞❡ di,j t♦♠❛❞❛ s♦❜r❡ t♦❞♦s ♦sN(N −1)/2 ♣❛r❡s ❞❡ ✈ért✐❝❡s r❡s✉❧t❛ ♥♦ ♠❡♥♦r ❝❛♠✐♥❤♦ ♠é❞✐♦l✳

❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ l ♥✉♠❛ r❡❞❡ ❛❧❡❛tór✐❛ ❞♦ t✐♣♦ ár✈♦r❡✳ P❛r❛ ✐ss♦✱ ♣❛rt✐♠♦s ❞♦ ♣r❡ss✉♣♦st♦ q✉❡ ❛ r❡❞❡ t❡♥❤❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♠é❞✐❛z✳ ❙❡♥❞♦ ❛ss✐♠✱ ✉♠

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✽

sít✐♦ q✉❛❧q✉❡r ♣♦❞❡ ✈✐s✐t❛r✱ ❡♠ ♠é❞✐❛✱ ♦✉tr♦s ③ sít✐♦s ❛❢❛st❛❞♦s ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣❛ss♦ ✭✈❡r ✜❣✉r❛ ✶✳✹✮✳ ●❡♥❡r❛❧✐③❛♥❞♦✱ ♣♦❞❡✲s❡ ❢❛❧❛r q✉❡ ❛♣ós l ♣❛ss♦s ❡st❡ sít✐♦✱ ✈✐s✐t❛rá ♦✉tr♦s zl sít✐♦s✳ ❆ ❡st✐♠❛t✐✈❛ ❞❛ ❞✐stâ♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ r❡❞❡✱ ✐st♦ é✱ ♦ ♠❡♥♦r ❝❛♠✐♥❤♦ ♠é❞✐♦✱ é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ r❡❧❛çã♦N ∼zl✳ ❉❡ss❛ r❡❧❛çã♦✱ ❝❤❡❣❛✲s❡ ❛ ❊q✉❛çã♦ ✶✳✻✿

l≈ lnN

lnz ✭✶✳✻✮

◆❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ♦ ❢❛t♦ ❞❡ l r❡❧❛❝✐♦♥❛r✲s❡ ❝♦♠ lnN t♦r♥❛ ❡①♣❧í❝✐t♦ ♦ ♣♦rq✉ê ❞❛ ♣❡q✉❡♥❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣❛r❡s ❞❡ sít✐♦s ♥❛ r❡❞❡✳ ❊ss❛ ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ❛❧❣✉♠❛s r❡❞❡s r❡❛✐s ❡ é✱ ❝♦♠✉♠❡♥t❡✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✏❡❢❡✐t♦ ❞❡ ♠✉♥❞♦ ♣❡q✉❡♥♦✑✳

✶✳✶ ❘❡❞❡s ❘❡❛✐s

P❛r❛ q✉❡ s❡ ♣♦ss❛ ♠♦❞❡❧❛r ✉♠ ♣r♦❜❧❡♠❛ r❡❛❧ ❝♦♠♦ ✉♠❛ r❡❞❡ ❝♦♠♣❧❡①❛ ♥❡❝❡ss✐t❛✲s❡✱ ❛♥t❡s ❞❡ ♠❛✐s ♥❛❞❛✱ ✐❞❡♥t✐✜❝❛r ♦s ❡❧❡♠❡♥t♦s q✉❡ ❛ ❝♦♠♣õ❡ ❡✱ s♦❜r❡t✉❞♦✱ s❛❜❡r ❝♦♠♦ t❛✐s ❡❧❡♠❡♥t♦s ✐♥t❡r❛❣❡♠✳ ◆♦s ❛♥♦s r❡❝❡♥t❡s✱ ✐ss♦ ✈❡♠ s❡♥❞♦ ❢❡✐t♦ ❝♦♠ ✈ár✐♦s s✐st❡♠❛s r❡❛✐s✱ ♦❜❥❡✲ t✐✈❛♥❞♦ ✐❞❡♥t✐✜❝❛r ❡ ❝♦♠♣r❡❡❡♥❞❡r s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ❡✴♦✉ ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s✳ Pr♦❝✉✲ r❛♥❞♦ s❡❣✉✐r ❡ss❛ ❧ó❣✐❝❛✱ ✈❡r❡♠♦s✱ ❞❛q✉✐ ♣♦r ❞✐❛♥t❡✱ ❡①❡♠♣❧♦s ❞❡ s✐st❡♠❛s r❡❛✐s q✉❡ ❢♦r❛♠ ♠♦❞❡❧❛❞♦s s♦❜ ❛ ❢♦r♠❛ ❞❡ r❡❞❡s ❡ ❢❛r❡♠♦s ♦ r❡❧❛t♦ s♦❜r❡ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s t♦♣♦❧ó❣✐❝❛s ❛♣r❡s❡♥t❛❞❛s ♣❡❧♦s ♠❡s♠♦s✳

