rt rt - PowerPoint PPT Presentation

❈r✐t✐❝❛❧ ♣❡r❝♦❧❛t✐♦♥ ●❛❞② ❑♦③♠❛ ❖♥❧✐♥❡ ❖♣❡♥ Pr♦❜❛❜✐❧✐t② ❙❝❤♦♦❧✱ ✷✵✷✵

❋♦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❛♣❤ Z d ✱ d ≥ 2 ✳

❚❤❡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❛♣❤ Z d ✱ d ≥ 2 ✳ ❋♦r ❛ p ∈ [0 , 1] ❦❡❡♣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p ❛♥❞ ❞❡❧❡t❡ ✐t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1 − p ✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❢♦r ❡❛❝❤ ❡❞❣❡✳

P❡r❝♦❧❛t✐♦♥ ✲ ❞❡✜♥✐t✐♦♥s ❊①❛♠✐♥❡ t❤❡ ❣r❛♣❤ Z d ✱ 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♦♠❡ p c ∈ (0 , 1) ✭✏t❤❡ ❝r✐t✐❝❛❧ p ✑✮ s✉❝❤ t❤❛t ❢♦r p < p c ❛❧❧ ❝♦♠♣♦♥❡♥ts ✭✏❝❧✉st❡rs✑✮ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❣r❛♣❤ ❛r❡ ✜♥✐t❡✱ ✇❤✐❧❡ ❢♦r p > p c t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✜♥✐t❡ ❝❧✉st❡r✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ❛t ❛♥❞ ♥❡❛r p c ✐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❡♠ E p c ( | C (0) | ) = ∞ ✳ Pr♦♦❢✳ ❋✐① p ❛♥❞ ❞❡♥♦t❡ χ = E p ( | C (0) | ) ✳ ▲❡t 1 ε < 4 dχ. ❲❡ ✇✐❧❧ 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 e 1 , . . . , e n ✱ ❞❡♥♦t❡ ❜② E x,e 1 ,...,e n t❤❡ ❡✈❡♥t t❤❛t 0 ✐s ❝♦♥♥❡❝t❡❞ t♦ x ❜② ❛ ♣❛t❤ γ 1 ✐♥ p ✲♣❡r❝♦❧❛t✐♦♥ ❢r♦♠ 0 t♦ e − 1 t❤❡♥ e 1 ✐s s♣r✐♥❦❧❡❞✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ γ 2 ❢r♦♠ e + 1 t♦ e − 2 t❤❡♥ e 2 ✐s s♣r✐♥❦❧❡❞ ❛♥❞ s♦ ♦♥✳ ❲❡ ❡♥❞ ✇✐t❤ ❛ ♣❛t❤ γ n +1 ❢r♦♠ e n t♦ x ✳ ❲❡ r❡q✉✐r❡ ❛❧❧ t❤❡ γ i t♦ ❜❡ ❞✐s❥♦✐♥t✳ ❈❧❡❛r❧② 0 ↔ x ✐s p + ε ♣❡r❝♦❧❛t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐st s♦♠❡ e 1 , . . . , e n ✭♣♦ss✐❜❧② ❡♠♣t②✮ s✉❝❤ t❤❛t E x,e 1 ,...,e n ❤♦❧❞✳

❚❤❡♦r❡♠ E p c ( | C (0) | ) = ∞ ✳ e 1 e 2 0 x e 3

❚❤❡♦r❡♠ E p c ( | C (0) | ) = ∞ ✳ Pr♦♦❢✳ χ = E p ( | C (0) | ) ✱ ε < 1 / 4 dχ ✱ E x,e 1 ,...,e n ✐s t❤❡ ❡✈❡♥t t❤❛t ∃ γ i ❢r♦♠ e + i − 1 t♦ e − i ✱ ❞✐s❥♦✐♥t✱ ❛♥❞ ❛❧❧ e i ❛r❡ s♣r✐♥❦❧❡❞✳ ∞ � � P p + ε (0 ↔ x ) ≤ P ( E x,e 1 ,...,e n ) . e 1 ,...,e n n =0 ❇② t❤❡ ❇❑ ✐♥❡q✉❛❧✐t② ∞ � � P p (0 ↔ e − 1 ) P p ( e + 1 ↔ e − 2 ) · · · P ( e + n ↔ x ) ε n ≤ n =0 e 1 ,...,e n ❙✉♠♠✐♥❣ ♦✈❡r ❛❧❧ x ❣✐✈❡s ∞ � ε n � P p (0 ↔ e − 1 ) P p ( e + 1 ↔ e − 2 ) · · · P p ( e + χ ( p + ε ) ≤ n ↔ x ) . n =0 x,e 1 ,...,e n

