II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3

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II (Percolation) 12 9 27 ( 3-4 ) 1 [ ] 2 [ ] 3 [ ] 4 [ ] 1992 5 [ ] G Grimmett Percolation Springer-Verlag New-York 1989 6 [ ] 3 1 3 p H 2 3 2 FKG BK Russo 2 p H = p T (=: p c ) 3 2 Kesten p c =1/2 ( ) )

Percolation 1 1 11 Z 2 B 2 = {{x y}; x y Z 2 x y =1} 2 bond p (0 p 1) open 1 p closed b = (p) b = (p) b (ω) b B 2 = { b ; b B 2 } P p S = {b B 2 ; b =1} O C O (open cluster) kc O k C O p H : Hammersley (critical probability) θ(p) =P p (kc O k = ) : p H =inf{p [0 1]; θ(p) > 0} p T : Temperley χ(p) =E p [kc O k]= : np p (kc O k = n)+ P p (kc O k = ) n=1 p T =inf{p [0 1]; χ(p) = } p H p T p H = p T 11 Z 2 p H p T p H = p T = 1/2 p c p>p c 1 p p c 1 11 Z d (d 3) p c =1/2 θ(p) =P p (kc O k = ) p p 11 p H p T 12 (Ω FP) (Ω FP) (probability space) Ω F( 2 Ω ) Ω σ (σ-field); (2 Ω Ω ) (1) Ω F

Percolation 2 (2) A F A c F (3) A n F (n =1 2) S A n F P = P (dω) (Ω F) (probability measure) ie 1 ; P : F [0 1] (1) P (Ω) =1( P ( ) =0 ) (2) A n F (n =1 2) P ( S A n )= P P (A n )(σ ) (Ω FP) = (ω) :Ω R (random variable) { a} := {ω Ω; (ω) a} F ( a R) (Ξ B) = (ω) Ξ { B} F ( B B) i (Ω FP) (Ξ i B i ) (i =1 2n) { i } n i=1 (independent) P ( 1 B 1 n B n )=P ( 1 B 1 ) P ( n B n ) ( B i B i i=1n) n { n } n 1 N 1 { n } N n=1 µ(a) = P ( A) (distribution) F (x) = P ( x) (distribution function) 12 P P : F [0 ] P ( ) =0 P ( ) =0 P (Ω) =1 1 Ω = Ξ := {0 1} B2 3 ω = ω(b); B 2 {0 1} (0 1) Ω ; (cylinder set) A i1in b 1b n = {ω; ω(b 1 )=i 1 ω(b n )=i n } (b k B 2 i k = 0 or 1k =1n) C F = B(Ξ) :=σ(c) (C σ-field C σ-field) B(Ξ) = \ {G 2 Ω ; G C σ-field} P = P p cylinder set P p (A i 1+ +i n b 1b n )=p i 1 i n (1 p) (1 i 1)+ +(1 i n ) ( ) b B 2 (Ω b F b P bp ) Ω b = {ω(b) =1 ω(b) =0} F b =2 Ω b P bp (ω(b) =1)=p b = (p) b b (ω) =ω(b) b P p ( b =1)= P p (ω(b) = 1) = p = { b ; b B 2 } P p (ω) =ω ( )

Percolation 3 12 (Ω FP) (1) B n F (n 1) T B n F ³[ (2) B n F B n P Bn = lim P (B n) n ³\ (3) B n F B n P Bn = lim P (B n) n ³[ (4) B n F (n 1) P Bn P (B n ) µ P (5) (Borel-Cantelli ) B n F (n 1) P (Bn ) < P lim sup B n = 0 ie ³ n P lim inf n Bc n =1 lim sup B n = \ [ B n lim inf B n = [ \ B n n n N 1 n N N 1 n N θ(p) =P p (kc O k = ) p θ n (p) :=P p (kc O k n) θ(p) = lim n θ n (p) θ n (p) b B 2 Z b Z b (ω) [0 1] {Z b } Q Q(Z b p) =p = P p ( b =1) S(p) ={b B 2 ; Z b p} O C O (p) p θ n (p) =P p (kc O k n) =Q(kC O (p)k n) θ n (p) 13 1/3 p H 2/3 p H Peierls 12 Hammersley p H = inf{p [0 1]; θ(p) = P p (kc O k = ) > 0} 1/3 p H 2/3 γ = {x 0 b 1 x 1 b 2 b n x n } ( path) (i) b = {x i 1 x i } (ii) i 6= j b i 6= b j γ = {b 1 b n } 12 [p H 1/3 ] p<1/3 θ(p) =0 γ n ( ) P p (γ C O )=P p ( b =1 b γ) =p n γ 4 3 n 1 γ P p (γ C O ) 4 γ; 3 n 1 p n (< if p<1/3) {γ k } kc O k = N 1 k N; γ k C O p<1/3 Borel-Cantelli θ(p) =P p (kc O k = ) P p \ [ {γ k C O } =0 N 1 n=1 k N

