❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼■◆❆❙ ●❊❘❆■❙
■◆❙❚■❚❯❚❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❊❙❚❆❚❮❙❚■❈❆
❚ó♣✐❝♦s ❡♠ P❡r❝♦❧❛çã♦ ❞❡ ▲♦♥❣♦ ❆❧❝❛♥❝❡
❘♦❣❡r ❲✐❧❧✐❛♠ ❈â♠❛r❛ ❙✐❧✈❛
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼■◆❆❙ ●❊❘❆■❙
■◆❙❚■❚❯❚❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❊❙❚❆❚❮❙❚■❈❆
❚ó♣✐❝♦s ❡♠ P❡r❝♦❧❛çã♦ ❞❡ ▲♦♥❣♦ ❆❧❝❛♥❝❡
❘♦❣❡r ❲✐❧❧✐❛♠ ❈â♠❛r❛ ❙✐❧✈❛
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❊st❛tíst✐❝❛✳
❖r✐❡♥t❛❞♦r✿ ❇❡r♥❛r❞♦ ◆✉♥❡s ❇♦r❣❡s ❞❡ ▲✐♠❛ ❈♦✲♦r✐❡♥t❛❞♦r✿ ❘é♠② ❞❡ P❛✐✈❛ ❙❛♥❝❤✐s
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ✜♠ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛❣r❛❞❡ç♦✿
❆ ❉❡✉s✱ ♣♦r ♠❡ ❞❛r ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❡ss❡ tr❛❜❛❧❤♦✱ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ r❡♥♦✈❛♥❞♦ ♠✐♥❤❛s ❢♦rç❛s ❡ ♣❡❧❛ ❜❡♥çã♦ ❞❛ ✈✐❞❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❇❡r♥❛r❞♦ ◆✉♥❡s ❇♦r❣❡s ❞❡ ▲✐♠❛ ❡ ❘é♠② ❞❡ P❛✐✈❛ ❙❛♥❝❤✐s ♣❡❧❛ ♦r✐❡♥t❛çã♦✳ ➱ ✐♠♣♦ssí✈❡❧ ♠❡♥s✉r❛r ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ✈♦❝ês ♣❛r❛ ♦ ♠❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛❝❛❞ê♠✐❝♦✳ ❱❛❧❡✉ ♣❡❧❛s ❤♦r❛s q✉❡ ♣❛ss❛♠♦s ❞✐s❝✉t✐♥❞♦ ♠❛t❡♠át✐❝❛✦
❆♦s ♣r♦❢❡ss♦r❡s ●r❡❣ór✐♦ ❙❛r❛✈✐❛ ❆t✉♥❝❛r ❡ ▼✐❝❤❡❧ ❙♣✐r❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ t❛♠❜é♠ ❛♦s ❝♦❧❡❣❛s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯❋▼●✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ à ❋❆P❊▼■● ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
➚ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣r♦❢✳ ❱❧❛❞❛s ❙✐❞♦r❛✈✐❝✐✉s ✭■▼P❆✮✱ ♣r♦❢❛✳ ▼❛r✐❛ ❊✉❧á❧✐❛ ❱❛r❡s
✭❈❇P❋✮✱ ♣r♦❢✳ ❊♥r✐q✉❡ ❆♥❞❥❡❧ ✭❯♥✐✈❡rs✐té ❞❡ Pr♦✈❡♥❝❡✮✱ ♣r♦❢✳ ❆❞r✐❛♥ P❛❜❧♦ ❍✐♥♦❥♦s❛ ▲✉♥❛ ✭❯❋▼●✮ ❡ ♣r♦❢❛✳ ❉❡♥✐s❡ ❉✉❛rt❡ ❙❝❛r♣❛ ✭❯❋▼●✮✳ ❖❜r✐❣❛❞♦ ♣♦r ❛t❡♥❞❡r❡♠ ❛♦ ❝♦♥✈✐t❡✳
❆♦s ✈❡r❞❛❞❡✐r♦s ❛♠✐❣♦s q✉❡✱ ♠❡s♠♦ ♥ã♦ ✈❡r❜❛❧✐③❛♥❞♦ ✐ss♦✱ t♦r❝❡r❛♠ ♣♦r ♠✐♠ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❡ ✈✐❜r❛r❛♠ ❝♦♠✐❣♦ ❡♠ ❝❛❞❛ ❝♦♥q✉✐st❛✳ ❖❜r✐❣❛❞♦ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♣❛❝✐ê♥❝✐❛ q✉❡ ✈♦❝ês ♠❡ ♦❢❡r❡❝❡r❛♠ ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ✶✵ ❛♥♦s ❞❡ ❛❝❛❞❡♠✐❛✳
❆♦ ❘❛♠♦♥ ♣❡❧❛ ❛❥✉❞❛ ❝♦♠ ♦s ❞❡s❡♥❤♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❡ss❡ tr❛❜❛❧❤♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧❛s ❤♦r❛s ❣❛st❛s ❝♦♠✐❣♦ ♥❡ss❛ t❛r❡❢❛✳
❆♦s ♠❡✉s ❛♠❛❞♦s ♣❛✐s✱ ♥ã♦ s❡✐ ❝♦♠♦ ❛❣r❛❞❡❝❡r✳ ◗✉❛❧q✉❡r ♣❛❧❛✈r❛ ❝♦❧♦❝❛❞❛ ❛q✉✐ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❛❣r❛❞❡❝✐♠❡♥t♦ s❡r✐❛ ✐♥s✉✜❝✐❡♥t❡ ♣❛r❛ ❡①♣r❡ss❛r ❛ r❡❛❧ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ✈♦❝ês ♣❛r❛ q✉❡ ✐ss♦ ❛❝♦♥t❡❝❡ss❡✳ ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❝❤❛♠❛rã♦ s❡✉ ✜❧❤♦ ❞❡ ❞♦✉t♦r✱ ♠❛s ❡✉ s❡✐ q✉❡ ♦ ❞✐♣❧♦♠❛ é ♥❛ ✈❡r❞❛❞❡ ❞❡ ✈♦❝ês✳ ◆ã♦ ♣♦r ❝♦♥❤❡❝✐♠❡♥t♦s ❡st❛tíst✐❝♦s✱ ♠❛t❡♠át✐❝♦s ♦✉ ♣r♦❜❛❜✐❧íst✐❝♦s✱ ♠❛s ♣♦r ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ✈✐❞❛ ❡ ❞♦ q✉❡ r❡❛❧♠❡♥t❡ s✐❣♥✐✜❝❛♠ ❛s ♣❛❧❛✈r❛s ❛♠♦r✱ ❞♦❛çã♦ ❡ s❛❝r✐❢í❝✐♦✳ ❆ ✈♦❝ês✱ ♠✐♥❤❛ ❡t❡r♥❛ ❣r❛t✐❞ã♦✦
❘❡s✉♠♦
◆❡st❛ t❡s❡ ❡st✉❞❛♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ❡♠
Zd✱ d ≥ 2✳ ❊ss❡ ♠♦❞❡❧♦ é ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ sít✐♦s
❡♠ Zd✱ ♦♥❞❡ ❝❛❞❛ sít✐♦ ❡stá ♦❝✉♣❛❞♦ ♦✉ ✈❛③✐♦ ❞❡ ♠❛♥❡✐r❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡
p ❡ 1−p r❡s♣❡❝t✐✈❛♠❡♥t❡✱ p ∈[0,1]✳ ◆✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡
♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ♥♦ ❣r❛❢♦ Ld
K = (Zd,∪Kn=1En)✱ ♦♥❞❡ Ené ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧♦s ♣❛r❛❧❡❧♦s ❛
❛❧❣✉♠ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦ ❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ n ∈N ❡ ✉♠❛ ♣❛❧❛✈r❛ é ✉♠ ❡❧❡♠❡♥t♦ ξ ∈ {0,1}N
✳ ❖❜t❡♠♦s ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿
• ∀p ∈ (0,1)✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ K = K(p)✱ t❛❧ q✉❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s sã♦ ✈✐st❛s ❡♠
Ld
K q✉❛s❡ ❝❡rt❛♠❡♥t❡✳
• ❖❜t❡♠♦s ❛ ❡s❝❛❧❛ ❝♦rr❡t❛ ❞❛ ❝♦♥st❛♥t❡K(p)q✉❛♥❞♦p✈❛✐ ♣❛r❛ ③❡r♦✱ ❛ ✜♠ ❞❡ q✉❡ t♦❞❛s
❛s ♣❛❧❛✈r❛s s❡❥❛♠ ✈✐st❛s q✉❛s❡ ❝❡rt❛♠❡♥t❡✳
• ❖❜t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❡s❝❛❧❛ ❞❛ ❝♦♥st❛♥t❡K(p)q✉❛♥❞♦p✈❛✐ ♣❛r❛ ③❡r♦✱
q✉❛♥❞♦ ♦ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡ é ✈❡r q✉❛s❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s✳
❊♠ ✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ❡st✉❞❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♣❡r❝♦❧❛çã♦
θGk
(p) ❡ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ pc(Gk) ❡♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ sít✐♦s ❡♠
Gk = (Zd,E
1∪ Ek)✳ ❖❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
• lim
k→∞pc(G
k) =p
c(Z2d)✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② s♦♠❡ ❛s♣❡❝ts ♦❢ ❛ ❧♦♥❣ r❛♥❣❡ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧ ✐♥ Zd✱ d ≥ 2✳
❚❤✐s ♠♦❞❡❧ ✐s ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t s✐t❡ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧ ✐♥Zd✱ ✇❤❡r❡ ❡❛❝❤ s✐t❡ ✐s
✐♥❞❡♣❡♥❞❡♥t❧② ♦❝❝✉♣✐❡❞ ♦r ✈❛❝❛♥t ✇✐t❤ ♣r♦❜❛❜✐❧✐t②p♦r1−p r❡s♣❡❝t✐✈❡❧②✱p∈[0,1]✳ ❋✐rst❧②✱
✇❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦❢ ♣❡r❝♦❧❛t✐♦♥ ♦❢ ✇♦r❞s ✐♥ t❤❡ ❣r❛♣❤ Ld
K = (Zd,∪Kn=1En)✱ ✇❤❡r❡ En