❚❤❡♦r❡♠ E p c ( | C (0) | ) = ∞ ✳ Pr♦♦❢✳ ∞ � ε n � P p (0 ↔ e − 1 ) P p ( e + 1 ↔ e − 2 ) · · · P p ( e + χ ( p + ε ) ≤ n ↔ x ) . x,e 1 ,...,e n n =0 ❙✉♠♠✐♥❣ ♦✈❡r x ❣✐✈❡s ♦♥❡ χ ( p ) t❡r♠ ✇❤✐❝❤ ✇❡ ❝❛♥ t❛❦❡ ♦✉t ♦❢ t❤❡ s✉♠ ∞ � ε n χ ( p ) � P p (0 ↔ e − 1 ) P p ( e + 1 ↔ e − 2 ) · · · P p ( e + n − 1 ↔ e − = n ) . e 1 ,...,e n n =0 e + n ❤❛s 2 d ♣♦ss✐❜✐❧✐t✐❡s✳ ❙✉♠♠✐♥❣ ♦✈❡r e − n ❣✐✈❡s ❛♥♦t❤❡r χ t❡r♠✳ ❚❛❦✐♥❣ ❜♦t❤ ♦✉t ♦❢ t❤❡ s✉♠ ❣✐✈❡s ∞ ε n · 2 dχ ( p ) 2 � � P p (0 ↔ e − 1 ) · · · P p ( e + n − 2 ↔ e − = n − 1 ) . n =0 e 1 ,...,e n − 1

❚❤❡♦r❡♠ E p c ( | C (0) | ) = ∞ ✳ Pr♦♦❢✳ χ ( p ) = E p ( | C (0) | ) ✱ ε < 1 / 4 dχ ( p ) ✱ ∞ � � ε n P p (0 ↔ e − 1 ) P p ( e + 1 ↔ e − 2 ) · · · P p ( e + χ ( p + ε ) ≤ n ↔ x ) n =0 x,e 1 ,...,e n ∞ ε n · (2 d ) n χ ( p ) n +1 < ∞ . � = n =0 ❚❤✐s s❤♦✇s t❤❛t p + ε ≤ p c ✳ ❚❤❡ t❤❡♦r❡♠ ✐s t❤❡♥ ♣r♦✈❡❞ ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ❛r❣✉♠❡♥t ❛❧s♦ ❣✐✈❡s 1 χ ( p ) ≥ ∀ p < p c . 4 d ( p c − p ) ❚❤✐s ✐s s❤❛r♣ ♦♥ ❛ tr❡❡ ❜✉t ♥♦t ✐♥ ❣❡♥❡r❛❧✳

❆♥❞ ✇❡ ❤❛✈❡ ❢♦r s♦♠❡ ♥✉♠❜❡r t❤❛t ❞❡♣❡♥❞s ♦♥ ✳ ❆ ❝❛❧❝✉❧❛t✐♦♥ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢ s❤♦✇s t❤❛t ❋♦r ❛ s❡t S ⊂ Z d ❞❡♥♦t❡ ❜② ∂S t❤❡ s❡t ♦❢ x ∈ S ✇✐t❤ ❛ ♥❡✐❣❤❜♦✉r y �∈ S ✳ ❚❤❡♦r❡♠ ▲❡t S ⊂ Z d ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0 ✳ ❚❤❡♥ S � P p c (0 ← → x ) ≥ 1 . x ∈ ∂S Pr♦♦❢ s❦❡t❝❤✳ ▲❡t x ∈ Z d ✳ ■❢ 0 ↔ x t❤❡♥ t❤❡r❡ ❡①✐sts 0 = y 1 , . . . , y n = x s✉❝❤ ❛♥❞ ♦♣❡♥ ♣❛t❤s γ i s✉❝❤ t❤❛t ✶ γ i ✐s ❢r♦♠ y i t♦ y i +1 ❛♥❞ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ y i + S ✳ ✷ ❚❤❡ γ i ❛r❡ ❞✐s❥♦✐♥t✳

❋♦r ❛ s❡t S ⊂ Z d ❞❡♥♦t❡ ❜② ∂S t❤❡ s❡t ♦❢ x ∈ S ✇✐t❤ ❛ ♥❡✐❣❤❜♦✉r y �∈ S ✳ ❚❤❡♦r❡♠ ▲❡t S ⊂ Z d ❜❡ s♦♠❡ ✜♥✐t❡ s❡t ❝♦♥t❛✐♥✐♥❣ 0 ✳ ❚❤❡♥ S � P p c (0 ← → x ) ≥ 1 . x ∈ ∂S x 0