✶✳✶✳✶ ❖ ▼✉♥❞♦ é P❡q✉❡♥♦

❊♠ ✶✾✻✼✱ ❙t❛♥❧❡② ▼✐❧❣r❛♠✱ ✉♠ ♣s✐❝ó❧♦❣♦ s♦❝✐❛❧ ❡ ♣r♦❢❡ss♦r ❡♠ ❍❛r✈❛r❞ s❡ ❧❛♥ç♦✉ ❡♠ ♠❛✐s ✉♠ ❞❡ s❡✉s ❡①♣❡r✐♠❡♥t♦s s♦❝✐❛✐s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ ▼✐❧❣r❛♠ ❛❣♦r❛ ❡r❛ ❡♥❝♦♥tr❛r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞✉❛s ♣❡ss♦❛s q✉❛✐sq✉❡r ♥♦s ❊❯❆✳ P❛r❛ ❞❛r ✐♥í❝✐♦ ❛ ❡ss❡ ❡①♣❡r✐♠❡♥t♦ ❞✉❛s ♣❡ss♦❛s ❢♦r❛♠ ❡s❝♦❧❤✐❞❛s ❝♦♠♦ ❛❧✈♦✳ ❆ ♣r✐♠❡✐r❛ ❢♦✐ ❛ ❡s♣♦s❛ ❞❡ ✉♠ ❡st✉❞❛♥t❡ ❞❡ t❡♦❧♦❣✐❛ ❡♠ ❙❤❛r♦♥✱ ▼❛ss❛❝❤✉s❡tts✱ ❛ s❡❣✉♥❞❛ ❢♦✐ ✉♠ ❝♦rr❡t♦r ❡♠ ❇♦st♦♥✳ ❖ ♣ró①✐♠♦ ♣❛ss♦ ❢♦✐ ❡s❝♦❧❤❡r ❛s ❝✐❞❛❞❡s ❞❡ ❲✐❝❤✐t❛✱ ❑❛♥s❛s ❡ ❖♠❛❤❛✱ t♦❞❛s ❡❧❛s ♥♦ ◆❡❜r❛s❦❛✱ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ♦ ❡st✉❞♦✳

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✾

❡s❝♦❧❤✐❞♦s✱ ❞❡ ❲✐❝❤✐t❛ ❡ ❖❤❛♠❛ ♣❡❞✐♥❞♦ ❛ ❡❧❡s q✉❡ ♣❛rt✐❝✐♣❛ss❡♠ ❞❡ ✉♠ ❡st✉❞♦ ❞❡ ❝♦♥t❛t♦ s♦❝✐❛❧ ♥❛ s♦❝✐❡❞❛❞❡ ❛♠❡r✐❝❛♥❛✳ ❆ ❝❛rt❛ ❝♦♥t✐♥❤❛ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❞✉❛s ♣❡ss♦❛s ❡s❝♦❧❤✐❞❛s ❝♦♠♦ ❛❧✈♦s✱ ❥✉♥t♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s ✐♥str✉çõ❡s✿

