(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law

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1 I (Radom Walks ad Percolatios) ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.)

2 (Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law of Large Numbers) (Radom Walks) 4 2. (Markov Chais) d (d-dimesioal Radom Walks) (Oe-dimesioal ati-symmetric Radom Walks) (Percolatios) /3 p H 2/ ( 2.2(iii))

3 (Basic of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F,P),, X,. (Ω, F,P) (probability space) Ω ( ω Ω ) F( 2 Ω ) Ω σ (σ-field); (2 Ω Ω ) (i) Ω F (ii) A F A c F (iii) A F ( =, 2,...) A F P = P (dω) (Ω, F) (probability measure), i.e., ; P : F [0, ]. (i) P (Ω) = (ii) A F ( =, 2,...) P ( A )= P (A )(σ ). (Ω, F,P),. (i) σ-., F σ- A, B, A F F, A B, A \ B, A B := (A \ B) (B \ A), A. lima = lim sup A := A, lima = lim if A := A F. N N N N (lim = if sup, lim = sup if.) (ii) P ( ) =0, A, B F; A B P (A) P (B) ( ). (iii) A k F (k =, 2,...,) P ( A k)= ( ) P (A k)( ). (iv) A F, A P A = lim P (A ). ( ) (v) A F, A P A = lim P (A ). ( ) (vi) A F ( ) P A P (A ). ( ) (vii) (Borel-Catelli ) A F ( ), P (A ) < P lim sup A = 0, i.e., ( ) P lim if Ac =. (Ω, F,P) X = X(ω) :Ω R {X a} := {ω Ω; X(ω) a} F ( a R). (radom variable). X k S = {a j } j, {X = a j } F ( j ). X k (Ω, F,P) (k =, 2,...,). {X k } (idepedet) P (X a,,x a )=P (X a ) P (X a ) ( a k R,k =,...,).

4 {X k } k N {X k } N. X k S = {a j } j, : P (X = b,,x = b )=P (X = b ) P (X = b ) (b k S, k =,...,). µ(a) =P (X A) X (distributio), F (x) =P (X x) X (distributio fuctio)..2, (Expectatios, Meas) X Z := Z {± }. X (expectatio, mea) EX = E[X] = XdP = X(ω)P (dω). () X 0 EX := P (X = )+ P (X = ). =0 (P (X = ) =0 P (X = ) =0. P (X = ) > 0 EX =.) (2) X X + := X 0, X := ( X) 0 ( X ± 0, X = X + X.) EX := EX + EX.,. EX = Z P (X = ), f : Z R, Ef(X) = f()p (X = ). (.) Z ;f()>0 ;f()<0 X, V (X) :=E[(X EX) 2 ]=E[X 2 ] E[X] 2.. (Chebichev ) p. a>0, P ( X a) E[ X p ] a p. [ ] P ( X a) =P ( X p a p ) p =. E X = P ( X = ) a Ω P ( X = ) a a P ( X = ) =ap ( X a)..2 X,...,X Z, E[Xk 2 ] < (k =,...,). X,...,X, E[X j X k ]=E[X j ]E[X k ](j k). 0(E[X k ]=0) ( ) 2 E X k = E[Xk 2 ]. [ ] () j k P (X j = m, X k = ) =P (X j = m)p (X k = ) E[X j X k ]= m, mp (X j = m, X k = ) = m, mp (X j = m)p (X k = ) =E[X j ]E[X k ]. ( ) 2 (2) X k = X j X k () j k E[X j X k ]=E[X j ]E[X k ]=0. X 2 k + j k 2