✐s t❤❡ s❡t ♦❢ ❡❞❣❡s ♦❢ ❧❡♥❣t❤ n ∈ N ♣❛r❛❧❧❡❧ t♦ s♦♠❡ ❝♦♦r❞✐♥❛t❡ ❛①✐s ❛♥❞ ❛ ✇♦r❞ ✐s ❥✉st ❛♥
❡❧❡♠❡♥t ξ∈ {0,1}N
✳ ❲❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts✿
• ∀p∈(0,1)✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t K =K(p)✱ s✉❝❤ t❤❛t ❛❧❧ ✇♦r❞s ❝❛♥ ❜❡ s❡❡♥ ♦♥ Ld K
❛❧♠♦st s✉r❡❧②✳
• ❲❡ ♦❜t❛✐♥ t❤❡ s❝❛❧✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢K(p)✇❤❡♥p ❣♦❡s t♦ ③❡r♦✱ s♦ t❤❛t ❛❧❧ ✇♦r❞s ❝❛♥ ❜❡
s❡❡♥ ❛❧♠♦st s✉r❡❧②✳
• ❲❡ ♦❜t❛✐♥ ❛ ♣❛rt✐❛❧ r❡s✉❧t ❢♦r t❤❡ s❝❛❧✐♥❣ ❜❡❤❛✈✐♦✉r ♦❢K(p)✇❤❡♥p❣♦❡s t♦ ③❡r♦✱ ✇❤❡♥
t❤❡ ❡✈❡♥t ♦❢ ✐♥t❡r❡st ✐s t♦ s❡❡ ❛❧♠♦st ❛❧❧ ✇♦r❞s ✐♥st❡❛❞ ♦❢ ❛❧❧ ✇♦r❞s✳
▲❛t❡r✱ ✇❡ st✉❞② t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♣❡r❝♦❧❛t✐♦♥ ♣r♦❜❛❜✐❧✐t② θGk
(p)❛♥❞ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t
pc(Gk)✐♥ ❛♥ ✐♥❞❡♣❡♥❞❡♥t s✐t❡ ♣❡r❝♦❧❛t✐♦♥ ♠♦❞❡❧ ✐♥Gk = (Zd,E1∪Ek)✳ ❲❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣
r❡s✉❧t✿
• lim
k→∞pc(G
k) = p
c(Z2d)✳
■♥tr♦❞✉çã♦
❯♠ ❞♦s ♠♦❞❡❧♦s ♣r♦❜❛❜✐❧íst✐❝♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s é ♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❊ss❡ ♠♦❞❡❧♦ é t✐❞♦ ♣♦r ♠✉✐t♦s ❝♦♠♦ ✉♠❛ ❢♦♥t❡ ❞❡ ♣r♦❜❧❡♠❛s ✐♥t❡r❡ss❛♥t❡s✱ ♠❛s ♥ã♦ r❛r❛♠❡♥t❡ ❞❡ ❞✐❢í❝✐❧ s♦❧✉çã♦✳ ■♥tr♦❞✉③✐❞♦ ❡♠ ✶✾✺✼ ♣♦r ❇r♦❛❞❜❡♥t ❡ ❍❛♠♠❡rs❧❡②✱ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❡s❝r❡✈❡r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ✉♠ ✢✉✐❞♦ ♣♦r ✉♠ ♠❡✐♦ ♣♦r♦s♦✳
❉❛❞♦ ✉♠ ❣r❛❢♦ G = (V,E)✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡
❞❡ sít✐♦s ❡♠ G✱ ❛tr✐❜✉✐♥❞♦ ❛ ❝❛❞❛ ✈ért✐❝❡ v ∈ V ❞♦✐s ❡st❛❞♦s ❞✐st✐♥t♦s✱ ♦❝✉♣❛❞♦ ❡ ✈❛③✐♦✳
❊ss❡s ❡st❛❞♦s sã♦ ❛tr✐❜✉í❞♦s ❞❡ ♠❛♥❡✐r❛ ✐♥❞❡♣❡♥❞❡♥t❡ s❡❣✉♥❞♦ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ❞❡ ♣❛râ♠❡tr♦ p✳ P♦❞❡♠♦s t❛♠❜é♠ ❛tr✐❜✉✐r ❛♦s ♣♦ssí✈❡✐s ❡st❛❞♦s✱
♦❝✉♣❛❞♦ ❡ ✈❛③✐♦✱ ♦s ✈❛❧♦r❡s ♥✉♠ér✐❝♦s ✶ ❡ ✵ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s ❛ss✐♠ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧
Ω = {0,1}V ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ❡♠ Ω✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r P
p✳ ❊ss❡
♠♦❞❡❧♦ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❞❡s❝r✐t♦s ❢♦r♠❛❧♠❡♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✶✳
❱ár✐❛s ✈❛r✐❛♥t❡s ❞❡ss❡ ♠♦❞❡❧♦ ❢♦r❛♠ ♣r♦♣♦st❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❡ ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s ❢♦r❛♠ ❛tr❛í❞♦s ♣❛r❛ ❛ ár❡❛✱ q✉❡ t❡✈❡ ❜♦❛ ♣❛rt❡ ❞♦s s❡✉s ♣r♦❜❧❡♠❛s ♠❛✐s ✐♥tr✐❣❛♥t❡s r❡s♦❧✈✐❞♦s ♥❛ ❞é❝❛❞❛ ❞❡ ✽✵✳ ◆❡ss❛ t❡s❡ ✈❡rs❛r❡♠♦s s♦❜r❡ ✉♠❛ ❞❡ss❛s ✈❛r✐❛♥t❡s✱ ❛ s❛❜❡r ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ ❊st❡ ♠♦❞❡❧♦ é ❞❡s❝r✐t♦ ❞❡t❛❧❤❛❞❛♠❡♥t❡ ♥❛ ❙❡çã♦ ✶✳✸✳ ❖s ❈❛♣ít✉❧♦s ✷ ❡ ✸ sã♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ♦r✐❣✐♥❛✐s ❞❡st❛ t❡s❡ ❡ sã♦ ♦s t❡♠❛s ❞♦s tr❛❜❛❧❤♦s ❬✷✵❪ ❡ ❬✷✶❪✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
◆♦ ❈❛♣ít✉❧♦ ✷ ❡st✉❞❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❡♠ ♠♦❞❡❧♦s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❇❡♥❥❛♠✐♥✐ ❡ ❑❡st❡♥ ❡♠ ❬✸❪✱ tr❛❜❛❧❤♦ ❡♠ q✉❡ ♦s ❛✉t♦r❡s ❛♣r❡s❡♥t❛♠ ✈ár✐♦s r❡s✉❧t❛❞♦s ❡ ❡①❡♠♣❧♦s ✐♥t❡r❡ss❛♥t❡s✳ P♦st❡r✐♦r♠❡♥t❡✱ ✈ár✐♦s ♦✉tr♦s ❛✉t♦r❡s tr❛❜❛❧❤❛r❛♠ ♥❡ss❡ ♣r♦❜❧❡♠❛ ❡♠ ❝♦♥t❡①t♦s ❞✐❢❡r❡♥t❡s✳
❯♠❛ ♣❛❧❛✈r❛ é ✉♠❛ s❡q✉ê♥❝✐❛ ❜✐♥ár✐❛ ✐♥✜♥✐t❛ ξ∈Ξ = {0,1}N
✽
ω∈Ω✱ ♦ ♦❜❥❡t✐✈♦ é ❡♥t❡♥❞❡r ❡♠ q✉❡ ❝✐r❝✉♥stâ♥❝✐❛s ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s q✉❡ sã♦ ✈✐st❛s ❛♦
❧♦♥❣♦ ❞❡ ❛❧❣✉♠ ❝❛♠✐♥❤♦ ❡♠G ♥❛ ❝♦♥✜❣✉r❛çã♦ ω é ❣r❛♥❞❡ ❡♠ ❛❧❣✉♠ s❡♥t✐❞♦✳
❈♦♥s✐❞❡r❡ ♦ ❣r❛❢♦ Zd = (Zd,E)✱ ♦♥❞❡ E é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧♦s ❡♥tr❡ ♦s ✈✐③✐♥❤♦s ♠❛✐s
♣ró①✐♠♦s ❞❡ Zd✳ ❊♠ ❬✸❪✱ ♦s ❛✉t♦r❡s ❞❡✜♥❡♠ Sv(ω)✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛❧❛✈r❛s q✉❡ sã♦ ✈✐st❛s
♥❛ ❝♦♥✜❣✉r❛çã♦ω ❛ ♣❛rt✐r ❞❡ v ∈ V ❛♦ ❧♦♥❣♦ ❞❡ ❛❧❣✉♠ ❝❛♠✐♥❤♦ ❛✉t♦✲❡✈✐t❛♥t❡✱ ♦✉ s❡❥❛✱ ✉♠❛
❝❛♠✐♥❤♦ s❡♠ ✐♥t❡rs❡çõ❡s ❡ t❛♠❜é♠ S∞(ω) = Sv∈VSv(ω)✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛❧❛✈r❛s q✉❡ sã♦
✈✐st❛s ♥❛ ❝♦♥✜❣✉r❛çã♦ω❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ ✈ért✐❝❡ ❛♦ ❧♦♥❣♦ ❞❡ ❛❧❣✉♠ ❝❛♠✐♥❤♦ ❛✉t♦✲❡✈✐t❛♥t❡✳
◆❡ss❡ ♠❡s♠♦ tr❛❜❛❧❤♦ ❡❧❡s ♠♦str❛♠✱ ❡♥tr❡ ♦✉tr♦s r❡s✉❧t❛❞♦s✱ q✉❡ ❛✮ ❙❡ d≥10✱ t❡♠♦s
P1
2{ω∈Ω :S∞ = Ξ ❡♠ Z
d}= 1.
❜✮ ❙❡ d≥40✱ t❡♠♦s
P1 2
( [
v∈Zd
{ω∈Ω :Sv(ω) = Ξ ❡♠ Zd} )
= 1.
❖❜s❡r✈❡ q✉❡ ❡♠ ❜✮✱ t♦❞❛s ❛s ♣❛❧❛✈r❛s sã♦ ✈✐st❛s ❛ ♣❛rt✐r ❞❡ ✉♠ ú♥✐❝♦ ✈ért✐❝❡ ✭q✉❡ ❞❡♣❡♥❞❡ ❞❛ ❝♦♥✜❣✉r❛çã♦✮ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳
❯♠ ❞♦s ♥♦ss♦s ♦❜❥❡t✐✈♦s ♥❡ss❛ t❡s❡ é ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❛♣❧✐❝❛❞♦ ❛ ♠♦❞❡❧♦s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡st✉❞❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❡♠ Zd✱ ❝♦♥s✐❞❡r❛♥❞♦ ♥ã♦ s♦♠❡♥t❡ ❝♦♥❡①õ❡s ❡♥tr❡ ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s✱ ♠❛s
t❛♠❜é♠ ❝♦♥❡①õ❡s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ P❛r❛ ✐ss♦✱ s❡❥❛Ld
K = (Zd,∪Kn=1En)✱ ♦♥❞❡ En é ♦ ❝♦♥❥✉♥t♦
❞❡ ❡❧♦s ♣❛r❛❧❡❧♦s ❛ ❛❧❣✉♠ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦ ❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ n ∈ N✳ ❖ ♣r✐♠❡✐r♦ t❡♦r❡♠❛
q✉❡ ♣r♦✈❛r❡♠♦s é ♦ s❡❣✉✐♥t❡✿
❚❡♦r❡♠❛ ✶ P❛r❛ t♦❞♦ p∈(0,1)✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ K =K(p)✱ t❛❧ q✉❡
Pp{ω ∈Ω :S0(ω) = Ξ ❡♠ LdK}>0
♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡
Pp (
[
v∈Zd
{ω∈Ω :Sv(ω) = Ξ ❡♠ LdK} )
= 1.