Percolation 4 p H 1/3 [p H 2/3 ] p>2/3 θ(p) > 0 Z 2 (Z 2 ) := {(m +1/2n+1/2); m n Z} (Z 2 ) b (B 2 ) b B 2 1 1 Y b := b {Y b ; b (B 2 ) } = {Y b ; b B 2 } N 1 V N := {(m n) Z 2 ; m n := max( m n ) N} V N p>2/3 N = N(p) S = {b B 2 ; b =1} P p (S V N ) 1 (1) 2 kc O k < C O (B 2 ) (closed path) γ O V N P p (S V N ) = P p (V N (B 2 ) γ ) P p (Y b =0 b γ ) γ ; V N kγ k = k V N k 4(2N +1) = 8N +4 ( 8N) γ [0k] {0} ( {(j +1/2 1/2); 1 j k} ) γ k 4 3 k 1 P p (S V N ) 4k 3 k 1 (1 p) k p>2/3 N 0 ( ) N = N(p) (1) (1) { b ; b 6 V N } { b ; b V N } P p {b =1 b V N } {S V N } = P p ( b =1 b V N )P p (S V N ) 1 2 P p( b =1 b V N )= p4n 2 > 0 2 P p (kc O k = ) ( ) p>2/3 θ(p) > 0 p H 2/3 13 [0 p<1/3 θ(p) =0 p H 1/3] [2/3 < p 1 θ(p) > 0 p H 2/3] 14 p >2/3 P k3 k 1 (1 p) k < k 1 k 8N 15 θ(p) Z b S(p) ={b B 2 ; Z b p} θ(p + h) =P S(p + h) h 0 S(p + h) S(p) ie T S(p + h) =S(p) h>0 ( ) θ(p + h) θ(p) (h 0) θ h (p) (p ) θ n (p) θ(p) (n ) 16 f n (x) [0 1] f n f ( ) f(x) [0 1]

Percolation 5 2 3 FKG BK Russo Ξ = ξ : B 2 {0 1} ª [ξ η ξ(b) η(b) ( b B 2 )] f : Ξ R ( ) ξ η f(ξ) f(η) (f(ξ) f(η) ) A B(Ξ) ( ) 1 A ( ) 21 A B(Ξ): [ξ A ξ η η A] x =(x 1 x 2 ) Z 2 x := x 1 x 2 =max{x 1 x 2 } V N = {b = {x y} B 2 ; x y N} ξ Ξ S N (ξ) ={b V N ; ξ(b) =1} f(ξ) =ks N k S(ξ) ={b B 2 ; ξ(b) =1} O C O (ξ) f(ξ) =kc O (ξ)k A = { P n i=1 ξ(b i) k} A = { P n i=1 ξ(b i) k} 21 FKG [Fortuin Kesten Ginibre] Harris Harris-FKG 60 Harris [2] 72 Fortuin Kesten Ginibre 3 [1] FKG 21 (FKG ) p [0 1] Ξ fg : E p [f()g()] E p [f()]e p [f()] fg 1 f(ξ) =f(ξ(b)) (g ) ( )=f(1)g(1)pf(0)g(0)(1 p) ( )=[f(1)p + f(0)(1 p)][g(1)p + g(0)(1 p)] ( ) ( )=[f(1) f(0)][g(1) g(0)]p(1 p) 0 (fg b1 bn ) 22 22 BK [van den Berg and Kesten] A B Ξ {b 1 b n } A B {b 1 b n } ξ 0 A ξ 00 B ξ

Percolation 6 22 (BK ) A B p [0 1] : P p ( A B) P p ( A)P p ( B) FKG (P p) ( 0 P p) 2 BK 1 ep = P p P p ( ) {b 1b n} A e = A Ξ B0 e = B =1n Bk e [(ξ ξ 0 ) Bk e ξ 0 b 1 b k ξ b k+1 b n B ( )] P p( A B) = e P (( e ) e A e B0) P p( A)P p( B) = e P (( 0 ) e A e Bn) k =0 1N : ep (( 0 ) e A e Bk ) e P (( 0 ) e A e Bk+1 ) 23 Russo ξ Ξ A Ξ b B 2 ξ A (pivotal) 1 ξ δ b (ξ) (ξ b ) A A c b b : N(A) =N(A; ξ) =#{b B 2 ; b ξ A pivotal} = 1 {b ξ A pivotal} b B 2 A = { P n ξ(b P n i=1 i) k} ξ(b i=2 i)=k 1 ξ b 1 A 23 (Russo ) A b 1 b n : d dp P p( A) =E p [N(A; )] (p [0 1]) P p ( A) p [ ] b i open p i [0 1] p =(p 1 p n ) P p i =1n 1 p i p 0 i ( ) p 0 p i p 0 i P p 0( A) P p ( A) =(p 0 i p i )P p (b i A ) ( b i ( A) b i A ( bi =1) ( A) b i b i ( bi =1) P p 0 P p ) p i p 0 i P p ( A) =P p (b i A ) p i 1 pivot;

