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❈r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥

• ❛❞② ❑♦③♠❛

❖♥❧✐♥❡ ❖♣❡♥ Pr♦❜❛❜✐❧✐t② ❙❝❤♦♦❧✱ ✷✵✷✵

P❡r❝♦❧❛t✐♦♥ ✲ ❞❡✜♥✐t✐♦♥s

❊①❛♠✐♥❡ t❤❡ ❣r❛♣❤ Zd✱ d ≥ 2✳ ❋♦r ❛ ❦❡❡♣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ❞❡❧❡t❡ ✐t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❢♦r ❡❛❝❤ ❡❞❣❡✳ ❚❤❡r❡ ❡①✐sts s♦♠❡ ✭✏t❤❡ ❝r✐t✐❝❛❧ ✑✮ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❝♦♠♣♦♥❡♥ts ✭✏❝❧✉st❡rs✑✮ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❛r❡ ✜♥✐t❡✱ ✇❤✐❧❡ ❢♦r t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✜♥✐t❡ ❝❧✉st❡r✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ❛t ❛♥❞ ♥❡❛r ✐s ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❡①❝❡♣t ✐❢ ♦r ✳ ❚❤✐s ♠✐♥✐❝♦✉rs❡ ✇✐❧❧ ❢♦❝✉s ♦♥ r❡❝❡♥t ❛❞✈❛♥❝❡s ❛r♦✉♥❞ t❤✐s ♣r♦❜❧❡♠✱ ✇✐t❤ ♣❛rt✐❝✉❧❛r ❡♠♣❤❛s✐s ♦♥ t❤❡ ❣r♦✇✐♥❣ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❆✐③❡♥♠❛♥✲❑❡st❡♥✲◆❡✇♠❛♥ ❛r❣✉♠❡♥t✳ ✭❜✉t ✇❡ ✇✐❧❧ ♦♥❧② ❣❡t t♦ ✐t ✐♥ t❤❡ s❡❝♦♥❞ ❤♦✉r✮

P❡r❝♦❧❛t✐♦♥ ✲ ❞❡✜♥✐t✐♦♥s

❊①❛♠✐♥❡ t❤❡ ❣r❛♣❤ Zd✱ d ≥ 2✳ ❋♦r ❛ p ∈ [0, 1] ❦❡❡♣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p ❛♥❞ ❞❡❧❡t❡ ✐t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1 − p✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❢♦r ❡❛❝❤ ❡❞❣❡✳ ❚❤❡r❡ ❡①✐sts s♦♠❡ ✭✏t❤❡ ❝r✐t✐❝❛❧ ✑✮ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❝♦♠♣♦♥❡♥ts ✭✏❝❧✉st❡rs✑✮ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❛r❡ ✜♥✐t❡✱ ✇❤✐❧❡ ❢♦r t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✜♥✐t❡ ❝❧✉st❡r✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ❛t ❛♥❞ ♥❡❛r ✐s ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❡①❝❡♣t ✐❢ ♦r ✳ ❚❤✐s ♠✐♥✐❝♦✉rs❡ ✇✐❧❧ ❢♦❝✉s ♦♥ r❡❝❡♥t ❛❞✈❛♥❝❡s ❛r♦✉♥❞ t❤✐s ♣r♦❜❧❡♠✱ ✇✐t❤ ♣❛rt✐❝✉❧❛r ❡♠♣❤❛s✐s ♦♥ t❤❡ ❣r♦✇✐♥❣ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❆✐③❡♥♠❛♥✲❑❡st❡♥✲◆❡✇♠❛♥ ❛r❣✉♠❡♥t✳ ✭❜✉t ✇❡ ✇✐❧❧ ♦♥❧② ❣❡t t♦ ✐t ✐♥ t❤❡ s❡❝♦♥❞ ❤♦✉r✮