✾
K(p) ♣❛r❛ q✉❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s s❡❥❛♠ ✈✐st❛s q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❖ s❡❣✉♥❞♦ r❡s✉❧t❛❞♦ q✉❡
❡st❛❜❡❧❡❝❡r❡♠♦s r❡s♦❧✈❡ ❡ss❡ ♣r♦❜❧❡♠❛ ❡ ❢♦r♥❡❝❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡K(p)q✉❛♥❞♦
p ✈❛✐ ♣❛r❛ ③❡r♦✳
❚❡♦r❡♠❛ ✷ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ λ0 ∈ 21d,−6 ln(1−pc(Zd))
t❛❧ q✉❡ s❡ K(p) = ⌊λ p⌋✱
❡♥tã♦
lim
p→0Pp
( [
v∈Zd
{ω∈Ω :Sv(ω) = Ξ ❡♠ LdK} )
=
(
0 s❡ λ < λ0,
1 s❡ λ > λ0.
❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ é ❢❡✐t❛ ❡♠ ❞✉❛s ♣❛rt❡s✳ ❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ♠♦str❛r❡♠♦s q✉❡ s❡ λ0 é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ p∗ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛❝✐♠❛ é ✐❣✉❛❧ ❛ ✶✱ ∀p∈(0, p∗)✳ ❊ss❡ é ♦ ❝♦♥t❡ú❞♦ ❞♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r✳
▲❡♠❛ ✶ ❙❡ K =K(p) = 2⌊λ
p⌋✱ ❡♥tã♦ ♣❛r❛ λ >−3 ln(1−pc(Zd)) t❡♠♦s q✉❡
lim
p→0Pp
( [
v∈Zd
{ω ∈Ω :Sv(ω) = Ξ ❡♠ LdK} )
= 1.
❊♠ s❡❣✉✐❞❛ ♠♦str❛r❡♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✿
▲❡♠❛ ✷ ❙❡ K =K(p) = ⌊λ
p⌋ ❝♦♠ λ <
1
2d✱ ❡♥tã♦
lim
p→0Pp
( [
v∈Zd
{ω ∈Ω :Sv(ω) = Ξ ❡♠ LdK} )
= 0.
❈♦♠❜✐♥❛♥❞♦ ♦s ▲❡♠❛s ✶ ❡ ✷✱ ♦❜t❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳
❙❡❥❛ ~1 = (1,1, . . .)✳ ❈♦♠ ✉♠❛ ♣❡q✉❡♥❛ ♠♦❞✐✜❝❛çã♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✱
♦❜t❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡K(p)✱ q✉❛♥❞♦p✈❛✐ ♣❛r❛ ③❡r♦✱ q✉❛♥❞♦ ♦ ❡✈❡♥t♦ ❡♠
q✉❡stã♦ é {ω∈Ω :~1é ✈✐st❛ ❡♠ω ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ v}✳
❈♦r♦❧ár✐♦ ✶ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ λ0 ∈ 21d,−2 ln(1−pc(Zd))
t❛❧ q✉❡ s❡ K(p) =⌊λ p⌋✱
❡♥tã♦
lim
p→0Pp
( [
v∈Zd
{ω ∈Ω :→1 é ✈✐st❛ ❡♠ ω ❡♠ LdK ❛ ♣❛rt✐r ❞❡ v}
)
=
(
0 s❡ λ < λ0
✶✵
▼✉❞❛♥❞♦ ✉♠ ♣♦✉❝♦ ❛ ❞✐r❡çã♦ ❡♠ ♥♦ss❛ ❞✐s❝✉ssã♦✱ ♦❜s❡r✈❡ q✉❡ ♠❡s♠♦ q✉❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s s❡❥❛♠ ✈✐st❛s ❡♠ ✉♠ ❣r❛❢♦G❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✵✱ é ♣♦ssí✈❡❧ q✉❡ q✉❛s❡ t♦❞❛s ♣❛❧❛✈r❛s
s❡❥❛♠ ✈✐st❛s q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❆q✉✐ q✉❛s❡ t♦❞❛s s❡ r❡❢❡r❡ à ♠❡❞✐❞❛µα =
∞
Y
i=1
νi ❡♠ Ξ✱ ♦♥❞❡
♣❛r❛ ❝❛❞❛ i
νi(1) = 1−νi(0) =α, ♣❛r❛ ❛❧❣✉♠ 0< α <1.
◆♦t❡ q✉❡ s♦❜ν ❛sξi sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s ❝♦♠
❞✐str✐❜✉✐çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ❝♦♠ ♣❛râ♠❡tr♦ α✳
◆♦ss♦ ♦❜❥❡t✐✈♦ ❡r❛ ♦❜t❡r ✉♠ t❡♦r❡♠❛ s✐♠✐❧❛r ❛♦ ❚❡♦r❡♠❛ ✷✳ ❊♥tr❡t❛♥t♦✱ t✐✈❡♠♦s s✉❝❡ss♦ ♣❛r❝✐❛❧ ♥❡ss❛ ❞✐r❡çã♦✱ ♦❜t❡♥❞♦ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✸ ❙❡❥❛ 0< α <1✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ǫ >0 ❡ K =K(p)< 21dpα1−ǫ✱ t❡♠♦s
lim
p→0Pp
( [
v∈Zd
{ω ∈Ω : µα(Sv(ω)) = 1 ❡♠ LdK} )
= 0.
◆♦ ❈❛♣ít✉❧♦ ✸✱ q✉❡ s❡ ❜❛s❡✐❛ ❡♠ ❬✷✶❪✱ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♣❡r❝♦❧❛çã♦ θGk
(p) ❡ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ pc(Gk) ❡♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡
❞❡ sít✐♦s ❡♠ Gk = (Zd,E
1 ∪ Ek)✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❡ss❡ ♣r♦❜❧❡♠❛ ♥✉♠ ❝♦♥t❡①t♦ ♠❛✐s
❣❡r❛❧✱ q✉❡ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ ❛ s❡❣✉✐r✳
❙❡❥❛~k= (k1, k2, . . . , kn)✱ ki ∈ {2,3, . . .}✱i= 1, . . . , n✳ ❈♦♥s✐❞❡r❡ ♦ ❣r❛❢♦
G~k = (Zd,E1∪(∪n
i=1Ek1×···×ki)),
✐st♦ é✱G~k éZd ❞❡❝♦r❛❞♦ ❝♦♠ t♦❞♦s ❡❧♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1, k
1, k1×k2, . . . , k1×k2× · · · ×kn
♣❛r❛❧❡❧♦s ❛ ❛❧❣✉♠ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦✳ ◗✉❛♥❞♦ n = 1✱ t❡♠♦s ♦ ❣r❛❢♦ Gk ♠❡♥❝✐♦♥❛❞♦
❛♥t❡r✐♦r♠❡♥t❡✳
❖ r❡s✉❧t❛❞♦ q✉❡ ♦❜t❡♠♦s ❞✐③ r❡s♣❡✐t♦ ❛♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❡♠G~k✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♠♦str❛♠♦s q✉❡ ♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ss❡ ♠♦❞❡❧♦ ❝♦♥✈❡r❣❡✱
q✉❛♥❞♦ki → ∞,∀i= 1, . . . n✱ ♣❛r❛ ♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡
❡♠ Zd(n+1)✳ ❋♦r♠❛❧♠❡♥t❡✱ ♠♦str❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✹
lim
ki→∞,∀i
✶✶
❊♠ s❡❣✉✐❞❛ ❣♦st❛rí❛♠♦s ❞❡ ✈❡r✐✜❝❛r s❡ ❡①✐st❡ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛ ❢✉♥çã♦ θGk
(p) ♥❛ ✈❛r✐á✈❡❧ k✱ ♣❛r❛ p ✜①♦✳ ❊♠ r❡❧❛çã♦ ❛ ❡ss❡ ♣r♦❜❧❡♠❛ ❛♣r❡s❡♥t❛♠♦s ✉♠❛
❙✉♠ár✐♦
✶ ▼♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❡ ❛❧❣✉♠❛s ✈❛r✐❛çõ❡s ✶✸
✶✳✶ ▼♦❞❡❧♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ P❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸ P❡r❝♦❧❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹ ❈♦♥str✉çã♦ ❞✐♥â♠✐❝❛ ❞♦ ❛❣❧♦♠❡r❛❞♦ ❞❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✺ P❡r❝♦❧❛çã♦ ❡♠ ▲❛❥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷ P❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❡♠ ♠♦❞❡❧♦s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✸✷ ✷✳✶ ❚♦❞❛s ♣❛❧❛✈r❛s ♣♦❞❡♠ s❡r ✈✐st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ◗✉❛❧ ❛ ❡s❝❛❧❛ ❞❛ ❝♦♥st❛♥t❡ K(p)q✉❛♥❞♦ p ✈❛✐ ♣❛r❛ ③❡r♦❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷✳✷✳✶ P❛r❛ ✈❡r t♦❞❛s ❛s ♣❛❧❛✈r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✳✷ P❛r❛ ✈❡r q✉❛s❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✸ P♦♥t♦ ❈rít✐❝♦ ❡ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ P❡r❝♦❧❛çã♦ ❡♠ ▼♦❞❡❧♦s ❞❡ ▲♦♥❣♦
❆❧❝❛♥❝❡ ✺✵
✸✳✶ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷ Pr♦✈❛s ❞♦s ❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸ ❯♠❛ ❝♦♥❥❡❝t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❈❛♣ít✉❧♦