Percolation 7 p P p( A) = = n P p ( A) p i p=(pp) n P p (b i A ) i=1 i=1 = E p [N(A; )] 3 2 p H = p T (=: p c ) 2 Hammersley : p H =inf{p [0 1]; θ(p) :=P p (kc O k = ) > 0} Temperley p T =inf{p [0 1]; χ(p) :=E p [kc O k]= } p H p T 3 ( 2 ) 4 Mensikov 5 ( 1 ) : C O C O {kc O k = } = { C O = } θ(p) =P p ( C O = ) χ(p) :=E p [ C O ] p T =inf{p [0 1]; χ(p) = } ( C O /2 kc O k 4 C O ) A A 31 p H = p T : Menshikov p H p T p<p H χ(p) < 31 ( χ(p) < p p T ) [ ] ξ N(A : ) =N(A : ξ) =N(A) x C Z 2 x r x (C) =sup{ y x ; y C} π n (p) =P p (A n ) (A n = {ξ Ξ; r 0 (C O (ξ)) n}) Russo d dp π n(p) =E p [N(A n )] = π n(p) E p [N(A n ) A n ] p = A n b A n A n b =1 :

Percolation 8 { b A n } { b =1} = { b A n } A n b A n b b P p ( b A n )p = P p ({ b A n } A n ) E p [N(A n )] = P p ( b A n ) b V n = 1 P p ( b A n A n )P p (A n ) p b V n = π n(p) E p [N(A n ) A n ] p 0 <p 1 <p 2 < 1 (p 1 p 2 ) : π n (p 2 )=π n (p 1 )exp µz p2 p 1 E p [N(A n ) A n ] dp p (2) ½ ¾ 1 k n (p) =max k; k π [n/k] (p) ([n/k] n/k ;[ ] ) n 0< α < β <p H π n (α) π n (β)exp β α 4β k n(β) (3) ( BK ) n 0 = n[1/π n (β)] π n 0(α) π n (β)exp β α 4β (π n(β) 1 1) (4) 2 ( ) p<p H K>0 N n N π n (p) exp[ Kn 1/3 ] (Mensikov ) ( β <p H π n (β) 0 π n (β) C/n (4) π n 2(α) C 0 exp[ Kn] π n (β) a/ log n (4) (4) (3) ) V N = {x Z 2 ; x = N} p<p H θ(p) =0 χ(p) =E p [ C O ]= x Z 2 E p [1 {x CO } ]= x Z 2 P p (x C O )= N=0 x V N P p (x C O ) 8Nπ N (p) Mensikov p H p T p H = p T N=0

Percolation 9 32 p H = p T : Aizenman-Barsky BK (2 ) 4 41 C =( )( ) B(Ξ) =σ(c) P p cylinder set A i 1i n b 1 b n = {ω; ω(b 1 )=i 1 ω(b n )=i n } (b k B 2 i k = 0 or 1) P p (A b1 b n )=p i1+ +in (1 p) (1 i1)+ +(1 in) B(Ξ) ( n ) [ A(C) := A k ; A k Ck =1nn 1 k=1 ( ) B(Ξ) =σ(a(c)) 41 ( ) B B(Ξ) ²>0 A A(C); P p (A4B) <² A4B := (A \ B) (B \ A): A(C) 3{kC O k n} {kc O k = } B(Ξ) x Z 2 τ x : Ξ Ξ; τ x ξ := ξ( + x) (translation) B B(Ξ) τ x B = {τ x ξ; ξ B} = {ξ; τ x ξ B} ( P) Ξ µ = P 1 = P ( ) x Z 2 B B(Ξ)µ(τ x B)=µ(B) µ B B(Ξ); τ x B = B ( x Z 2 ) µ(b) =P ( B) =0or1 µ (ergodic) 41 (p) P p B Ξ := { x Z 2 ; kc x k = } [ ] cylinderset P p A B Ξ τ x (A4B) =(τ x A)4(τ x B) ( ) B B(Ξ); τ x B = B ( x Z 2 ) A A(C); P p (A4B) <² (B B 0 )4(A A 0 ) (B4A) (B 0 4A 0 ) P p (B4(A τ x A)) = P p ((B τ x B)4(A τ x A)) P p (B4A)+P p (τ x (A4B)) 2² (P p )