❈❖▼❖ P❆❘❚■❈■P❆❘ ❉❖ ❊❙❚❯❉❖

✭✶✮ ❈❛❞❛ ♣❡ss♦❛ q✉❡ r❡❝❡❜❡r ♦ ❝❛❞❡r♥♦ ❞❡✈❡ ❛❞✐❝✐♦♥❛r s❡✉ ♥♦♠❡ ♥❡❧❡✳ ■st♦ s❡r✈❡ ♣❛r❛ r❛str❡❛r ❞❡ q✉❡♠ ♣❛rt✐✉ ♦ ❡♥✈❡❧♦♣❡✱ ❡✈✐t❛♥❞♦ q✉❡ ❡st❡ ✈♦❧t❡ ♣❛r❛ ✉♠❛ ♠❡s♠❛ ♣❡ss♦❛✳

✭✷✮ ❙❡ ✈♦❝ê ♥ã♦ ❝♦♥❤❡❝❡ ❛ ♣❡ss♦❛ ❛❧✈♦✱ ♥ã♦ t❡♥t❡ ❝♦♥t❛❝tá✲❧❛ ❞✐r❡t❛♠❡♥t❡✳ ❆♦ ✐♥✈és ❞✐ss♦✱ r❡♣❛ss❡ ❡st❡ ❝❛❞❡r♥♦ ♣❛r❛ ❛❧❣✉é♠ q✉❡ ✈♦❝ê ❝♦♥❤❡❝❡ ❡ ❛❝r❡❞✐t❡ q✉❡✱ ❡st❡✱ t❡♥❤❛ ♠❛✐♦r ❝❤❛♥❝❡ ❞❡ ❝♦♥❤❡❝❡r ❛ ♣❡ss♦❛ ❛❧✈♦✳

✭✸✮ ❙❡ ✈♦❝ê ❝♦♥❤❡❝❡ ❛ ♣❡ss♦❛ ❛❧✈♦✱ ❡♥✈✐❡ ❡st❡ ❞♦❝✉♠❡♥t♦ ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❡❧❡✭❛✮✳ ✭✹✮ ❆♦ ❡♥✈✐❛r ♦ ❝❛❞❡r♥♦✱ ❝❛❞❛ ♣❡ss♦❛ ❞❡✈❡ r❡t✐r❛r ✉♠❛ ♣á❣✐♥❛ ❞❡❧❡✱ ♣r❡❡♥❝❤ê✲❧❛ ❡ r❡♠❡tê✲❧❛ ❛♦ ❝✐❡♥t✐st❛✳ ❊st❛✱ s❡♠ ❞ú✈✐❞❛✱ é ✉♠❛ ❢❛s❡ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❞♦ ❡st✉❞♦✱ ♣♦✐s ❡❧❛ ♣❡r♠✐t❡✱ ❛ ▼✐❧❣r❛♠✱ ❛❝♦♠♣❛♥❤❛r ♦ ♣r♦❣r❡ss♦ ❞❛ ❡①♣❡r✐ê♥❝✐❛ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❥❡t♦ ❞♦ ❝❛❞❡r♥♦ ❛té ❝❤❡❣❛r ❛ ♣❡ss♦❛ ❛❧✈♦✳ ◆❛s ♠ã♦s ❞❛ ♣❡ss♦❛ ❛❧✈♦ ♦ ❝❛❞❡r♥♦ ❞❡✈❡✱ ❛❣♦r❛✱ s❡r ❡♥✈✐❛❞♦ ❛♦ Ps✐❝♦❧ó❣♦ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ❡①♣❡r✐♠❡♥t♦✳

❆ ❋✐❣✉r❛ ✶✳✸ ♠♦str❛ ♦ ▼❛♣❛ ❞♦s ❊st❛❞♦s ❯♥✐❞♦s ❡ ❛ ❋✐❣✉r❛ ✶✳✹ ♠♦str❛ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ❞❛ ❡①♣❡r✐ê♥❝✐❛✳