1
▼♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❡ ❛❧❣✉♠❛s ✈❛r✐❛çõ❡s
◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ sít✐♦s✳ P❛rt❡ ❞❛ ♥♦t❛çã♦ ✉t✐❧✐③❛❞❛ ♥❡st❛ t❡s❡ t❛♠❜é♠ s❡rá ✐♥tr♦❞✉③✐❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❛ss✐♠ ❝♦♠♦ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ♣❡r❝♦❧❛çã♦✱ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s✳ ❖ ❝❛♣ít✉❧♦ s❡ ❞✐✈✐❞❡ ❡♠ ❝✐♥❝♦ s❡çõ❡s✳ ◆❛ ♣r✐♠❡✐r❛ ❞❡❧❛s ✐♥tr♦❞✉③✐r❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ sít✐♦s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ss❡ ♠♦❞❡❧♦✳ ◆❛ s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ s❡çõ❡s ❛♣r❡s❡♥t❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆❛ q✉❛rt❛ ❡ q✉✐♥t❛ s❡çõ❡s ❛♣r❡s❡♥t❛r❡♠♦s ❞✉❛s té❝♥✐❝❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦ ❞❡❝♦rr❡r ❞❡ss❡ tr❛❜❛❧❤♦✳
✶✳✶ ▼♦❞❡❧♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s
❆ t❡♦r✐❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❇r♦❛❞❜❡♥t ❡ ❍❛♠♠❡rs❧❡② ✭✈❡❥❛ ❬✻❪✮ ❡♠ ✶✾✺✼✳ ❉❡s❞❡ ❡♥tã♦✱ ❛ t❡♦r✐❛ t❡♠ s✐❞♦ ❢♦♥t❡ ❞❡ ♣r♦❜❧❡♠❛s ✐♥t❡r❡ss❛♥t❡s ❡ ♠✉✐t♦s ❛rt✐❣♦s ❡ ❧✐✈r♦s q✉❡ tr❛t❛♠ ❞♦ ❛ss✉♥t♦ ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s✳ ❖ ♣r♦♣ós✐t♦ ♦r✐❣✐♥❛❧ ❡r❛ ❡st✉❞❛r ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s t❛✐s ❝♦♠♦ ♦ ✢✉①♦ ❞❡ ✉♠ ✢✉✐❞♦ ❛tr❛✈és ❞❡ ✉♠ ♠❡✐♦ ♣♦r♦s♦ ❞❡s♦r❞❡♥❛❞♦✳
❯♠ ♣r♦❝❡ss♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ sít✐♦s ❞❡ ♣❛râ♠❡tr♦p∈[0,1]é ❞❡✜♥✐❞♦ ❝♦♠♦
s❡❣✉❡✳ ❙❡❥❛G = (V,E)✉♠ ❣r❛❢♦ ✐♥✜♥✐t♦✱ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✱ ❝♦♥❡①♦✱ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s
V(G) =V ❡♥✉♠❡rá✈❡❧ ❡ ❝♦♥❥✉♥t♦ ❞❡ ❡❧♦sE(G) = E✳ P❛r❛ ♦s ♥♦ss♦s ♣r♦♣ós✐t♦s✱ ❝♦♥s✐❞❡r❛♠♦s V =Zd✱ d ≥ 2✱ ❡ E = {hx, yi ∈ Zd×Zd : ∃! i t❛❧ q✉❡ |xi−yi| = 1e|xj −yj| = 0, ∀j 6= i}✳
❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ s❡♠♣r❡ q✉❡ ❢❛❧❛r♠♦s s♦❜r❡ ♦ ❣r❛❢♦ Zd✱ ❡st❛r❡♠♦s✱ ❝♦♠ ✉♠ ❝❡rt♦ ❛❜✉s♦
❞❡ ♥♦t❛çã♦✱ ♥♦s r❡❢❡r✐♥❞♦ ❛♦ ❣r❛❢♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❡ ❝♦♥❥✉♥t♦ ❞❡ ❡❧♦s ❞❡✜♥✐❞♦s ❛❝✐♠❛✳ ■♥tr♦❞✉③✐♠♦s ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❡st❡ ♠♦❞❡❧♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❛ ❝❛❞❛ ✈ért✐❝❡ v ∈ V
❛ss♦❝✐❛♠♦s✱ ❞❡ ♠❛♥❡✐r❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ωv✱ q✉❡ ❛ss✉♠❡ ♦s ✈❛❧♦r❡s ✶ ♦✉
✵ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡sp ❡1−pr❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ωv = 1 ❞✐③❡♠♦s q✉❡v ❡stá ♦❝✉♣❛❞♦ ❡ s❡
✶✹
ωv = 0 ❞✐③❡♠♦s q✉❡ v ❡stá ✈❛③✐♦✳ ❖ ❡s♣❛ç♦ ❛♠♦str❛❧ é ♦ ❝♦♥❥✉♥t♦ Ω = {0,1}V✳ ❈❤❛♠❛♠♦s
Ω❞❡ ❡s♣❛ç♦ ❞❛s ❝♦♥✜❣✉r❛çõ❡s ❡ ❞❡♥♦t❛♠♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❡st❡ ❡s♣❛ç♦ ♣♦rω= (ωv :v ∈ V)✳
❘❡st❛✲♥♦s ❞❡✜♥✐r ✉♠❛ σ✲á❧❣❡❜r❛ ❛♣r♦♣r✐❛❞❛ ❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ P♦❞❡ s❡r
♠♦str❛❞♦ q✉❡ ♦s ❝✐❧✐♥❞r♦s ✜♥✐t♦✲❞✐♠❡♥s✐♦♥❛✐s ❢♦r♠❛♠ ✉♠❛ s❡♠✐✲á❧❣❡❜r❛✱ ❞❡♥♦t❛❞❛ ♣♦r C✳
P❛r❛ t♦❞♦ ❝✐❧✐♥❞r♦
C(j, a0, ..., an) ={ω ∈Ω : ωj+i =ai, 0≤i≤n, ai ∈ {0,1}}
❞❡✜♥✐♠♦s
µ:C 7−→[0,1],
♦♥❞❡
µ(C(j, a0, ..., an)) = Y
i:ai=1
p Y
i:ai=0
(1−p).
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❊①t❡♥sã♦ ❞❡ ❍❛❤♥ ❑♦❧♠♦❣♦r♦✈✱ ❡st❛ ♠❡❞✐❞❛ µ ♣♦❞❡ s❡r ❡①t❡♥❞✐❞❛ ❞❡
♠❛♥❡✐r❛ ú♥✐❝❛ ❛ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ Pp✱ ❞❡✜♥✐❞❛ ❡♠ F = σ(C)✱ ♦✉ s❡❥❛✱ ❛
♠❡♥♦r σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧❛ s❡♠✐✲á❧❣❡❜r❛ ❞♦s ❝✐❧✐♥❞r♦s ✜♥✐t♦✲❞✐♠❡♥s✐♦♥❛✐s ❡✱ ❛❧é♠ ❞✐ss♦✱ Pp(A) = µ(A), ∀A ∈ C✳ ❉❡st❛ ❢♦r♠❛✱ ❝♦♥str✉í♠♦s ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ (Ω,F, Pp)✱
q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛ ❢✉♥❞❛♠❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ♣❡r❝♦❧❛çã♦✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ❝❛♠✐♥❤♦ ❡♠G é ✉♠❛ s❡q✉ê♥❝✐❛ ✭✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛✮ v1, v2, . . . ❞❡ ✈ért✐❝❡s ❞❡G t❛❧ q✉❡✱∀i∈N✱ < vi, vi+1 >∈ E✳ ❉❛❞❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ω ∈Ω✱ ✉♠ ❝❛♠✐♥❤♦ é ♦❝✉♣❛❞♦
❡♠ ω s❡ t♦❞♦s ♦s s❡✉s ✈ért✐❝❡s ❡stã♦ ♦❝✉♣❛❞♦s ❡♠ ω✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❝❛♠✐♥❤♦ (v1, v2, . . .) é ❛✉t♦✲❡✈✐t❛♥t❡ s❡ vi 6=vj ♣❛r❛ i6=j✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❉❛❞❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ω ∈ Ω✱ ❞✐③❡♠♦s q✉❡ ♦ ✈ért✐❝❡ x ❡stá ❝♦♥❡❝t❛❞♦ ❛♦
✈ért✐❝❡ y ❡♠ ω s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ♦❝✉♣❛❞♦ (v1, v2, . . . , vk) ❡♠ ω t❛❧ q✉❡ v1 =x ❡ vk =y✱
v1 6=vk✳ ◗✉❛♥❞♦ ✐ss♦ ♦❝♦rr❡✱ ❡s❝r❡✈❡♠♦s x←→y✳
❖❜s❡r✈❡ q✉❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝❛♠✐♥❤♦ ♦❝✉♣❛❞♦ ❝♦♥❡❝t❛♥❞♦ ♦s ✈ért✐❝❡s x ❡ y
♥❡❝❡ss❛r✐❛♠❡♥❡ ✐♠♣❧✐❝❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝❛♠✐♥❤♦ ♦❝✉♣❛❞♦ ❛✉t♦✲❡✈✐t❛♥t❡ ❝♦♠ ❛ ♠❡s♠❛ ♣r♦♣r✐❡❞❛❞❡✳
❉❡♥♦t❛♠♦s ♣♦rCx(ω)♦ ❛❣❧♦♠❡r❛❞♦ ♦❝✉♣❛❞♦ q✉❡ ❝♦♥té♠x✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s
❞❡G✱ q✉❡ ❡stã♦ ❝♦♥❡❝t❛❞♦s ❛♦ ✈ért✐❝❡ x ♣♦r ❝❛♠✐♥❤♦s ♦❝✉♣❛❞♦s ♥❛ ❝♦♥✜❣✉r❛çã♦ ω✳ ❖s ❡❧♦s
❞❡Cx(ω) sã♦ ♦s ❡❧♦s ✐♥❝✐❞❡♥t❡s ❛♦s ✈ért✐❝❡s ❞❡ Cx(ω)✳ ❋♦r♠❛❧♠❡♥t❡ ❡s❝r❡✈❡♠♦s
✶✺
❯♠❛ ❞❛s q✉❛♥t✐❞❛❞❡s ♣r✐♥❝✐♣❛✐s ❡♠ ♣❡r❝♦❧❛çã♦ é ❛ ❢✉♥çã♦ θx(p)✱ q✉❡ r❡♣r❡s❡♥t❛ ❛
♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ✈ért✐❝❡ x ♣❡rt❡♥❝❡r ❛ ✉♠ ❛❣❧♦♠❡r❛❞♦ ♦❝✉♣❛❞♦ ✐♥✜♥✐t♦✳ ❉❡✜♥✐♠♦s
❡♥tã♦
θx : [0,1]→[0,1],
♦♥❞❡
θx(p) =Pp{ω∈Ω :|Cx(ω)|=∞}.