Percolation 10 P p (B) P p (A τ x A) P p (B4(A τ x A)) 2² A x A τ x A P p (A τ x A)=P p (A)P p (τ x A)=P p (A) 2 P p (B) P p (A) 2 2² P p (B) P p (A) P p (B4A) <² P p (B) P p (B) 2 P p (B) P p (A) 2 + P p (A) 2 P p (B) 2 2² +2² =4² ²>0 P p (B) =P p (B) 2 ie P p (B) =0or1 41 A B Ξ τ x (A4B) =(τ x A)4(τ x B)(x Z 2 ) 42 (S(ξ) Ξ open bonds ) Ξ := {ξ Ξ; S(ξ) } = [ {ξ Ξ; kc x (ξ)k = } x Z 2 42 θ(p) > 0 P p (Ξ )=1 N = N (ξ) :=#{S(ξ) } [ ] τ x Ξ = Ξ P p P p (Ξ )=0or1 θ(p) > 0 Ξ {kc O k = } P p (Ξ ) > 0 P p (Ξ )=1 P p (Ξ )=1 θ(p) =0 x Z 2 τ x C O = C x P p (kc x k = ) =0 P p (Ξ ) θ(p) > 0 x Z 2 P p (kc x k = ) =0 42 (Newman-Schulman) p [0 1] P p (N = k) =1 k {0 1 } [ ] p =0 1 p (0 1) 1 k< P p (N = k) =1 k =1 ξ {N = k} S(ξ) k 1 I 1 I k lim P p(v n I j 6= j =1k)=P p ( n [ {V n I j 6= j =1k}) =1 ²>0 n 0 ; P p (V n0 I j 6= j =1k) > 1 ² 0<²<1 n 0 n=1 A = A n0 := {ξ Ξ; N (ξ) =k V n0 I j 6= j =1k} V n0 1 Ξ n0 = {0 1} Vn 0 ϕ Ξ n0 A(ϕ) :={ξ A; ξ = ϕ on V n0 } A = [ A(ϕ) ϕ Ξ n0 ( )

Percolation 11 ξ Ξ ϕ Ξ n0 ξ ϕ Ξ ξ ϕ = ϕ on V n0 =ξ on Vn c 0 A(ϕ) ={ξ Ξ; ξ = ϕ on V n0 } {ξ Ξ; ξ ϕ A} {ξ ϕ A} V n0 2 ϕ Ξ n0 2 kv n k 0 α(p) :=p (1 p) 1 ² < P p (A) = = ϕ ϕ ϕ Ξ n0 P p ({ξ = ϕ on V n0 } {ξ ϕ A}) P p (ξ = ϕ on V n0 )P p (ξ ϕ A) ϕ Ξ n0 2 kvn 0 k α(p) kvn 0 k max{p p (ξ ϕ A); ϕ Ξ n0 } P p (ξ ϕ A) (2α(p)) kvn 0 k (1 ²) {ξ =1onV n0 } {ξ ϕ A} {N =1} P p (N =1) P p (ξ =1onV n0 )P p (ξ ϕ A) µ kvn0 k p (1 ²) > 0 2α(p) {N =1} P p (N =1)=1 43 (Burton-Keane) P p (N = ) =0 ie 6 1 0 1 0 P p 5 51 Kesten : p c =1/2 p c 1/2 p c 1/2 duality ( ) (Z 2 ) Z 2 +1/2 (B 2 ) b B 2 b (B) Y b = b Y = {Y b } B 2 open bond (b; b =1) (B 2 ) closed bond (b ; Y b 51 (Harris) =0) duality θ µ 1 =0 ie p c 1 2 2 p =1/2 [ ] θ(1/2) > 0 S = {b B 2 ; b =1} S = {b (B 2 ); Y b =0}