P❡r❝♦❧❛t✐♦♥ ✲ ❞❡✜♥✐t✐♦♥s

❊①❛♠✐♥❡ t❤❡ ❣r❛♣❤ Zd✱ d ≥ 2✳ ❋♦r ❛ p ∈ [0, 1] ❦❡❡♣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p ❛♥❞ ❞❡❧❡t❡ ✐t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1 − p✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❢♦r ❡❛❝❤ ❡❞❣❡✳ ❚❤❡r❡ ❡①✐sts s♦♠❡ pc ∈ (0, 1) ✭✏t❤❡ ❝r✐t✐❝❛❧ p✑✮ s✉❝❤ t❤❛t ❢♦r p < pc ❛❧❧ ❝♦♠♣♦♥❡♥ts ✭✏❝❧✉st❡rs✑✮ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❛r❡ ✜♥✐t❡✱ ✇❤✐❧❡ ❢♦r p > pc t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✜♥✐t❡ ❝❧✉st❡r✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ❛t ❛♥❞ ♥❡❛r pc ✐s ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❡①❝❡♣t ✐❢ d = 2 ♦r d > 6✳ ❚❤✐s ♠✐♥✐❝♦✉rs❡ ✇✐❧❧ ❢♦❝✉s ♦♥ r❡❝❡♥t ❛❞✈❛♥❝❡s ❛r♦✉♥❞ t❤✐s ♣r♦❜❧❡♠✱ ✇✐t❤ ♣❛rt✐❝✉❧❛r ❡♠♣❤❛s✐s ♦♥ t❤❡ ❣r♦✇✐♥❣ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❆✐③❡♥♠❛♥✲❑❡st❡♥✲◆❡✇♠❛♥ ❛r❣✉♠❡♥t✳ ✭❜✉t ✇❡ ✇✐❧❧ ♦♥❧② ❣❡t t♦ ✐t ✐♥ t❤❡ s❡❝♦♥❞ ❤♦✉r✮

❚❤❡♦r❡♠

Epc(|C (0)|) = ∞✳

Pr♦♦❢✳

❋✐① p ❛♥❞ ❞❡♥♦t❡ χ = Ep(|C (0)|)✳ ▲❡t ε < 1 4dχ. ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t ❛t p + ε t❤❡r❡ ✐s ♥♦ ✐♥✜♥✐t❡ ❝❧✉st❡r✳ ❈♦♥s✐❞❡r p + ε ♣❡r❝♦❧❛t✐♦♥ ❛s ✐❢ ✇❡ t❛❦❡ p✲♣❡r❝♦❧❛t✐♦♥ ❛♥❞ t❤❡♥ ✏s♣r✐♥❦❧❡✑ ❡❛❝❤ ❡❞❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ε✳ ❋♦r ❛ ✈❡rt❡① x ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❞✐r❡❝t❡❞ ❡❞❣❡s e1, . . . , en✱ ❞❡♥♦t❡ ❜② Ex,e1,...,en t❤❡ ❡✈❡♥t t❤❛t 0 ✐s ❝♦♥♥❡❝t❡❞ t♦ x ❜② ❛ ♣❛t❤ γ1 ✐♥ p✲♣❡r❝♦❧❛t✐♦♥ ❢r♦♠ 0 t♦ e−

1 t❤❡♥

e1 ✐s s♣r✐♥❦❧❡❞✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ γ2 ❢r♦♠ e+

1 t♦ e− 2 t❤❡♥ e2 ✐s

s♣r✐♥❦❧❡❞ ❛♥❞ s♦ ♦♥✳ ❲❡ ❡♥❞ ✇✐t❤ ❛ ♣❛t❤ γn+1 ❢r♦♠ en t♦ x✳ ❲❡ r❡q✉✐r❡ ❛❧❧ t❤❡ γi t♦ ❜❡ ❞✐s❥♦✐♥t✳ ❈❧❡❛r❧② 0 ↔ x ✐s p + ε ♣❡r❝♦❧❛t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐st s♦♠❡ e1, . . . , en ✭♣♦ss✐❜❧② ❡♠♣t②✮ s✉❝❤ t❤❛t Ex,e1,...,en ❤♦❧❞✳