❖❜s❡r✈❡ q✉❡|Cx(ω)|=∞s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦s
x =v1, v2, . . . t❛✐s q✉❡ < vi, vi+1 >∈ E ❡ vi é ♦❝✉♣❛❞♦ ♣❛r❛ t♦❞♦ i ∈N✳ ◗✉❛♥❞♦ ❡①✐st❡ t❛❧
❝❛♠✐♥❤♦✱ ❞✐③❡♠♦s q✉❡ ♦ ✈ért✐❝❡x ♣❡r❝♦❧❛✳ P❡❧❛ ✐♥✈❛r✐â♥❝✐❛ tr❛♥s❧❛❝✐♦♥❛❧ ❞❡Zd ❡ ❞❛ ♠❡❞✐❞❛
Pp✱ ♥ã♦ ❤á ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ❡♠ s✉♣♦r q✉❡xé ❛ ♦r✐❣❡♠ ❞♦ ❣r❛❢♦✳ ◆❡ss❡ ❝❛s♦✱ ❡s❝r❡✈❡♠♦s
C ♣❛r❛ ♦ ❛❣❧♦♠❡r❛❞♦ ❞❛ ♦r✐❣❡♠ ❡ θ(p) ❡♠ ❧✉❣❛r ❞❡θ0(p)✳
❙❛❜❡♠♦s q✉❡ ♦ ♣❛râ♠❡tr♦p♠❡❞❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ sít✐♦ ❡st❛r ♦❝✉♣❛❞♦✳ ❊①❛♠✐♥❡♠♦s
♦ ♠♦❞❡❧♦s ♥♦s ❝❛s♦s ❡♠ q✉❡p❛ss✉♠❡ ✈❛❧♦r❡s ❡①tr❡♠♦s✳ ❖❜s❡r✈❡ q✉❡θ(0) = 0✱ ♣♦✐s ♥❡ss❡ ❝❛s♦
♥ã♦ ❤á sít✐♦s ♦❝✉♣❛❞♦s✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱|C(ω)|= 0 q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ P♦r
♦✉tr♦ ❧❛❞♦✱ ✈❡♠♦s q✉❡θ(1) = 1✱ ♣♦✐s ♥❡ss❡ ❝❛s♦ t♦❞♦s sít✐♦s ❡stã♦ ♦❝✉♣❛❞♦s ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡
✶✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ♦r✐❣❡♠ ❡stá ❝♦♥❡❝t❛❞❛ ❛ t♦❞♦s ♦s sít✐♦s q✉❛s❡ ❝❡rt❛♠❡♥t❡✳
❯♠ ❢❛t♦ ✐♥t✉✐t✐✈♦✱ ♣♦ré♠ ♥ã♦ tr✐✈✐❛❧✱ é q✉❡ ❛ ❢✉♥çã♦ θ(p) é ♠♦♥ót♦♥❛ ♥ã♦ ❞❡❝r❡s❝❡♥t❡
❡♠ p ♥♦ ✐♥t❡r✈❛❧♦ [0,1]✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❢❛t♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✶✷❪✳ ❉❡ss❛ ❢♦r♠❛✱ é
♥❛t✉r❛❧ ❞❡✜♥✐r
pc = sup{p:θ(p) = 0}.
pc é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ♠♦❞❡❧♦ ❡ é t❛❧ q✉❡
θ(p)
(
= 0 s❡ p < pc,
>0 s❡ p > pc.
❯♠ ♣r♦❝❡ss♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❡❧♦s ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❞❡ ❢♦r♠❛ s✐♠✐❧❛r à ❝♦♥str✉çã♦ ❛❝✐♠❛✳ ❉❡ ❢♦r♠❛ ❣❡r❛❧✱ ❞✐③❡♠♦s q✉❡ ♦ ❡❧♦ e ❡stá ❛❜❡rt♦ ♦✉ ❢❡❝❤❛❞♦ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p ❡ 1−p
r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡ ❛s ❞❡✜♥✐çõ❡s ❛❝✐♠❛ sã♦ ❡st❛❜❡❧❡❝✐❞❛s ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳ P❛r❛ ✉♠❛ ❞❡s❝r✐çã♦ ❞❡t❛❧❤❛❞❛ ❞❡st❛ ❝♦♥str✉çã♦✱ ✈❡❥❛ ❬✶✷❪✳
❉❡♥♦t❛♠♦s ♣♦r ps
c(G) ♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ♥♦ ❣r❛❢♦ G
❡ pe
c(G) ♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❡❧♦s ♥❡ss❡ ♠❡s♠♦ ❣r❛❢♦✳ ◗✉❛♥❞♦ ♦
❝♦♥t❡①t♦ ❡st✐✈❡r ❝❧❛r♦✱ ♦♠✐t✐r❡♠♦s ♦ í♥❞✐❝❡ ❡ ❞❡♥♦t❛r❡♠♦s ❛♣❡♥❛s ♣♦r pc ♦✉pc(G)✳
✶✻
s❡❥❛✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✱ ✈á❧✐❞♦ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ❡ ❞❡ ❡❧♦s✱ ❡ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✶✷❪✳
❚❡♦r❡♠❛ ✶✳✶✳ ❙❡ d≥2 ❡♥tã♦ 0< ps
c(Zd)<1✳
P❛r❛ ✉♠ ♣r♦❝❡ss♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❡❧♦s ❡♠ Z2✱ s❛❜❡✲s❡ q✉❡ pe
c = 12 ✭✈❡❥❛ ❙❡çã♦ ✶✶✳✸ ❡♠ ❬✶✷❪✮✳ ❊ss❡ ❢❛t♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❚❡♦r❡♠❛ ✶✳✸✸ ❡♠ ❬✶✷❪✱ ❣❛r❛♥t❡ q✉❡
ps
c(Z2)>
1 2.
❊♠ ❬✷✼❪ ❢♦✐ ♠♦str❛❞♦ q✉❡
psc(Z2)≤0,679492.
❉❡ ❢❛t♦✱ r❡s✉❧t❛❞♦s ❞❡ s✐♠✉❧❛çõ❡s ✐♥❞✐❝❛♠ q✉❡
psc(Z2)≈0,59.
❖ ✈❛❧♦r ❡①❛t♦ ❞❡ ps
c ♥ã♦ s❡rá ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♥ós✱ ❜❛st❛ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ q✉❡0< psc <1✳
❯♠ ❢❛t♦ ❞✐❣♥♦ ❞❡ ♠❡♥çã♦ é q✉❡ s❡ ❛ ♦r✐❣❡♠ ♣❡r❝♦❧❛ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦s✐t✐✈❛✱ ❡♥tã♦ ❡①✐st❡ ❛❣❧♦♠❡r❛❞♦ ✐♥✜♥✐t♦ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ P❛r❛ ✈❡r ✐ss♦ ❝♦♥s✐❞❡r❡ ♦ ❡✈❡♥t♦
[
x∈Zd
{ω ∈Ω :|Cx(ω)|=∞},
✐st♦ é✱ ♦ ❡✈❡♥t♦ ❡①✐st❡ ❛❣❧♦♠❡r❛❞♦ ✐♥✜♥✐t♦ ❡ ❛ ❢✉♥çã♦
ψ : [0,1]→[0,1],
♦♥❞❡
ψ(p) =Pp (
[
x∈Zd
{ω ∈Ω :|Cx(ω)|=∞} )
.
◆♦t❡ q✉❡ ♦ ❡✈❡♥t♦ ❛❝✐♠❛ é ❝❛✉❞❛❧✱ ♣♦✐s ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ❡st❛❞♦ ❞❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ sít✐♦s✳ ❚❡♠♦s ❡♥tã♦✱ ♣❡❧❛ ▲❡✐ ✵✲✶ ❞❡ ❑♦❧♠♦❣♦r♦✈✱ q✉❡ψ(p) = 0 ♦✉ψ(p) = 1✳ ❖❜s❡r✈❡ q✉❡
ψ(p)≤ X
x∈Zd
✶✼
▲♦❣♦✱ s❡ θ(p) = 0✱ ❡♥tã♦ ψ(p) = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♥♦t❡ q✉❡
{ω∈Ω :|C(ω)|=∞} ⊂ [
x∈Zd
{ω ∈Ω :|Cx(ω)|=∞}.
P♦rt❛♥t♦
Pp (
[
x∈Zd
{ω∈Ω :|Cx(ω)|=∞ )
≥Pp{ω∈Ω :|C(ω)|=∞},
❡ θ(p)>0✐♠♣❧✐❝❛ q✉❡ ψ(p) = 1✳ ❉❡ss❛ ❢♦r♠❛ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✶✳✷✳
ψ(p) =
(
0 s❡ θ(p) = 0,
1 s❡ θ(p)>0.
❆ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ❡stá ♥♦ ❢❛t♦ ❞❡ q✉❡✱ ❡♠ ❞✉❛s ♦✉ ♠❛✐s ❞✐♠❡♥sõ❡s✱ ❡①✐st❡♠ ❞✉❛s ❢❛s❡s ❞✐st✐♥t❛s ❞♦ ♣r♦❝❡ss♦ ❞❡ ♣❡r❝♦❧❛çã♦✳ ❙❡ p < pc✱ ♦✉ s❡❥❛✱ ♥❛ ❢❛s❡ s✉❜❝rít✐❝❛✱ ❝❛❞❛
✈ért✐❝❡ ❞❡ Zd ❡stá✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡✱ ❡♠ ✉♠ ❛❣❧♦♠❡r❛❞♦ ♦❝✉♣❛❞♦ ✜♥✐t♦ ♦✉ ✈❛③✐♦✳ ❉❡st❛
❢♦r♠❛✱ t♦❞♦s ❛❣❧♦♠❡r❛❞♦s ♦❝✉♣❛❞♦s sã♦ ✜♥✐t♦s ♦✉ ✈❛③✐♦s ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✶✳ ◆♦ ❡♥t❛♥t♦✱ s❡ p > pc✱ ♦✉ s❡❥❛✱ ♥❛ ❢❛s❡ s✉♣❡r❝rít✐❝❛✱ ❝❛❞❛ ✈ért✐❝❡ ❡stá✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦s✐t✐✈❛✱ ❡♠
✉♠ ❛❣❧♦♠❡r❛❞♦ ♦❝✉♣❛❞♦ ✐♥✜♥✐t♦✱ ❞❡ ❢♦r♠❛ q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❛❣❧♦♠❡r❛❞♦ ♦❝✉♣❛❞♦ ✐♥✜♥✐t♦ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳
❊♠ Zd é ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r q✉❡✱ s❡ ❡①✐st❡ ❛❣❧♦♠❡r❛❞♦ ✐♥✜♥✐t♦✱ ❡st❡ é ú♥✐❝♦ q✉❛s❡
❝❡rt❛♠❡♥t❡✳ P❛r❛ ✈❡r ✐ss♦✱ ❝♦♥s✐❞❡r❡ ♦ ❡✈❡♥t♦
Ik(ω) = {ω∈Ω :❊①✐st❡♠ ❡①❛t❛♠❡♥t❡ ❦ ❛❣❧♦♠❡r❛❞♦s ✐♥✜♥✐t♦s ❡♠ω},
0 ≤ k ≤ ∞. ❚❡♠✲s❡ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ♥❛ ❙❡çã♦ ✽✳✷ ❞❡
❬✶✷❪✳
❚❡♦r❡♠❛ ✶✳✸✳ ❙❡ ♣ é t❛❧ q✉❡ θ(p)>0✱ ❡♥tã♦
Pp{ω ∈Ω :I1(ω)}= 1.