5 .3 (Weak Law of Large Numbers), /2.,..3 ( (Weak Law of Large Numbers)) X,X 2,... EX = m v := sup V (X ) < ɛ>0, ( ) ( ) lim P X k m ɛ =0, i.e., lim P X k m <ɛ =. [ ] {X } { X = X m} ( ). X k m = (X k m), X X m = 0, i.e., E[X ]=0 V (X )=E[X] 2, ( ) 2 E X k = E[Xk 2 ]= V (X k ) sup V (X )=v. ɛ>0, ( P ) X k ɛ = P ( ) X k ɛ E[( X k) 2 ] ɛ 2 2 v ɛ 2 2 v = ɛ 2 0 ( ).., P. ( lim ) X k = m =. X,X, ɛ>0, P ( X X ɛ) 0( ), X X i pr., X X. P (X X) =, X X, P -a.s., X X. (a.s. almost surely ).2, i.e., X X, P -a.s. X X i pr.. ( P (X X) = ( { P X X < } ) ( { = P X X } ) =0 k k k N N = k, lim P N ( N { X X k } ) = P k N N ( N N { X X k } ) =0 k, ε =/k, lim P ( X X /k) =0 m, N ; N,P( X X /k) < /m.) 3

6 2 (Radom Walks),,.,,. d, d. Z d ( j =(j,...,j d )) d (lattice). (X,P) d (simple radom walk),, 2d,. Y = X X ( ) {X 0,Y,Y 2,...}, {Y }, P (Y = k) =/(2d) ( k =), =0( k ). k =(k,...,k d ), k = k k2 d. {X 0,Y,Y 2,...}, k 0,k,...,k Z, P (X 0 = k 0,Y = k,...,y = k )=P(X 0 = k 0 )P (Y = k ) P(Y = k ). Z d {p k } k Z d (p k 0, p k =), (X,P), d. P (Y = k) =p k (,k Z d ). P j (X = k,...,x = k ):=P(X = k,...,x = k X 0 = j) P j (X,P j ) j d. 2. P (A B) :=P (A B)/P (B) P (B) > 0. A, B F P (A B) =P (A). 2. (Markov Chais),,.. 2. S,,.,,,.,.,,,,.. S, S (X,P)=(X (ω),p(dω)) ( =0,, 2,...) (Markov Chai) : 4

7 (M) [ ],j 0,j,...,j,k S, P (X + = k X 0 = j 0,X = j,...,x = j )=P (X + = k X = j ).. (M2) [ ],j,k S, P (X + = k X = j) =P (X = k X 0 = j).,. X 0 µ = {µ j }; µ j = P (X 0 = j) (iitial distributio),, j S, P (X 0 = j) = P P j, (X,P j ) j. ( P (X 0 = j) > 0, P j ( ) :=P ( X 0 = j),.) 2.2 P j ( ): P j (X = k,...,x = k ) := P (X = k,...,x = k X 0 = j) = P (X m+ = k,...,x m+ = k X m = j) (m 0). 0,j,k S, j,k = P (X = k X 0 = j), Q () =( j,k ) ( ) (-step trasitio probability (trasitio matrix)),, Q () Q =(q j,k ),, ( ). 2.. (i) j,k 0, k q() j,k =(j S), (ii), j 0,j,...,j S P (X 0 = j 0,X = j,...,x = j )=µ j0 q j0,j q j,j, (iii) m,, j,...,j m,k 0,k,...,k S P (X + = j,...,x +m = j m X 0 = k 0,X = k,...,x = k )=q k,j q j,j 2 q jm,j m. (iv) Q (0) = I := (δ jk )( ), Q () = Q ( ),, δ jk =(j = k), = 0 (j k). 2.2 µ = {µ j } X. P (X = k) = j S µ j j,k., j S (recurrece time): T j : T j = if{ ; X = j} (= if { } = ). j (recurret) j (trasiet) def P j (T j < ) =, def P j (T j < ) < 5