❚❤❡♦r❡♠

Epc(|C (0)|) = ∞✳

e1 e3 x e2

❚❤❡♦r❡♠

Epc(|C (0)|) = ∞✳

Pr♦♦❢✳

χ = Ep(|C (0)|)✱ ε < 1/4dχ✱ Ex,e1,...,en ✐s t❤❡ ❡✈❡♥t t❤❛t ∃γi ❢r♦♠ e+

i−1 t♦ e− i ✱ ❞✐s❥♦✐♥t✱ ❛♥❞ ❛❧❧ ei ❛r❡ s♣r✐♥❦❧❡❞✳

Pp+ε(0 ↔ x) ≤

∞

• n=0
• e1,...,en

P(Ex,e1,...,en). ❇② t❤❡ ❇❑ ✐♥❡q✉❛❧✐t② ≤

∞

• n=0
• e1,...,en

Pp(0 ↔ e−

1 )Pp(e+ 1 ↔ e− 2 ) · · · P(e+ n ↔ x)εn

❙✉♠♠✐♥❣ ♦✈❡r ❛❧❧ x ❣✐✈❡s χ(p+ε) ≤

∞

• n=0

εn

• x,e1,...,en

Pp(0 ↔ e−

1 )Pp(e+ 1 ↔ e− 2 ) · · · Pp(e+ n ↔ x).

❚❤❡♦r❡♠

Epc(|C (0)|) = ∞✳

Pr♦♦❢✳

χ(p+ε) ≤

∞

• n=0

εn

• x,e1,...,en

Pp(0 ↔ e−

1 )Pp(e+ 1 ↔ e− 2 ) · · · Pp(e+ n ↔ x).

❙✉♠♠✐♥❣ ♦✈❡r x ❣✐✈❡s ♦♥❡ χ(p) t❡r♠ ✇❤✐❝❤ ✇❡ ❝❛♥ t❛❦❡ ♦✉t ♦❢ t❤❡ s✉♠ =

∞

• n=0

εnχ(p)

• e1,...,en

Pp(0 ↔ e−

1 )Pp(e+ 1 ↔ e− 2 ) · · · Pp(e+ n−1 ↔ e− n ).

e+

n ❤❛s 2d ♣♦ss✐❜✐❧✐t✐❡s✳ ❙✉♠♠✐♥❣ ♦✈❡r e− n ❣✐✈❡s ❛♥♦t❤❡r χ t❡r♠✳

❚❛❦✐♥❣ ❜♦t❤ ♦✉t ♦❢ t❤❡ s✉♠ ❣✐✈❡s =

∞

• n=0

εn · 2dχ(p)2

• e1,...,en−1

Pp(0 ↔ e−

1 ) · · · Pp(e+ n−2 ↔ e− n−1).

❚❤❡♦r❡♠

Epc(|C (0)|) = ∞✳

Pr♦♦❢✳

χ(p) = Ep(|C (0)|)✱ ε < 1/4dχ(p)✱ χ(p + ε) ≤

∞

• n=0

εn

• x,e1,...,en

Pp(0 ↔ e−

1 )Pp(e+ 1 ↔ e− 2 ) · · · Pp(e+ n ↔ x)

=

∞

• n=0

εn · (2d)nχ(p)n+1 < ∞. ❚❤✐s s❤♦✇s t❤❛t p + ε ≤ pc✳ ❚❤❡ t❤❡♦r❡♠ ✐s t❤❡♥ ♣r♦✈❡❞ ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ❛r❣✉♠❡♥t ❛❧s♦ ❣✐✈❡s χ(p) ≥ 1 4d(pc − p) ∀p < pc. ❚❤✐s ✐s s❤❛r♣ ♦♥ ❛ tr❡❡ ❜✉t ♥♦t ✐♥ ❣❡♥❡r❛❧✳

❋♦r ❛ s❡t S ⊂ Zd ❞❡♥♦t❡ ❜② ∂S t❤❡ s❡t ♦❢ x ∈ S ✇✐t❤ ❛ ♥❡✐❣❤❜♦✉r y ∈ S✳

❚❤❡♦r❡♠

▲❡t S ⊂ Zd ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0✳ ❚❤❡♥

• x∈∂S

Ppc(0

S

← → x) ≥ 1.

Pr♦♦❢ s❦❡t❝❤✳

▲❡t x ∈ Zd✳ ■❢ 0 ↔ x t❤❡♥ t❤❡r❡ ❡①✐sts 0 = y1, . . . , yn = x s✉❝❤ ❛♥❞ ♦♣❡♥ ♣❛t❤s γi s✉❝❤ t❤❛t

✶ γi ✐s ❢r♦♠ yi t♦ yi+1 ❛♥❞ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ yi + S✳ ✷ ❚❤❡ γi ❛r❡ ❞✐s❥♦✐♥t✳