✶✽
❉❡s✐❣✉❛❧❞❛❞❡ ❋❑●
❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❋❑● s❡rá ✉t✐❧✐③❛❞❛ ♠❛✐s t❛r❞❡ ♥❡st❛ t❡s❡✳ ❆ s❡❣✉✐r ❞❡✜♥✐♠♦s ♦ q✉❡ é ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝r❡s❝❡♥t❡ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❈♦♥s✐❞❡r❡ ω ❡ ω′ ∈ Ω✳ ❉✐③❡♠♦s q✉❡ ω ω′ s❡ ω(x)≤ ω′(x) ∀x ∈ V✱ ✐st♦ é✱ s❡ ❝❛❞❛ ✈ért✐❝❡ q✉❡ ❡stá ♦❝✉♣❛❞♦ ❡♠ ω t❛♠❜é♠ ❡stá ♦❝✉♣❛❞♦ ❡♠ ω′✳
❉❡✜♥✐çã♦ ✶✳✺✳ ❆ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ X é ❝❤❛♠❛❞❛ ❝r❡s❝❡♥t❡ s❡ X(ω) ≤ X(ω′) s❡♠♣r❡ q✉❡
ω ω′✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♦ ❡✈❡♥t♦ A é ❝❤❛♠❛❞♦ ❝r❡s❝❡♥t❡ s❡ s✉❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ I
A
❢♦r ❝r❡s❝❡♥t❡✳
❊①❡♠♣❧♦ ✶ ❙❡ v1 ❡ v2 sã♦ ✈ért✐❝❡s ❞❡ Zd✱ ❡♥tã♦ ♦ ❡✈❡♥t♦ {ω ∈ Ω : v1 ←→ v2 ❡♠ ω} é ❝r❡s❝❡♥t❡✳
❙❡A ❡B sã♦ ❡✈❡♥t♦s ❝r❡s❝❡♥t❡s✱ ❡♥tã♦✱ ∀p∈(0,1)✱ é ✐♥t✉✐t✐✈❛ ❛ ✐❞é✐❛ ❞❡ q✉❡
Pp{A|B} ≥Pp{A},
♣♦✐s ❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦ ❝r❡s❝❡♥t❡ B ♥❛ ❝♦♥✜❣✉r❛çã♦ ω0✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ sít✐♦s ❛❜❡rt♦s ❡ ♣♦rt❛♥t♦ ❛✉♠❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦ ❝r❡s❝❡♥t❡A ❡♠ ω0✳ ➱ ♣r❡❝✐s❛♠❡♥t❡ ✐ss♦ ♦ q✉❡ ❞✐③ ♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳
❚❡♦r❡♠❛ ✶✳✹✳ ❉❡s✐❣✉❛❧❞❛❞❡ ❋❑●
❛✮ ❙❡❥❛♠ ❳ ❡ ❨ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❝r❡s❝❡♥t❡s ❝♦♠ Ep(X2)<∞ ❡ Ep(Y2)<∞✳ ❊♥tã♦
Ep(XY)≥Ep(X)Ep(Y).
❜✮ ❙❡❥❛♠ A ❡ B ❡✈❡♥t♦s ❝r❡s❝❡♥t❡s✳ ❊♥tã♦
Pp{A∩B} ≥Pp{A}Pp{B}. ✭✶✳✶✳✶✮
❆ ♣r♦✈❛ ❞❛ ♣❛rt❡ ❜✮ ❞♦ t❡♦r❡♠❛ ❛❝✐♠❛ é ❢❛❝✐❧♠❡♥t❡ ♦❜t✐❞❛ t♦♠❛♥❞♦ X =IA ❡ Y = IB✳
P❛r❛ ❛ ♣r♦✈❛ ❞❛ ♣❛rt❡ ❛✮ ✈❡❥❛ ❬✶✷❪✳
✶✾
✶✳✷ P❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s
❖ ♣r♦❜❧❡♠❛ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ■t❛✐ ❇❡♥❥❛♠✐♥✐ ❡ ❍❛rr② ❑❡st❡♥ ❡♠ ❬✸❪✳ ❖ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r ❢♦r♠✉❧❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s ✉♠❛ ❣r❛❢♦ G = (V,E) ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V ✐♥✜♥✐t♦ ❡♥✉♠❡rá✈❡❧✳
❘❡❛❧✐③❛♠♦s ✉♠ ♣r♦❝❡ss♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ♥❡st❡ ❣r❛❢♦✳ ❈♦♠♦ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♠❡s♠♦ ❡s♣❛ç♦ ♣r❡✈✐❛♠❡♥t❡ ❞❡✜♥✐❞♦✱ ♦✉ s❡❥❛✱(Ω,F, Pp)✱ ♦♥❞❡
Ω ={0,1}V✱ F é ❛σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❝✐❧✐♥❞r♦s ❡♠ Ω ❡ P
p =Qv∈Vµ(v)é ♦ ♣r♦❞✉t♦ ❞❡
♠❡❞✐❞❛s ❞❡ ❇❡r♥♦✉❧❧✐ ❝♦♠ ♣❛râ♠❡tr♦ p✳
❉❡♥♦t❡ Ξ = {0,1}N
✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❜✐♥ár✐❛ s❡♠✐✲✐♥✜♥✐t❛ ξ = (ξ1, ξ2, . . .) ∈ Ξ s❡rá ❝❤❛♠❛❞❛ ✉♠❛ ♣❛❧❛✈r❛✳ ❉❛❞♦s ✉♠❛ ♣❛❧❛✈r❛ ξ ∈ Ξ✱ ✉♠ ✈ért✐❝❡ v ∈ V ❡ ✉♠❛ ❝♦♥✜❣✉r❛çã♦
ω ∈Ω✱ ✐♥tr♦❞✉③✐♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✻✳ ❉✐③❡♠♦s q✉❡ ❛ ♣❛❧❛✈r❛ ξ é ✈✐st❛ ♥❛ ❝♦♥✜❣✉r❛çã♦ ω ❛ ♣❛rt✐r ❞♦ ✈ért✐❝❡ v s❡
❡①✐st✐r ✉♠ ❝❛♠✐♥❤♦ ❛✉t♦✲❡✈✐t❛♥t❡ (v =v0, v1, v2. . .) t❛❧ q✉❡ ωvi =ξi✱ ∀i= 1,2, . . .✳
◆♦t❡ q✉❡ ♦ ❡st❛❞♦ ❞❡ v é ✐♥❞✐❢❡r❡♥t❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ♥♦
♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡s❝r✐t♦ ❛♥t❡r✐♦r♠❡♥t❡ ❡stá✈❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r s♦❜ q✉❛✐s ❝✐r❝✉♥stâ♥❝✐❛s ❛ ♣❛❧❛✈r❛ →1 = (1,1, . . .)é ✈✐st❛ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦s✐t✐✈❛✳
❆♦ ❡st✉❞❛r ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s✱ ♥♦ ❢✉♥❞♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♥❛ ♦❝♦rrê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❝❛♠✐♥❤♦s (v0, v1, . . .)t❛✐s q✉❡ ωvi =ξi✱ i≥1✱ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ♣r❡s❝r✐t❛(ξi)i≥1 ∈Ξ✳
●♦st❛rí❛♠♦s t❛♠❜é♠ ❞❡ ❞❡t❡r♠✐♥❛r ❡♠ q✉❡ ❝✐r❝✉♥stâ♥❝✐❛s ❛ ❝♦❧❡çã♦ ❞❡ s❡q✉ê♥❝✐❛s ξ q✉❡
sã♦ ✈✐st❛s é ❣r❛♥❞❡ ❡♠ ❛❧❣✉♠ s❡♥t✐❞♦ q✉❡ s❡rá ❞❡✜♥✐❞♦ ♣♦st❡r✐♦r♠❡♥t❡✳ ❈♦♠ ❡st❡ ✐♥t✉✐t♦ ✐♥tr♦❞✉③✐♠♦s ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿
❋✐①❛❞♦s ω ∈Ω❡ v ∈ V✱ ❝♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s ❛❧❡❛tór✐♦s
Sv(ω) = {ξ∈Ξ :ξ é ✈✐st❛ ❡♠ ω ❛ ♣❛rt✐r ❞❡ v}
❡
S∞(ω) :=
[
v∈V
Sv(ω) ={ξ ∈Ξ :ξé ✈✐st❛ ❡♠ ω ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ ✈ért✐❝❡}.
❖ ♣r✐♠❡✐r♦ ❞❡❧❡s é ❢♦r♠❛❞♦ ♣❡❧❛s ♣❛❧❛✈r❛s ξ ∈Ξ q✉❡ sã♦ ✈✐st❛s ♥❛ ❝♦♥✜❣✉r❛çã♦ ω ❛ ♣❛rt✐r
✷✵
q✉❛✐s ❝✐r❝✉♥stâ♥❝✐❛s ♦s ❡✈❡♥t♦s
A1 ={ω∈Ω :S∞(ω) = Ξ}
❡
A2 ={ω∈Ω :∃v ∈V t❛❧ q✉❡ Sv(ω) = Ξ}
♦❝♦rr❡♠ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❆♥t❡s ❞❡ ❝♦♥s✐❞❡r❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ❡✈❡♥t♦s ❛❝✐♠❛✱ ❡♥✉♥❝✐❛♠♦s ❛ Pr♦♣♦s✐çã♦ ✷ ❞❡ ❬✸❪✱ q✉❡ ❣❛r❛♥t❡ ❛ ♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❞♦s ♠❡s♠♦s✳
Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡ G é ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✱ ❡♥tã♦ ♦s ❡✈❡♥t♦s A1 ❡ A2 sã♦ ♠❡♥s✉rá✈❡✐s ❝♦♠ r❡❧❛çã♦ ❛ F✱ ♦♥❞❡ F é ❛ σ✲ á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r {ωv :v ∈ V}✳
❊♠ ❬✸❪✱ ♦s ❛✉t♦r❡s ♠♦str❛♠ q✉❡ ♦s ❡✈❡♥t♦s A1 ❡A2 ❞❡✜♥✐❞♦s ❛❝✐♠❛ ♦❝♦rr❡♠ P1
2 ✲ q✉❛s❡
❝❡rt❛♠❡♥t❡ ❡♠ ✉♠ ♠♦❞❡❧♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ❡♠Zd ♣❛r❛d≥10❡d≥40
r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦t❡ q✉❡ ♥♦ ❡✈❡♥t♦A2✱ ♦ ✈ért✐❝❡v✱ ❛ ♣❛rt✐r ❞♦ q✉❛❧ s❡ ✈ê t♦❞❛s ❛s ♣❛❧❛✈r❛s✱
♣♦❞❡ ♠✉❞❛r ❞❡ ❝♦♥✜❣✉r❛çã♦ ♣❛r❛ ❝♦♥✜❣✉r❛çã♦✳
▼❡s♠♦ q✉❡ ❡♠ ✉♠ ❣r❛❢♦ G ♥ã♦ s❡❥❛ ♣♦ssí✈❡❧ ✈❡r t♦❞❛s ♣❛❧❛✈r❛s✱ ♦✉ s❡❥❛✱ s❡
Pp{ω ∈ Ω : S∞(ω) = Ξ} = 0✱ ❛✐♥❞❛ ❛ss✐♠ s❡r✐❛ ♣♦ssí✈❡❧ ✈❡r q✉❛s❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s ♥❡ss❡ ❣r❛❢♦✳ ❆q✉✐✱ q✉❛s❡ t♦❞❛s s❡ r❡❢❡r❡ à ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣r♦❞✉t♦µα =
∞
Y
i=1
νi ❡♠ Ξ✱ ♦♥❞❡
♣❛r❛ ❝❛❞❛ i
νi(1) = 1−νi(0) =α, ♣❛r❛ ❛❧❣✉♠ 0< α <1. ✭✶✳✷✳✶✮
◆♦t❡ q✉❡ s♦❜ µα ❛s ξi sã♦ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s
❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ❝♦♠ ♣❛râ♠❡tr♦ α✳ ❉❡st❛ ❢♦r♠❛ t❡♠♦s ❞❡✜♥✐❞♦ ♦✉tr♦ ❡s♣❛ç♦
❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ (Ξ,B, µα)✱ ♦♥❞❡ B é ❛ σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❝✐❧✐♥❞r♦s ❡♠ Ξ✳ P❛r❛ t♦❞❛
ξ∈Ξ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦
τp : Ξ→[0,1]
τp(ξ) = Pp{ω ∈Ω :ξ é ✈✐st❛ ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ v}.