Percolation 12 open-closed 1/2 P 1/2 (S S )=1 (0 <P 1/2 (kc O k = ) P 1/2 ( x Z 2 ; kc x k = ) =P 1/2 ( x (Z 2 ) ; kc x k = ) = 1 ) n V n = {x R 2 ; x = n} 0 <²<1/4 n P 1/2 S S V n > 1 ² 4 A + n (r) (ora + n (l)a + n (u)a + n (d)) V n or S \ V n FKG P 1/2 (A + n (r)) = P 1/2 (A + n (l)) = P 1/2 (A + n (u)) = P 1/2 (A + n (d)) > 1 ² i = r l u d P 1/2 (A + n (i)) FKG P 1/2 (A + n (i) c ) 4 P 1/2 (A + n (r) c A + n (l) c A + n (u) c A + n (d) c ) <² 4 S V n U n = {b = {u v } (B ); n u v n 1} V n A n (r)a n (l)a n (u)a n (d) P 1/2 A + n (r) A + n (l) A n (u) A n (d) > 1 4² V n S S θ(1/2) = 0 52 p c 1 2 [ ] T (n) =[0n+1] [0n] B 2 S(n) = 1 2 n+ 1 1 2 2 n+ 1 (B 2 ) 2 1/2 a n = P 1/2 (T (n) open path T (n) ) = P 1/2 (S(n) closed path S(n) ) = 1 2 p c > 1/2 Menshikov [p <p c K >0 N N; π n (p) P p (O V n ) exp[ Kn 1/3 ]( n N)] M>0 π n (1/2) M exp[ Kn 1/3 ]

Percolation 13 (A B A B ) P 1/2 (x y) M exp[ K y x 1/3 ] a n P 1/2 (x y) M(n +1) 2 exp[ K(n +1) 1/3 ] 0 x (T (n) ) y (T (n) ) (n ) a n =1/2 6 61 [ 43 P p (N = ) =0 ie ] ξ {N = }x Z 2 x C x (ξ) 3 x (ξ ) (encounter point) P p (N = ) =1 ( P p (N = ) < 1 =0 ) P p ( ) > 0 42 n µ S(ξ) \ Vn V n P p 1 3 2 ( B V n ) H = {x 1 x 2 x 3 } ( V n 3 ) B(H) :={ξ B; x i H V n S } H V n B(H) =B V n H 3 A(H) B(H) [ ]= S H V n (A(H) B(H)) ( ) β(p) :=p (1 p) P (A(H)) β(p) kvnk P p ( ) β(p) kvnk P p (B(H)) β(p) kvnk 1 2 > 0 H V n λ := P p ( ) > 0 lim P p(v n (λ/2) V n )=1 (5) n ( ) V n k V n S(ξ) \ V n k +2 V n P p (V n V n 2) = 1 n (λ/2) V n V n P p (N = ) =0 [ (5) ] A = { } ² = λ/2 (λ = P p (A)) lim P 1 p n 1 A (τ x ξ) P p (A) V n >² =0 (6) x V n

Percolation 14 ([1 A (τ x ξ)=1 x ξ ] ) : (P p ) A B B(Ξ) lim P p(a (τ x B)) = P p (A)P p (B) x [ ] P p = P A B (A B C ) x A τ x B A B ²>0 A 0 B 0 C; P (A4A 0 ) <²P(B4B 0 ) <² P P ((A τ x B)4(A 0 τ x B 0 )) P (A4A 0 )+P ((τ x B)4(τ x B 0 )) < 2² x Z 2 P (A τ x B) P (A 0 τ x B 0 ) < 2² x P (A 0 τ x B 0 )= P (A 0 )P (τ x B 0 ) lim sup P p (A (τ x B)) P p (A)P p (B) < 4² (7) x ²>0 61 (7) P p (5) Chebyshev [Chebyshev ] (Ω FP) ²>0 P ( >²) E[ 2 ]/² 2 62 ( E[ 2 ; >²]=E[ 2 1 { >²} ] ) (6) A = B = { } ²>0 L>0; x L P p (A (τ x A)) P p (A) 2 <² E p 1 2 1 A (τ x ξ) P p (A) 1 V n = E V n 2 p [(1 A (τ x ξ) P p (A))(1 A (τ y ξ) P p (A))] x V n xy V n P p = 1 V n 2 xy V n P p ((τ x A) (τ y A)) = P p (A (τ x y A)) Pp ((τ x A) (τ y A)) P p (A) 2 ) x V n y V n y x <L y x L Pp (A (τ x y A)) P p (A) 2 Pp (A (τ x y A)) P p (A) 2 + ² V n y V n y V n ; x y <L x V n V n 2 (x y x L y (2L 1) 2 ) 1 V n 2 xy V n Pp ((τ x A) (τ y A)) P p (A) 2 4L2 V n + ² n lim sup n E p 1 2 1 A (τ x ξ) P p (A) V n ² x V n