❆♥❞ ✇❡ ❤❛✈❡ ❢♦r s♦♠❡ ♥✉♠❜❡r t❤❛t ❞❡♣❡♥❞s ♦♥ ✳ ❆ ❝❛❧❝✉❧❛t✐♦♥ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢ s❤♦✇s t❤❛t

❋♦r ❛ s❡t S ⊂ Zd ❞❡♥♦t❡ ❜② ∂S t❤❡ s❡t ♦❢ x ∈ S ✇✐t❤ ❛ ♥❡✐❣❤❜♦✉r y ∈ S✳

❚❤❡♦r❡♠

▲❡t S ⊂ Zd ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0✳ ❚❤❡♥

• x∈∂S

Ppc(0

S

← → x) ≥ 1.

x

❚❤❡♦r❡♠

▲❡t S ⊂ Zd ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0✳ ❚❤❡♥

• x∈∂S Ppc(0

S

← → x) ≥ 1.

Pr♦♦❢ s❦❡t❝❤✳

▲❡t x ∈ Zd✳ ■❢ 0 ↔ x t❤❡♥ t❤❡r❡ ❡①✐sts 0 = y1, . . . , yn = x s✉❝❤ ❛♥❞ ♦♣❡♥ ♣❛t❤s γi s✉❝❤ t❤❛t

✶ γi ✐s ❢r♦♠ yi t♦ yi+1 ❛♥❞ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ yi + S✳ ✷ ❚❤❡ γi ❛r❡ ❞✐s❥♦✐♥t✳

❆♥❞ ✇❡ ❤❛✈❡ n ≥ r|x|S✳ ❆ ❝❛❧❝✉❧❛t✐♦♥ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢ s❤♦✇s t❤❛t P(0 ↔ x) ≤

• n≥r|x|

y∈∂S

Ppc(0

S

← → y) n . ■❢ t❤❡ ✈❛❧✉❡ ✐♥ t❤❡ ♣❛r❡♥t❤❡s✐s ✐s s♠❛❧❧❡r t❤❛♥ ✶ t❤❡♥ P(0 ↔ x) ❞❡❝❛②s ❡①♣♦♥❡♥t✐❛❧❧② ✐♥ |x|✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✳

❚❤❡♦r❡♠

▲❡t S ⊂ Zd ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0✳ ❚❤❡♥

• x∈∂S Ppc(0

S

← → x) ≥ 1. ❆ ❢✉❧❧ ♣r♦♦❢ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❍✳ ❉✉♠✐♥✐❧✲❈♦♣✐♥ ❛♥❞ ❱✳ ❚❛ss✐♦♥✱ ❆ ♥❡✇ ♣r♦♦❢ ♦❢ t❤❡ s❤❛r♣♥❡ss ♦❢ t❤❡ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❢♦r ❇❡r♥♦✉❧❧✐ ♣❡r❝♦❧❛t✐♦♥ ♦♥ Zd✱ ▲✬❊♥s❡✐❣♥❡♠❡♥t ▼❛t❤é♠❛t✐q✉❡✱ ✻✷✭✶✴✷✮ ✭✷✵✶✻✮✱ ✶✾✾✲✷✵✻✳ ■t ✐s t❤❡ ❜❛s✐s ❢♦r ❛ ♥❡✇✱ s✐❣♥✐✜❝❛♥t❧② s✐♠♣❧❡r ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭▼❡♥s❤✐❦♦✈❆✐③❡♥♠❛♥✲❇❛rs❦②✮

❋♦r ❛♥② p < pc χ(p) < ∞✳ ✭r❡❝❛❧❧ t❤❛t χ(p) = Ep(|C (0)|) ❛♥❞ t❤❛t ✇❤❛t ✇❡ ♣r♦✈❡❞ ❜❡❢♦r❡ ✐s χ(pc) = ∞✮✳

❚❤❡♦r❡♠

▲❡t S ⊂ Zd ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0✳ ❚❤❡♥

• x∈∂S Ppc(0

S

← → x) ≥ 1. ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s✿

▲❡♠♠❛ ✭❑✲◆❛❝❤♠✐❛s✱ ✷✵✶✶✮

❋♦r ❛♥② x ∈ ∂Λn✱ Λn := [−n, n]d✱ Ppc(0

Λn

← → x) ≥ c exp(−C log2 n).