✷✶
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡ G é ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✱ ❡♥tã♦ τp(.) é B✲♠❡♥s✉rá✈❡❧✳ ❙❡ G =Zd ♦✉ ✉♠❛
ár✈♦r❡ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛✱ ❡♥tã♦ τp(.) t♦♠❛ s♦♠❡♥t❡ ♦s ✈❛❧♦r❡s ✵ ♦✉ ✶✳
❉❡✜♥✐çã♦ ✶✳✼✳ ❉✐③❡♠♦s q✉❡ ❛ ♣❛❧❛✈r❛ ξ ♣❡r❝♦❧❛ s❡ τp(ξ) = 1✳
❖✉tr❛ ♦❜❡r✈❛çã♦ ✐♠♣♦rt❛♥t❡✱ ❝✉❥❛ ❞❡♠♦str❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✸❪✱ é q✉❡ ❡♠ ♠✉✐t♦s ❣r❛❢♦s τp(ξ) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ♥❡♥❤✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ξ1, ξ2, . . . , ξn✳ ■st♦
é✱ s❡ τp(ξ) > 0✱ ξ = (ξ1, ξ2, . . . , ξn, . . .)✱ ❡♥tã♦ τp( ˆξ) > 0✱ ♦♥❞❡ ξˆ= ( ˆξ1,ξˆ2, . . . ,ξˆn, ξn+1, . . .)✳ ❈♦♠♦ τp t♦♠❛ s♦♠❡♥t❡ ♦s ✈❛❧♦r❡s ✵ ♦✉ ✶✱ t❡♠♦s τp( ˆξ) = 1 ♣❛r❛ t♦❞❛ ξˆq✉❡ ❞✐❢❡r❡ ❞❡ ξ ❡♠
❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❙❡♥❞♦ ❛ss✐♠✱ τp é ✉♠❛ ✈❛r✐á✈❡❧ ❝❛✉❞❛❧✱ ❧♦❣♦
♣❡❧❛ ▲❡✐ ✵✲✶ ❞❡ ❑♦❧♠♦❣♦r♦✈ t❡♠♦s q✉❡
µα{ξ :τp(ξ) = 1}= 0 ♦✉1, ∀α∈(0,1).
❙❡ µα{ξ∈Ξ :τp(ξ) = 1}= 1✱ ❞✐③❡♠♦s q✉❡ ❛ ♣❛❧❛✈r❛ ❛❧❡❛tór✐❛ ♣❡r❝♦❧❛✳
❖✉tr♦ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛♦ ❝♦♥s✐❞❡r❛r♠♦s ♦ ❝♦♥❥✉♥t♦
Λ :={(ξ, ω) :ξé ✈✐st❛ ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ ✈ért✐❝❡ v ♥❛ ❝♦♥✜❣✉r❛çã♦ω}.
P❡❧❛ Pr♦♣♦s✐çã♦ ✶ ❡♠ ❬✸❪✱ t❡♠♦s q✉❡Λé ✉♠ ❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ s❡❣✉♥❞♦ ❛σ✲á❧❣❡❜r❛ F ×B✳
❙❡❥❛ IΛ ❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ ❞♦ ❝♦♥❥✉♥t♦ Λ✳ ❚❡♠♦s q✉❡
Z
Ξ
Z
Ω
IΛdPpdµα = Z
Ξ
τp(ξ)dµα = Z
{ξ∈Ξ:τp(ξ)=1}
dµα =µα{ξ∈Ξ :τp(ξ) = 1}. ✭✶✳✷✳✷✮
P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s
Z
Ω
Z
Ξ
IΛdµαdPp = Z
Ω
µα(S∞(ω))dPp = Z
{ω∈Ω:µα(S∞(ω))=1}
dPp
= Pp{ω∈Ω :µα(S∞(ω)) = 1}. ✭✶✳✷✳✸✮
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐✱ t❡♠♦s q✉❡ ❛s ❡①♣r❡ssõ❡s ❡♠ ✭✶✳✷✳✷✮ ❡ ✭✶✳✷✳✸✮ sã♦ ✐❣✉❛✐s✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ ❛ ♣❛❧❛✈r❛ ❛❧❡❛tór✐❛ ♣❡r❝♦❧❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s ❞❡ ♠❡❞✐❞❛ µα ✐❣✉❛❧ ❛ ✶ ♣♦❞❡ s❡r ✈✐st♦ ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♠ ✈ért✐❝❡ ❡♠ Zd✱ Pp ✲ q✉❛s❡
✷✷
Pp ✲ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❱❡❥❛♠♦s ♦s s❡❣✉✐♥t❡s ❡①❡♠♣❧♦s✿
❊①❡♠♣❧♦ ✷ ❈♦♥s✐❞❡r❡ ♦ ❝❛s♦ ❡♠ q✉❡ G =T = (V,E) é ❛ r❡❞❡ tr✐❛♥❣✉❧❛r✳ ◆❡ss❡ ❝❛s♦✱
V = Z2 ❡ E =E(Z2) ❛❝r❡s❝✐❞♦ ❞❡ ✉♠ ❡❧♦ ❞✐❛❣♦♥❛❧ ❡♠ ❝❛❞❛ ❢❛❝❡ ❞❡ T ♥♦ s❡♥t✐❞♦ s✉❞♦❡st❡
♥♦r❞❡st❡✳ ❋♦r♠❛❧♠❡♥t❡✱ E sã♦ ♦s ❡❧♦s ❡♥tr❡ ♣❛r❡s(i1, j1) ❡ (i2, j2)t❛✐s q✉❡
|i1−i2|+|j1−j2|= 1
♦✉
i2 =i1+ 1, j2 =j1−1.
◆❡ss❡ ❣r❛❢♦ t❡♠♦sps
c(T) = 12✱ ❡ ❡♠ ❬✶✻❪ ♦s ❛✉t♦r❡s ♠♦str❛♠ q✉❡ ❛ ♣❛❧❛✈r❛ →
1 ♥ã♦ é ✈✐st❛ ❡♠
T q✉❛♥❞♦ p= 12 q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡
P1
2{ω ∈Ω :S∞(ω) = Ξ ❡♠ T } = 0.
◆♦ ❡♥t❛♥t♦✱ ❛ ♣❛❧❛✈r❛ ✭✶✱✵✱✶✱✵✱✳✳✳✮ é ✈✐st❛ ❡♠ T q✉❛♥❞♦ p= 12 q✉❛s❡ ❝❡rt❛♠❡♥t❡✱ ✈❡❥❛ ❬✷✽❪✳
❉❡ ❢❛t♦✱ ❡♠ ❬✶✽❪ ♦s ❛✉t♦r❡s ♠♦str❛♠ q✉❡✱ ♣❛r❛ t♦❞♦ 0< α <1✱ q✉❛s❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s sã♦
✈✐st❛s ❡♠T P1
2 ✲ q✉❛s❡ ❝❡rt❛♠❡♥t❡✳ ❆q✉✐ q✉❛s❡ t♦❞❛s s❡ r❡❢❡r❡ à ♠❡❞✐❞❛ ❞❛❞❛ ♥❛ ❡①♣r❡ssã♦
✭✶✳✷✳✶✮✳
❊①❡♠♣❧♦ ✸ ❈♦♥s✐❞❡r❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ ❡❧♦s ❡♠ Z2✳ ❈♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✶✷❪✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❡st❡ ♠♦❞❡❧♦ ❛ss♦❝✐❛♠♦s✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ❛ ❝❛❞❛ ❡❧♦e ❞❡Zd✉♠❛
✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛X(e) t❛❧ q✉❡
Pp{ω∈Ω :X(e) = 1}=p= 1−Pp{ω∈Ω :X(e) = 0}.
◆❡ss❡ ❝❛s♦✱ ♦ s✐❣♥✐✜❝❛❞♦ ❞❛ ❢r❛s❡ ✑ξé ✈✐st❛ ❛ ♣❛rt✐r ❞❡v✑ ❞❡✈❡r✐❛ s❡r q✉❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦
❛✉t♦✲❡✈✐t❛♥t❡ (v0 = v, v1, . . .) ❡♠ Zd✱ ❝♦♠❡ç❛♥❞♦ ❡♠ v✱ t❛❧ q✉❡ X(ei) = ξi✱ ♦♥❞❡ ei é ♦ ❡❧♦
❡♥tr❡ ♦s ✈ért✐❝❡s vi−1 ❡ vi✳ ❊♠ Z2 t❡♠♦s pec = 12✱ ❡ s❛❜❡✲s❡ q✉❡ q✉❛♥❞♦ p= 1
2✱ ❛ ♣❛❧❛✈r❛ →
1
♥ã♦ é ✈✐st❛ ❡♠Z2 q✉❛s❡ ❝❡rt❛♠❡♥t❡ ✭✈❡❥❛ ❙❡çã♦ ✶✶✳✸ ❞❡ ❬✶✷❪✮✳ P♦rt❛♥t♦✱
P1
2{ω ∈Ω :S∞(ω) = Ξ ❡♠ Z
2}= 0.