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✶✵

❋✐❣✉r❛ ✶✳✸✿ ▼❛♣❛ ❞♦s ❊st❛❞♦s ❯♥✐❞♦s✳ ❖s ❡st❛❞♦s ❡♠ ✈❡r♠❡❧❤♦ r❡♣r❡s❡♥t❛♠ ❛ ♦r✐❣❡♠ ❞❛s ❝♦rr❡✲ s♣♦♥❞ê♥❝✐❛s ❡ ♦ ❡st❛❞♦ ❡♠ ✈❡r❞❡ r❡♣r❡s❡♥t❛ ♦ ❞❡st✐♥♦ ✜♥❛❧✳ ❋✐❣✉r❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ r❡❢✳ ❬✶✹❪✳

❈❛♣ít✉❧♦ ✶✳ ◆♦♠❡♥❝❧❛t✉r❛ ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✶✶

✶✳✶✳✷ ❘❡❞❡s s♦❝✐❛✐s

❯♠❛ r❡❞❡ s♦❝✐❛❧ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❡ss♦❛s ♦✉ ❣r✉♣♦s ❞❡ ♣❡ss♦❛s q✉❡ ♣♦ss✉❡♠ ❛❧❣✉♠ t✐♣♦ ❞❡ ❝♦♥t❛t♦ ♦✉ ✐♥t❡r❛çã♦ ❡♥tr❡ ❡❧❛s✳ ❆s ♣❡ss♦❛s ❢❛③❡♠ ♦ ♣❛♣❡❧ ❞❡ ✈ért✐❝❡ ❡ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ❡❧❛s sã♦ ❛s ❧✐❣❛çõ❡s✳ ❆s r❡❞❡s s♦❝✐❛✐s s❡❣✉r❛♠❡♥t❡ ❡stã♦ ❡♥tr❡ ❛s ♠❛✐s ❡st✉❞❛❞❛s✳

✶✳✶✳✸ ❘❡❞❡s ❞❡ ✐♥❢♦r♠❛çã♦

❘❡❞❡s ❞❡ ✐♥❢♦r♠❛çã♦✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ r❡❞❡s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ sã♦ ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❡ ❜❛s❡s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❢♦r♠❛❧✱ ❝♦♠♦ ❛s ❝✐t❛çõ❡s ❞❡ ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s✱ ❛ ❲♦r❧❞ ❲✐❞❡ ❲❡❜✱ ♦s r❡❣✐str♦s ❞❡ ♣❛t❡♥t❡s✱ ❛ ❡str✉t✉r❛ ❞❛s ❧✐♥❣✉❛❣❡♥s✱ ❡t❝✳ ❆s ❝✐t❛çõ❡s ❞❡ ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s sã♦ ♠♦❞❡❧❛❞❛s ❝♦♠♦ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦✱ ♣r❛t✐❝❛♠❡♥t❡ ❛❝í❝❧✐❝♦✱ ❞❛❞♦ q✉❡ ❛s ♣✉❜❧✐❝❛çõ❡s ♠❛✐s r❡❝❡♥t❡s ❝✐t❛♠ ❛q✉❡❧❛s ♠❛✐s ❛♥t✐❣❛s ❡ ♥✉♥❝❛ ♦ ✐♥✈❡rs♦✳ ◆♦ ♦✉tr♦ ❡①tr❡♠♦✱ ❛ ❲❡❜ ❢♦r♠❛ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ ❝♦♠ ❢♦rt❡ ♣r❡s❡♥ç❛ ❞❡ ❝✐❝❧♦s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣❡r♠✐t❡ ❛ ✉♠ s✐t❡ ❛❞✐❝✐♦♥❛r ❧✐❣❛çõ❡s ❛ ♦✉tr♦s ❞❡ ❝♦♥str✉çã♦ ♣♦st❡r✐♦r✳

✶✳✶✳✹ ❘❡❞❡s t❡❝♥♦❧ó❣✐❝❛s