Percolation 15 ² 0 lim n E p 1 2 1 A (τ x ξ) P p (A) V n =0 x V n Chebyshev (6) 62 p H = p T : Menshikov p H p T p<p H χ(p) < [ ] 0 < α < β <p H [ (3) π n (α) π n (β)exp β α 4β k n(β) (n À 1) ] A n = {O V n } ξ A n O V n N(A n ; ξ) =k ( n) b 1 b k A n O V n 2 ρ 1 O b 1 2 i k ρ i b i 1 b i : 61 r 1 r k 0; r 1 + + r k n k P p (ρ 1 = r 1 ρ k 1 = r k 1 ρ k r k A n ) (1 π rk +1(p))P p (ρ 1 = r 1 ρ k 1 = r k 1 A n ) ρ k N(A n ) k [ ] b k 1 V n y k 1 b k 1 G k 1 BK P p ({ρ 1 = r 1 ρ k 1 = r k 1 } A n ) P p ({ρ 1 = r 1 ρ k 1 = r k 1 ρ k r k } A n ) P p {yk 1 V n } {y k 1 y k 1 + V rk +1} G k = P p (G k 1 = G y k 1 = y)p p {y V n outside G} {y y + V rk +1 outside G} (Gy) P p (G k 1 = G y k 1 = y)p p y V n outside G P p y y + V rk +1 outside G (Gy) P p {Gk 1 = G y k 1 = y} {y V n outside G} π rk +1(p) (Gy) = P p ({ρ 1 = r 1 ρ k 1 = r k 1 } A n ) π rk +1(p) : 62 1 k n 0 p 1 E p [N(A n ) A n ] k(1 π [n/k] (p)) k

Percolation 16 [ ] ρ 1 ρ k 1 0 [n/k] 1 E p [N(A n ) A n ] kp p (N(A n ) k A n ) h n i h n i kp p ³ρ 1 1ρ k 1 A n k h k n i h n i k(1 π [n/k] (p))p p ³ρ 1 1ρ k 1 1 A n k k k(1 π [n/k] (p)) k k = k n (p) :=max k; k 1/π [n/k] (p) ª π [n/k] (p) 1/k k 2 (n À 1 ) (1 1/k) k 1/4 (k 2 ) µ E p [N(A n ) A n ] k 1 1 k k k 4! Ã Z β (2) π n (β) =π n (α)exp E p [N(A n ) A n ] dp α p (3) π n (α) π n (β)exp β α 4β k n (p) k n(β) n 0< α < β <p H 63 k n (p) in n in p p<p H lim k n(p) = n [ ] π [n/k] (p) n p k n (p) k n (p)+1> 1/π [n/(kn(p)+1)](p) p<p H π m (p) 0(m ) k n (p) [ (4) π n 0(α) π n (β)exp β α 4β (π n(β) 1 1) (n 0 = n[1/π n (β)]) ] (3) n n 0 ( n) π n 0(β) π n (β) k n 0(β) (π n (β) 1 1) k =[π n (β) 1 ] n 0 = kn π [n0 /k](β) =π n (β) k =[π [n0 /k](β) 1 ] π [n0 /k](β) 1 k n 0(β) k k n 0(β) k =[π n (β) 1 ] π n (β) 1 1 [ (4) Mensikov ] 7 FKG BK FKG n 1 I = {1n} Ξ = {0 1} I B(Ξ) =2 Ξ (Ξ ) Ξ f [ξ η f(ξ) f(η)] 71 (FKG ) µ (Ξ B(Ξ)) µ(ξ) :=µ({ξ}) > 0(ξ Ξ) µ(ξ)µ(η) µ(ξ η)µ(ξ η) (ξ η Ξ) (8) Ξ fg : E µ [fg] E µ [f]e µ [g]