P♦❞❡r✐❛ s❡ ❡s♣❡r❛r q✉❡ q✉❛s❡ t♦❞❛s ❛s ♣❛❧❛✈r❛s ♣♦❞❡r✐❛♠ s❡r ✈✐st❛s q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠Z2✳ P♦ré♠ ❡ss❡ ♣r♦❜❧❡♠❛ ❝♦♥t✐♥✉❛ ❡♠ ❛❜❡rt♦ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳
✷✸
✉♠ ❣r❛❢♦ G t❛❧ q✉❡ pc(G)< 12✳
▲❡♠❛ ✶✳✶✳ ❙❡❥❛ G = (V(G),E(G)) ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ✐♥✜♥✐t♦
❡♥✉♠❡rá✈❡❧ ❡ ❝♦♥s✐❞❡r❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞❡ sít✐♦s ✐♥❞❡♣❡♥❞❡♥t❡✱ ❝♦♠ ♣❛râ♠❡tr♦
p ∈ [0,1]✱ ♥♦ ❣r❛❢♦ G✳ ❉❡✜♥❛ ♦ ❡✈❡♥t♦ Zξ(v) = {ω ∈ Ω : ξ é ✈✐st❛ ❡♠ G ❛ ♣❛rt✐r ❞❡ ✈}✳
❊♥tã♦✱ ❞❛❞❛ q✉❛❧q✉❡r ♣❛❧❛✈r❛ ξ∈Ξ ❡ q✉❛❧q✉❡r ✈ért✐❝❡ v0 ∈V(G)✱ t❡♠♦s q✉❡
Pp(Zξ(v0))≥min{Pp(Z→
0(v0)), Pp(Z→1(v0))}. ✭✶✳✷✳✹✮ ◆♦t❡ q✉❡✱ s❡ pc(G)< 12✱ ❡♥tã♦ ∀p∈(pc(G),1−pc(G))✱ t❡♠♦s
Pp{ω∈Ω :
→
1 é ✈✐st❛ ❛ ♣❛rt✐r ❞❡ v ∈ V(G)}>0 ✭✶✳✷✳✺✮
❡
Pp{ω ∈Ω :
→
0 é ✈✐st❛ ❛ ♣❛rt✐r ❞❡ v ∈ V(G)}>0. ✭✶✳✷✳✻✮
◆❡ss❡ ❝❛s♦✱ ♦ ▲❡♠❛ ✶✳✶ ✐♠♣❧✐❝❛ q✉❡
Pp{ω∈Ω :ξé ✈✐st❛ ❛ ♣❛rt✐r ❞❡ v ∈ G}>0 ∀ξ ∈Ξ, ✭✶✳✷✳✼✮
♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
τp(ξ) = 1 ∀ξ ∈Ξ, ✭✶✳✷✳✽✮
✐st♦ é✱
µα{ξ ∈Ξ :τp(ξ) = 1}= 1, ∀α∈(0,1). ✭✶✳✷✳✾✮
❖ ❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐ ✐♠♣❧✐❝❛ ❡♥tã♦ q✉❡
Pp{ω∈Ω :µα(S∞(ω)) = 1 ❡♠ G}= 1.
❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❛♥t❡ ♦❝♦rr❡ q✉❛♥❞♦ G = Zd✱ d ≥ 3✳ ◆♦t❡ q✉❡
pc(Zd+1)≤pc(Zd)✱ ♣♦✐s
{|C(ω)|=∞❡♠ Zd} ⊂ {|C(ω)|=∞❡♠Zd+1}.
❊♠ ❬✼❪✱ ❈❛♠♣❛♥✐♥♦ ❡ ❘✉ss♦ ♠♦str❛r❛♠ q✉❡ pc(Z3)< 12✳ ❉❡st❛ ❢♦r♠❛✱ t❡♠♦s q✉❡
✷✹
q✉❛♥❞♦ pc(Zd)< p <1−pc(Zd)✳
❖❜s❡r✈❡ q✉❡ ❛ ❊①♣r❡ssã♦ ✭✶✳✷✳✶✵✮ ♥ã♦ ✐♠♣❧✐❝❛ q✉❡
Pp{ω∈Ω :S∞(ω) = Ξ}= 1 ✭✶✳✷✳✶✶✮
❡♠ Zd✳
❯♠ ❡①❡♠♣❧♦ ❞❡ss❛ s✐t✉❛çã♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✸❪✱ ♦♥❞❡ ♦s ❛✉t♦r❡s ❝♦♥str♦❡♠ ✉♠ ❣r❛❢♦ ❡♠ q✉❡
P1
2{ω∈Ω :µ12(S∞(ω)) = 1, S∞ 6= Ξ}= 1.
◆♦t❡ q✉❡ ✐ss♦ t❛♠❜é♠ ♦❝♦rr❡ ♥❛ r❡❞❡ ❚r✐❛♥❣✉❧❛r✱ ❝♦♠♦ ✈✐st♦ ♥♦ ❊①❡♠♣❧♦ ✷✳ ❚❛❧✈❡③ ♦ ♣r♦❜❧❡♠❛ ❡♠ ❛❜❡rt♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❡♠ ♣❡r❝♦❧❛çã♦ ❞❡ ♣❛❧❛✈r❛s s❡❥❛ ♠♦str❛r ✭✶✳✷✳✶✶✮ q✉❛♥❞♦
d= 3✳
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡ P❡r❝♦❧❛çã♦ ❞❡ ▲♦♥❣♦ ❆❧❝❛♥❝❡✳
✶✳✸ P❡r❝♦❧❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡
❯♠❛ ♦✉tr❛ ✈❛r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣❡r❝♦❧❛çã♦ ✉s✉❛❧ é ♦ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❝♦♠ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ♦ q✉❛❧ ❞❡s❝r❡✈❡♠♦s ❛ s❡❣✉✐r✳ ❙❡❥❛ Ld = (V,E) ♦ ❣r❛❢♦ ❡♠
q✉❡V =Zd, d≥2✱ ❡
E ={hx, yi ⊂Zd×Zd:∃!i∈ {1, . . . , d}t❛❧ q✉❡ xi 6=yi ❡xj =yj,∀j 6=i},
✐st♦ é✱ E é ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ❡❧♦s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ❡♠ t♦❞❛s ❞✐r❡çõ❡s ♣❛r❛❧❡❧❛s ❛ ❛❧❣✉♠
❡✐①♦ ❝♦♦r❞❡♥❛❞♦✳ ❉❡✜♥❛
Ek={hv, ui ∈Zd×Zd:∃!i∈ {1, . . . , d} t❛❧ q✉❡ |vi−ui|=k ❡vj =uj,∀j 6=i}, ✭✶✳✸✳✶✮
✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧♦s ♣❛r❛❧❡❧♦s ❛ ❛❧❣✉♠ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦ ❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦k✱ ❡ ❝♦♥s✐❞❡r❡
♦ ❣r❛❢♦Ld
K = (Zd,∪Kn=1En)✳ ❖❜s❡r✈❡ q✉❡✱LdK ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡Ld❛♣❛❣❛♥❞♦ t♦❞♦s
♦s ❡❧♦s ❝✉❥♦s ❝♦♠♣r✐♠❡♥t♦s sã♦ ♠❛✐♦r❡s q✉❡K✳
❈♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❡r ♣❛❧❛✈r❛s ❡♠ Ld✱ d≥1✳ ❆✜r♠❛♠♦s q✉❡
Pp{ω∈Ω :µα(S∞(ω)) = 1 ❡♠ Ld}= 1, ∀p∈(0,1).
✷✺
τp(ξ) = 1 ∀ξ ∈Ξ❡ ∀p∈(0,1)♥♦ ❣r❛❢♦ L1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✳ ❆q✉✐✱ L1 é ♦
❣r❛❢♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V =Z ❡ ❝♦♥❥✉♥t♦ ❞❡ ❡❧♦s
E ={hx, yi ∈Z×Z:x6=y}.
❉❛❞♦ m∈N, m≥2 ✜①♦ ✱ ❝♦♥s✐❞❡r❡ ♦ ❡✈❡♥t♦
Bj ={ω ∈Ω :∃v1❡ v2 ∈[m(j−1), mj)t❛✐s q✉❡ ωv1 = 1❡ ωv2 = 0}, j ∈N.
❯♠ ❝á❧❝✉❧♦ s✐♠♣❧❡s ♠♦str❛ q✉❡
Pp{Bj}= 1−(1−p)m−pm ≡c(p)>0,∀j ∈N,∀p∈(0,1).
❈♦♠♦ ♦s ❡✈❡♥t♦s B′
js sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ X
j
Pp{Bj}= X
j
c(p) =∞,
♣♦❞❡♠♦s ❢❛③❡r ✉s♦ ❞♦ ▲❡♠❛ ❞❡ ❇♦r❡❧ ❈❛♥t❡❧❧✐ ❡ ❝♦♥❝❧✉✐r q✉❡
Pp{ω ∈Ω :Bj ✐♥✜♥✐t❛s ✈❡③❡s}= 1. ✭✶✳✸✳✷✮
P♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ τp(ξ) = 1, ∀ξ ∈ Ξ✳ P♦rt❛♥t♦✱ ♣❡❧♦
❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐✱ t❡♠♦s q✉❡
Pp{ω ∈Ω :µα(S∞(ω)) = 1 ❡♠ Ld}= 1, ∀p∈(0,1).
❉❡ ❢❛t♦✱ ❛ ❊q✉❛çã♦ ✭✶✳✸✳✷✮ ✐♠♣❧✐❝❛ q✉❡
Pp{ω ∈Ω :S∞(ω) = Ξ❡♠ Ld}= 1, ∀p∈(0,1).
❊♠ ❬✶✾❪ ❢♦✐ ❝♦♥❥❡❝t✉r❛❞♦ s❡ ❛ ✐♥✜♥✐t✉❞❡ ❞♦ t❛♠❛♥❤♦ ❞♦s ❡❧♦s é ❡ss❡♥❝✐❛❧ ♣❛r❛ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛❝✐♠❛ s❡❥❛ t♦t❛❧❄ ◆❡ss❡ ♠❡s♠♦ tr❛❜❛❧❤♦ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✶✮✱ ♦ ❛✉t♦r ♠♦str♦✉ q✉❡✱ ∀d≥2✱ ∀p∈(0,1) ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦K =K(p)✱ t❛❧ q✉❡
Pp (
[
v∈V
{ω∈Ω :ξé ✈✐st❛ ❡♠ LdK ❛ ♣❛rt✐r ❞❡v}
)