▲❡♠♠❛ ✭❈❡r❢✱ ✷✵✶✺✮

❋♦r ❛♥② x, y ∈ Λn✱ Ppc(x

Λ2n

← − → y) ≥ cn−C. ❆❧❧ ❝♦♥st❛♥ts c ❛♥❞ C ♠✐❣❤t ❞❡♣❡♥❞ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥✳

▲❡♠♠❛ ✭❈❡r❢✱ ✷✵✶✺✮

❋♦r ❛♥② x, y ∈ Λn✱ Ppc(x

Λ2n

← − → y) ≥ cn−C.

Pr♦♦❢✳

❆ss✉♠❡ ✜rst t❤❛t x − y = (2k, 0, . . . , 0)✱ k ≤ n✳ ❇② t❤❡ t❤❡♦r❡♠ t❤❡r❡ ❡①✐sts ❛ z ∈ ∂Λk s✉❝❤ t❤❛t P(0

Λk

← → z) ≥ 1 2d|∂Λk| ≥ c kd−1 . ❇② r♦t❛t✐♦♥ ❛♥❞ r❡✢❡❝t✐♦♥ s②♠♠❡tr② ✇❡ ♠❛② ❛ss✉♠❡ z ✐s ✐♥ s♦♠❡ ❢❛❝❡ ♦❢ Λk✱ ❢♦r ❡①❛♠♣❧❡ z1 = k✳ ▲❡t z ❜❡ t❤❡ r❡✢❡❝t✐♦♥ ♦❢ z ✐♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ✐✳❡✳ z = (−z1, z2, . . . , zd)✳ ❇② r❡✢❡❝t✐♦♥ s②♠♠❡tr② ✇❡ ❛❧s♦ ❤❛✈❡ P(0

Λk

← → z) ≥ ck1−d✳ ❚r❛♥s❧❛t✐♥❣ z t♦ x ❛♥❞ z t♦ y ❣✐✈❡s P(x

x+Λk

← − − → x + z), P(y

y+Λk

← − − → y + z) ≥ c kd−1 . ❇✉t x + z = y + z✦

▲❡♠♠❛ ✭❈❡r❢✱ ✷✵✶✺✮

❋♦r ❛♥② x, y ∈ Λn✱ Ppc(x

Λ2n

← − → y) ≥ cn−C.

Pr♦♦❢✳

❆ss✉♠❡ ✜rst t❤❛t x − y = (2k, 0, . . . , 0)✱ k ≤ n✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ z s✉❝❤ t❤❛t P(x

x+Λk

← − − → x + z), P(y

y+Λk

← − − → x + z) ≥ c kd−1 . ❙✐♥❝❡ x + Λk ⊂ Λ2n ❛♥❞ ❞✐tt♦ ❢♦r y + Λk ✇❡ ❝❛♥ ✇r✐t❡ P(x

Λ2n

← − → x + z), P(y

Λ2n

← − → x + z) ≥ c kd−1 . ❇② ❋❑● P(x

Λ2n

← − → y) ≥ P(x

Λ2n

← − → x + z, y

Λ2n

← − → y + z) ≥ c k2d−2 . Pr♦✈✐♥❣ t❤❡ ❧❡♠♠❛ ✐♥ t❤✐s ❝❛s❡✳

▲❡♠♠❛ ✭❈❡r❢✱ ✷✵✶✺✮

❋♦r ❛♥② x, y ∈ Λn✱ Ppc(x

Λ2n

← − → y) ≥ cn−C.

Pr♦♦❢✳

❆ss✉♠❡ ✜rst t❤❛t x − y = (2k, 0, . . . , 0)✱ k ≤ n✳ ❚❤❡♥ P(x

Λ2n

← − → y) ≥ ck2−2d ≥ cn2−2d✳ ❲✐t❤ ❛ s❧✐❣❤t❧② s♠❛❧❧❡r c✱ ✇❡ ❝❛♥ r❡♠♦✈❡ t❤❡ r❡q✉✐r❡♠❡♥t t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ x ❛♥❞ y ✐s ❡✈❡♥✳ ■❢ t❤❡② ❛r❡ ♥♦t ♦♥ ❛ ❧✐♥❡✱ ✇❡ ❞❡✜♥❡ x = x0, . . . , xd = y s✉❝❤ t❤❛t ❡❛❝❤ ❝♦✉♣❧❡ xi✱ xi+1 ❞✐✛❡r ❜② ♦♥❧② ♦♥❡ ❝♦♦r❞✐♥❛t❡✳ ❍❡♥❝❡ P(xi