Percolation 17 I n = I = {1n} n 1 Ξ n = {0 1} I n n =1 ie Ξ = {ξ = ξ(1) = 01} = {0 1} ( )=f(1)g(1)µ(1) + f(0)g(0)µ(0) ( )=[f(1)µ(1) + f(0)µ(0)][g(1)µ(1) + g(0)µ(0)] µ(1) + µ(0) = 1 µ(1)µ(1) 0 fg ( ) ( )=[f(1) f(0)][g(1) g(0)]µ(1)µ(0) 0 n n +1 ξ(n +1) ² =0 1 ξ Ξ n ξ ² Ξ n+1 ξ ² (j) =ξ(j) (1 j n) = ² (j = n +1) µ µ(ξ ²) =µ(η ² ) µ(ξ ² ) Ξ n ξ η Ξ (8) ( ); ξ Ξ n µ(ξ ²)µ(η ²) µ(ξ η ²)µ(ξ η ²) F G : Ξ n R F (ξ) =f(ξ ² ) G(ξ) =g(ξ ² ) F G Ξ ² =0 1 F (ξ)g(ξ)µ(ξ ²) F (ξ)µ(ξ ²) G(η)µ(η ²) ξ Ξ n ξ Ξ n ξη Ξ n f(ξ ² )g(ξ ² )µ(ξ ² ) f(ξ ² )µ(ξ ²) g(η ² )µ(η ²) µ(ξ ² ) ξ Ξ n ξ Ξ n ξ Ξ n ξ Ξ n ² =0 1 1 ²=0 bf(²) := ξ Ξ n H(ξ ² )= ξ Ξ n+1 H(ξ) ξ Ξ n f(ξ ² )µ(ξ ²) bµ := ξ Ξ n+1 f(ξ)g(ξ)µ(ξ) ξ Ξ n µ(ξ ² ) 1 bf(²)bg(²)bµ(²) ²=0 fbg b bµ ( ) n =1 ξ Ξ n+1 f(ξ)g(ξ)µ(ξ) = 1 1 bf(²)bµ(²) bg(δ)bµ(δ) ²=0 δ=0 f(ξ)µ(ξ) g(η)µ(η) ξ Ξ n+1 η Ξ n+1 n +1 bfbg bµ bµ f b φ(ξ) :=µ(ξ 1)/µ(ξ 0) Ξ n ξ η [φ(ξ) φ(η)]µ(ξ 0)µ(η 0) = µ(ξ 1)µ(η 0) µ(ξ 0)µ(η 1) = [µ(ξ 1 )µ(η 0 ) µ(ξ 0 )µ(η 1 )]/[bµ(1)bµ(0)]

Percolation 18 ξ 0 η 1 = ξ 1 ξ 0 η 1 = η 0 µ (8) ( ) 0 φ f bf(1) = ξ Ξ n f(ξ 1 )µ(ξ 1) ξ Ξ n f(ξ 0 )µ(ξ 1) = ξ Ξ n f(ξ 0 )φ(ξ)µ(ξ 0) f(ξ 0 ) ξ Ξ n µ( 0) Ξ n (8) φ bf(1) f(ξ 0 )µ(ξ 0) φ(η)µ(η 0) = f(0) b 1= f(0) b ξ Ξ n η Ξ n bf FKG (21) fg {b 1 b n } ξ Ξ = {0 1} I µ(ξ) =P p ( b1 = ξ(1) bn = ξ(n)) (8) ( ) fg f n g n : E p [ f f n 2 + g g n 2 ] 0 (n ) ( f n = f1 Vn Lebesgue ) Schwartz E p [ fg f n g n ] 2 E p [ f f n 2 ]E p [ g g n 2 ] 0 (n ) 71 µ (8) 72 f f n = f1 Vn Lebesgue E p [ f f n 2 ] 0 (n ) BK I = {1n} Ξ = {0 1} I A Ξ ξ A ξ η η A ξ Ξ S(ξ) ={i I; ξ(i) =1} A B Ξ A B := ξ Ξ; L S(ξ); η A eη B; S(η) =L S(eη) =S(η) \ L ª ξ Ξ L I L 0 ξ (L) ; ξ (L) (j) =ξ(j) (j L) = 0 (j I \ L) : A B = n o ξ Ξ; L S(ξ); ξ (L) A ξ (S(ξ)\L) B 72 (BK ) =( 1 n ) P [0 1] A B Ξ : P ( A B) P ( A)P ( B)