8 . j, T j, j (positive-recurret) def E j [T j ] <, j (ull-recurret) def E j [T j ]=, P j (T j < ) =. E j [T j ] T j P j, : E j [T j ]= mp j (T j = m)+ P j (T j = ). m= j (or,, ) (X ) (or,, ) {X } Q =(q j,k ) π = {π j } π (statioary distributio) def π k = j π jq j,k (k S), π (reversible distributio) def π k q k,j = π j q j,k (j, k S) (i) π,, X π. (ii) π, {X } :,j 0,...,j S, P (X 0 = j 0,...,X = j )=P(X 0 = j,...,x = j 0 ). {X } Q =(q j,k ) (irreducible) j, k,, j,k > 0.,,. (,,.), : 2.2 j, k S. (i) j : a) j,j =. =0 b) P j ({X } j )=. (ii) j : a) j,j <. =0 b) P j ({X } j )=0. (iii) {X },,,,,. (π j )[ k, j π jq j,k = π k ], π j =/E j [T j ] ( ). 6

9 (i), (ii) b), a), (iii). (iii). 2. O- {B k }, A, C, P (A B k)=p (A C) ( k ). P (A B k )=P(A C). O-2 m,, j,...,j m,k 0,k,...,k S P (X + = j,...,x +m = j m X 0 = k 0,X = k,...,x = k ) = P (X + = j,...,x +m = j m X = k ). 2. (i) j S P j ({X } j )=. (ii) j S P j ({X } j )=0..,,,., 0. m j T (m) j. P j (T (m) j T () j = T j, T (m) j = mi{ >T (m ) j ; X = j} (= if { } = ). < ) =P j (T j < ) m. s, t,, P j (T (m) j = s + t T (m ) j = s) =P j (T j = t) (, [ ]= P (X s+t = j, X s+u j ( u t )), {X u j} = k {X u S;k u j u = k u } {T (m ) j = s} {X,...,X s (= j)}, 2 O-, O-2.)., P j (T (m) j P j (T (m ) j = s, T (m) j P j (T (m) j < ) = P j (T (m ) j = = s + t) =P j (T (m ) j = s)p j (T j = t) s=m t= <T (m) j < ) P j (T (m ) j = P j (T (m ) j < )P j (T j < ) < ) =P j (T j < ) m. P j ({X } j ) = P j ( m P j (T j < ) =, 0. = s, T (m) j = s + t) {T (m) j < }) = lim m P j(t (m) j < ) = lim m P j(t j < ) m. [, ] 7

10 ,. j, k S, f (m) j,k := P j(t k = m) (m ) Q jk (s) := =0 j,k s ( s < ), F jk (s) := m= f (m) j,k s ( s ). { j,k } 0, {f (m) j,k } m (geeratig fuctios). F jk () = P j (T k < ). 2. j, k S, : j,k = m= f (m) j,k q m k,k ( ), Q jk (s) =δ jk + F jk (s)q kk (s) ( s < ). {T k = m} = {X m = k, X s k ( s m )} m= f (m) j,k q( m) k,k = = = P j (T k = m)p j (X = k X m = k) m= P j (T k = m)p j (X = k T k = m) m= P j (X = k, T k = m) m= = P j (X = k) = j,k. Q jk (s) = δ jk + = δ jk + j,k s f (m) j,k q( m) k,k s m= = δ jk + F jk (s)q kk (s). 2.2 j S =0 j,j =. Q jj (s)( F jj (s)) = ( s < ) F jj () = P j (T j < ) lim Q jj (s) = s =0 j,j s. : =0 j,j ( P j(t j < )) =. 8

11 2.6 j k j S =0 k,j < ( k S), k S; =0 k,j = j :. ( q() k,j = F kj() q() j,j.) 2.2 j j k [i.e., ; j,k > 0] P k(t j < ) =., i, j S P i (T j < ) =q i,j +. (, k S;k j q i,k P k (T j < ) P i (X = k, T j = ) =q i,k P k (T j = ) P i (T j < ) = P i (X = k, T j = ) k S.) i = j j, k ; q j,k > 0, P k (T j < ) =., k 2 ; q k,k 2 > 0, i.e, q (2) j,k 2 > 0, P k2 (T j < ) =., j,k > 0 (k,...,k ); q j,k q k,k 2 q k2,k 3 q k,k > 0, : j, j k =0 k,j =. j, k S j k k j j k. 2.3 j, k S; j k, j,, k.,,,. l, m 0; q (l) j,k > 0,q(m) k,j > 0. j, =0 q (l+m+) j,j q (l) j,k q() k,k q(m) k,j ( 0) Q jj (s) s l+m q (l) j,k q(m) k,j Q kk(s). lim Q jj (s) = s =0 j,j < k,k <, k. j, k. 9