Λ2n

← − → xi+1) ≥ cn2−2d✳ ❯s✐♥❣ ❋❑● ❛❣❛✐♥ ❣✐✈❡s P(x

Λ2n

← − → y) ≥ P(x0

Λ2n

← − → x1, x1

Λ2n

← − → x2, . . . , xd−1

Λ2n

← − → xd) ≥

d

• i=1

P(xi−1

Λ2n

← − → xi) ≥ c n2d2−2d .

s❦✐♣ t♦ ❞✐❛❣r❛♠

▲❡♠♠❛ ✭❈❡r❢✱ ✷✵✶✺✮

❋♦r ❛♥② x, y ∈ Λn✱ Ppc(x

Λ2n

← − → y) ≥ cn2d−2d2. ❚❤✐s ✇❛s r❡❝❡♥t❧② ✐♠♣r♦✈❡❞ t♦ cn−d2 ❜② ✈❛♥ ❞❡♥ ❇❡r❣ ❛♥❞ ❉♦♥✳ ❚❤❡✐r ♣r♦♦❢ ❤❛s ❛♥ ✐♥t❡r❡st✐♥❣ t♦♣♦❧♦❣✐❝❛❧ ❝♦♠♣♦♥❡♥t✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

▲❡t Λ ❜❡ ❛ ❜♦① ✐♥ Zd✱ ✇✐t❤ t❤❡ s✐❞❡ ❧❡♥❣t❤s ♥♦t ♥❡❝❡ss❛r✐❧② ❡q✉❛❧✳ ❆ ❝r♦ss✐♥❣ ✐s ❛♥ ♦♣❡♥ ♣❛t❤ ❢r♦♠ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ❜♦① t♦ t❤❡ ♦t❤❡r✳

❍❛r❞ ✇❛② ❊❛s② ✇❛②

❚❤❡♦r❡♠

▲❡t ❜❡ ❛♥ ❜♦① ✐♥ ✳ ❚❤❡♥ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣

Pr♦♦❢ ✭❑❡st❡♥❄ ❇♦❧❧♦❜ás✲❘✐♦r❞❛♥❄ ◆♦❧✐♥❄✮

■t ✐s ❡❛s✐❡r t♦ ❞r❛✇ ✐♥ s♦ ❧❡t ✉s ❞♦ t❤✐s✳ ▲❡t ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ♦❢ ❛♥ r❡❝t❛♥❣❧❡✳ ❲❡ ✜rst ❝❧❛✐♠ t❤❛t ✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

▲❡t Λ ❜❡ ❛ ❜♦① ✐♥ Zd✱ ✇✐t❤ t❤❡ s✐❞❡ ❧❡♥❣t❤s ♥♦t ♥❡❝❡ss❛r✐❧② ❡q✉❛❧✳ ❆ ❝r♦ss✐♥❣ ✐s ❛♥ ♦♣❡♥ ♣❛t❤ ❢r♦♠ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ❜♦① t♦ t❤❡ ♦t❤❡r✳

❚❤❡♦r❡♠

▲❡t Λ ❜❡ ❛♥ 2n × · · · × 2n × n ❜♦① ✐♥ Zd✳ ❚❤❡♥ Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) > c

Pr♦♦❢ ✭❑❡st❡♥❄ ❇♦❧❧♦❜ás✲❘✐♦r❞❛♥❄ ◆♦❧✐♥❄✮

■t ✐s ❡❛s✐❡r t♦ ❞r❛✇ ✐♥ d = 2 s♦ ❧❡t ✉s ❞♦ t❤✐s✳ ▲❡t p(a, b) ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ♦❢ ❛♥ a × b r❡❝t❛♥❣❧❡✳ ❲❡ ✜rst ❝❧❛✐♠ t❤❛t p(4n, n) ≤ 5p(2n, n)✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

❚❤❡♦r❡♠

▲❡t Λ ❜❡ ❛♥ 2n × · · · × 2n × n ❜♦① ✐♥ Zd✳ ❚❤❡♥ Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) > c