Percolation 19 (P) ( 0 P) ep = P P ea = A Ξ B0 e = B Ξ k =1n B e k [(ξ ξ 0 ) B e k ξ 0 1 k ξ k +1 n T k (ξ ξ 0 ):=(ξ 0 (1)ξ 0 (k) ξ(k +1)ξ(n)) B ] (T 0 (ξ ξ 0 )=ξt n (ξ ξ 0 )=ξ 0 ) ea B e k := n(ξ ξ 0 ) Ξ Ξ; L S(ξ) L 0 S(ξ 0 ); (ξ (L) ξ 0(L0) ) A e (ξ (S(ξ)\L) ξ 0(S(ξ0 )\L 0) ) B e o k ea B e k ea B e n o k = (ξ ξ 0 ) Ξ Ξ; L S(ξ); ξ (L) A T k (ξ (S(ξ)\L) ξ 0 ) B A e B e 0 = A B Ξ A e Bn e = A B P ( A B) = e P (( 0 ) e A e B 0 ) P( A)P ( B) = e P (( 0 ) e A e B n ) k =0 1n 1 ep (( 0 ) e A e B k ) e P (( 0 ) e A e B k+1 ) ( 0 ) µ ie µ(u V )= P e (( 0 ) U V ) µ( A e B e k ) µ( A e B e k+1 ) (ξ ξ 0 ) A e B e k ( ξ b ξ b0 ) Ξ Ξ (ξ ξ 0 ) ( A e B e k ) ( A e B e k+1 ) (ξ ξ 0 ) ( A e B e k ) \ ( A e B e k+1 ) ξ ξ 0 k +1 (ξ ξ 0 ) ( e A e B k ) \ ( e A e B k+1 ) ( b ξ b ξ 0 ) e (A e B k+1 ) \ ( e A e B k ) (9) ( ) b : A e B e k A e B e k+1 ;(ξ ξ 0 ) ( ξ b ξ b0 ) 1 1 ( ( A e B e k ) ( A e B e k+1 ) ( A e B e k ) \ ( A e B e k+1 ) 1 1 ) µ(( ξ b ξ b0 )) = P e ( j = ξ(j) b j 0 = ξ b0 (j)j =1n) Y = P ( j = ξ(j))p (j 0 = ξ 0 (j))p ( k+1 = ξ 0 (k +1))P (k+1 0 = ξ(k +1)) = j6=k+1 ny P ( j = ξ(j))p (j 0 = ξ 0 (j)) j=1 = µ((ξ ξ 0 )) : µ( e A e B k ) = (ξξ 0 ) ea eb k µ(( b ξ b ξ 0 )) = µ({( ξ b ξ b0 ); (ξ ξ 0 ) A e B e k }) µ( A e B e k+1 ) ( j j 0 ) (9) [(ξ ξ 0 ) ( A e B e k )\( A e B e k+1 ) ( ξ b ξ b0 ) ( A e B e k+1 )\( A e B e k )] ξ b ξ b0 ) e (A B e k+1 bξ ξ b0 ) / ( A e B e k L 0 S(ξ); ξ (L 0) A T k (ξ (S(ξ)\L 0) ξ 0 ) B L S(ξ) ξ (L) A T k+1 (ξ (S(ξ)\L) ξ 0 ) / B

Percolation 20 T k (ξ (S(ξ)\L0) ξ 0 ) BT k+1 (ξ (S(ξ)\L0) ξ 0 ) / B k +1 B 1=T k (ξ (S(ξ)\L0) ξ 0 )(k +1)=ξ (S(ξ)\L0) (k +1) 0=T k+1 (ξ (S(ξ)\L0) ξ 0 )(k +1)=ξ 0 (k +1) ξ (S(ξ)\L0) k +1 / L 0 ξ(k +1)=1 j L 0 j 6= k +1 ξ b ξ b(l 0) (j) = ξ(j)ξ(j) b =ξ (L0) (j) ie ξ b(l 0) = ξ (L0) ( A) T k+1 ( b ξ (S(bξ)\L 0) b ξ 0 )=T k (ξ (S(ξ)\L 0) ξ 0 ) B (10) ( ( ξ b ξ b0 ) A e B e k+1 ) j k +1 ( T k+1 ( ξ b(s(bξ)\l 0) ξ b0 )(j) = ξ b0 ξ 0 (j) (j k) (j) = ξ(k +1) (j = k +1) k +1 / L 0 ξ(k +1)=1 k +1 S(ξ) \ L 0 ( T k (ξ (S(ξ)\L0) ξ 0 ξ 0 (j) (j k) )(j) = ξ(k +1) (j = k +1) j>k+1 ξ(j) = ξ(j) b (j 6= k +1) T k+1 ( ξ b(s(bξ)\l0) ξ b0 )(j) = ξ b(s(bξ)\l0) (j) =ξ (S(ξ)\L0) (j) =T k (ξ (S(ξ)\L0) ξ 0 )(j) (10) ( ξ b ξ b0 ) A e B e k+1 ( ξ b ξ b0 ) / A e B e k L S( ξ) b ξ b(l) A T k ( ξ b(s(bξ)\l) ξ b0 ) / B L k +1 L ξ(k b +1)=ξ 0 (k +1)=0 k +1 / S( ξ) b k +1 / L ξ (L) = ξ b(l) A (10) T k ( b ξ (S(bξ)\L) b ξ 0 )=T k+1 (ξ (S(ξ)\L) ξ 0 ) / B ( ) ( ξ b ξ b0 ) / A e B e k 73 T k ( ξ b(s(bξ)\l) ξ b0 )=T k+1 (ξ (S(ξ)\L) ξ 0 ) / B [1] Fortuin C M Kesten P W and Ginibre J; Correlation inequalities on some partially ordered sets Communications in Mathematical Physics 22 89 103 (1972) [2] Harris T E; A lower bound for the critical probability in a certain percolation process Proceedings of the Cambridge Philosophical Society 56 13 20 (1960)

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