Pr♦♦❢✳

■t ✐s ❡❛s✐❡r t♦ ❞r❛✇ ✐♥ d = 2 s♦ ❧❡t ✉s ❞♦ t❤✐s✳ ▲❡t p(a, b) ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ♦❢ ❛♥ a × b r❡❝t❛♥❣❧❡✳ ❲❡ ✜rst ❝❧❛✐♠ t❤❛t p(4n, n) ≤ 5p(2n, n)✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐❢ s♦♠❡ ♣❛t❤ γ ❝r♦ss❡s ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ❛ 4n × n r❡❝t❛♥❣❧❡✱ ✐t ♠✉st ❝r♦ss ❡✐t❤❡r ♦♥❡ ♦❢ ✸ ❤♦r✐③♦♥t❛❧ r❡❝t❛♥❣❧❡s ♦r ♦♥❡ ♦❢ t✇♦ ✈❡rt✐❝❛❧ ♦♥❡s✳ ❲❡ ♥❡①t ❝❧❛✐♠ t❤❛t ✳ ❇✉t t❤❛t ♠❡❛♥s t❤❛t ❛♥❞ ✐♥❞✉❝t✐✈❡❧② t❤❛t ✳ ❚❤✉s✱ ✐❢ ❢♦r s♦♠❡ ✱ ✱ t❤❡♥ ✐t ❞❡❝❛②s ❡①♣♦♥❡♥t✐❛❧❧②✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ r❡s✉❧t t❤❛t ✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

❚❤❡♦r❡♠

▲❡t Λ ❜❡ ❛♥ 2n × · · · × 2n × n ❜♦① ✐♥ Zd✳ ❚❤❡♥ Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) > c

Pr♦♦❢✳

■t ✐s ❡❛s✐❡r t♦ ❞r❛✇ ✐♥ d = 2 s♦ ❧❡t ✉s ❞♦ t❤✐s✳ ▲❡t p(a, b) ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ♦❢ ❛♥ a × b r❡❝t❛♥❣❧❡✳ ❲❡ ✜rst ❝❧❛✐♠ t❤❛t p(4n, n) ≤ 5p(2n, n)✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐❢ s♦♠❡ ♣❛t❤ γ ❝r♦ss❡s ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ❛ 4n × n r❡❝t❛♥❣❧❡✱ ✐t ♠✉st ❝r♦ss ❡✐t❤❡r ♦♥❡ ♦❢ ✸ ❤♦r✐③♦♥t❛❧ r❡❝t❛♥❣❧❡s ♦r ♦♥❡ ♦❢ t✇♦ ✈❡rt✐❝❛❧ ♦♥❡s✳ ❲❡ ♥❡①t ❝❧❛✐♠ t❤❛t p(4n, 2n) ≤ p(4n, n)2✳

γ1 γ2

❇✉t t❤❛t ♠❡❛♥s t❤❛t ❛♥❞ ✐♥❞✉❝t✐✈❡❧② t❤❛t ✳ ❚❤✉s✱ ✐❢ ❢♦r s♦♠❡ ✱ ✱ t❤❡♥ ✐t ❞❡❝❛②s ❡①♣♦♥❡♥t✐❛❧❧②✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ r❡s✉❧t t❤❛t ✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

❚❤❡♦r❡♠

▲❡t Λ ❜❡ ❛♥ 2n × · · · × 2n × n ❜♦① ✐♥ Zd✳ ❚❤❡♥ Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) > c

Pr♦♦❢✳

■t ✐s ❡❛s✐❡r t♦ ❞r❛✇ ✐♥ d = 2 s♦ ❧❡t ✉s ❞♦ t❤✐s✳ ▲❡t p(a, b) ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ♦❢ ❛♥ a × b r❡❝t❛♥❣❧❡✳ ❲❡ ✜rst ❝❧❛✐♠ t❤❛t p(4n, n) ≤ 5p(2n, n)✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐❢ s♦♠❡ ♣❛t❤ γ ❝r♦ss❡s ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ❛ 4n × n r❡❝t❛♥❣❧❡✱ ✐t ♠✉st ❝r♦ss ❡✐t❤❡r ♦♥❡ ♦❢ ✸ ❤♦r✐③♦♥t❛❧ r❡❝t❛♥❣❧❡s ♦r ♦♥❡ ♦❢ t✇♦ ✈❡rt✐❝❛❧ ♦♥❡s✳ ❲❡ ♥❡①t ❝❧❛✐♠ t❤❛t p(4n, 2n) ≤ p(4n, n)2✳ ❇✉t t❤❛t ♠❡❛♥s t❤❛t p(4n, 2n) ≤ 25p(2n, n)2 ❛♥❞ ✐♥❞✉❝t✐✈❡❧② t❤❛t p(2k+1n, 2kn) ≤ 252k−1p(2n, n)2k✳ ❚❤✉s✱ ✐❢ ❢♦r s♦♠❡ n✱ p(2n, n) < 1

25✱ t❤❡♥ ✐t ❞❡❝❛②s ❡①♣♦♥❡♥t✐❛❧❧②✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡

r❡s✉❧t t❤❛t χ(pc) = ∞✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❈r♦ss✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s

❚❤❡♦r❡♠

▲❡t Λ ❜❡ ❛♥ 2n × · · · × 2n × n ❜♦① ✐♥ Zd✳ ❚❤❡♥ Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) > c ■t ✐s ♥❛t✉r❛❧ t♦ ❛s❦ ✐❢ t❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♣♣❡r ❜♦✉♥❞✱ ♥❛♠❡❧② ✐s ✐t tr✉❡ t❤❛t Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) ≤ 1 − c ❢♦r s♦♠❡ c > 0❄ ❚❤✐s ✐s tr✉❡ ✇❤❡♥ d = 2✳ ■t ✐s ❢❛❧s❡ ❢♦r d > 6✱ ✐♥ ❢❛❝t Ppc(Λ ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣) → 1 ❛s n → ∞. ■t ✐s ♥♦t ❦♥♦✇♥ ✐♥ ✐♥t❡r♠❡❞✐❛t❡ ❞✐♠❡♥s✐♦♥s✳ ■♥ ❞✐♠❡♥s✐♦♥s ✷ ❛♥❞ ❤✐❣❤✱ t❤❡r❡ ✐s ♥♦ s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❡❛s②✲✇❛② ❛♥❞ ❤❛r❞✲✇❛② ❝r♦ss✐♥❣✳ ■♥ ✐♥t❡r♠❡❞✐❛t❡ ❞✐♠❡♥s✐♦♥s t❤✐s ✐s ♥♦t ❦♥♦✇♥✳

s❦✐♣ t♦ ❞✐❛❣r❛♠

❖♥❡ ❛r♠ ❡①♣♦♥❡♥t

❚❤❡♦r❡♠

P(0 ↔ ∂Λn) > c/n(d−1)/2✳

Pr♦♦❢✳

❇② t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❜♦① [−n/2, n/2] × [−n, n] × · · · × [−n, n] ❤❛s ❛♥ ❡❛s②✲✇❛② ❝r♦ss✐♥❣ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st c✳ ✏❊❛s②✲✇❛②✑ ♠❡❛♥s ❢r♦♠ {n/2} × [−n, n]d−1 t♦ {−n/2} × [−n, n]d−1 s♦ ✐t ♠✉st ❝r♦ss 0 × [−n, n]d−1✳ ❚❤❡r❡❢♦r❡ t❤❡r❡ ❡①✐sts s♦♠❡ x ∈ {0} × [−n, n]d−1 s✉❝❤ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❝r♦ss✐♥❣ ♣❛ss t❤r♦✉❣❤ ✐t ✐s ❛t ❧❡❛st c/nd−1✳ ❇✉t ✐❢ ✐t ❞♦❡s✱ t❤❡♥ x ✐s ❝♦♥♥❡❝t❡❞ t♦ ❞✐st❛♥❝❡ ❛t ❧❡❛st n/2 ❜② t✇♦ ❞✐s❥♦✐♥t ♣❛t❤s✳ ❚❤❡ ❇❑ ✐♥❡q✉❛❧✐t② ✜♥✐s❤❡s t❤❡ ♣r♦♦❢✳ ■♥ d = 2 ❑❡st❡♥ ✐♠♣r♦✈❡❞ t❤✐s t♦ n−1/3✳

❉❡♣❡♥❞❡♥❝✐❡s ❞✐❛❣r❛♠

χ(pc) = ∞

• x∈∂Λn Ppc(0

Λn

← → x) ≥ 1 Ppc(❝r♦ss✐♥❣) > c Ppc(x

Λ2n

← − → y) > cn−C Ppc(0 ↔ ∂Λn) > cn(1−d)/2