12 . 2. Q jj (s)( F jj (s)) = F jj (s) =Q jj (s)/q jj(s) 2. j Q jj lim (s) s Q jj (s) 2 = F jj ( ) =E j[t j ] <.. Q kk (s) s l+m q (m) k,j q(l) j,k Q jj(s), Q jj (s) (l + m + )s l+m+ q (l+m+) =0 s l+m q (l) j,k q(m) k,j Q kk(s) Q kk (s) Q kk (s) 2 Q jj (s) s 3(l+m) (q (l) j,k )3 (q (m) k,j )3 Q jj (s) 2. j,j E k [T k ]=F kk( ) Q kk = lim (s) s Q kk (s) 2 < k. j, k. 2.8 k S E k [T k ]=F kk ( ) , 2.7 j, k S, q() j,k =. j, k S, q() j,k <., j, k S, q() j,k,. 2.2 d (d-dimesioal Radom Walks) (X,P) d., {p k } k Z d Z d, {X 0,X X 0,X 2 X,...}, P (X X = k) =p k (,k Z d ). ( p k =/(2d).), d. Q =(q j,k ) q j,k = p k j [,,, ] 2.9 (X,P) d. (i) X + X (X 0,X,...,X ), i.e., P (X + X = j, X 0 = k 0,X = k,...,x = k ) = P (X + X = j)p (X 0 = k 0,X = k,...,x = k ). k 0,k,...,k Z d, X + X X. (ii) P (X + = j X 0 = k 0,X = k,...,x = k )=P(X + = j X = k )=p j k. {X }, q j,k = p k j. 0

13 (iii). ( j k := j k + + j d k d j k, j = k.), Q =(q j,k )=(p k j ),,,., : 2.3 d (i) d =, 2 (i.e., E j [T j ]= P j (T j < )), (ii) d ( ), 0,0. q (2+) 0,0 =0, q (2) 0,0.. ( 2.2.) 2.4 d Q =(q j,k ) (i) d =, 2 q (2) 0,0 { / π (d =) /(π) (d =2) (ii) d =3 C q (2) 0,0 C 3/2. 2.4,, : (d =3 (3/π) 3 /4) q (2) 0,0 2 d d d/2 (π) d/2 ( ). a b ( ) def a /b ( ). 2.0 {a }, {b }, a b ( ) c,c 2 > 0; c b a c 2 b ( ).. [ (Stirlig s formula)]! 2π +/2 e ( ). 2.4 d =, : ( ) q (2) 2 0,0 = 2 2. d =2 q (2) 0,0 = j,k 0;j+k= (2)! (j!k!) = ( 2 ) j=0 ( ) k

14 , j=0 d =3 ( ) 2 = k ( ) 2. q (2) 0,0 = j,k,m 0;j+k+m= (2)! (j!k!m!) 2 6 2, 3 q (2) (2)! 0,0 c 3 6 2!. c = max j,k,m 0;j+k+m= (j!k!m!). c,,. c c3 +3/2 3/2 e (c>0 ). (), 3 (m!) 3 ( =3m) c (m!) 2 ((m + )!) ( =3m +) (m!) ((m + )!) 2 ( =3m +2) (2),, c,c 2 > 0 c +/2 e! c 2 +/2 e (2), (), d =3 ( ). [ ( 2.2).] d =, 2,. 2.5 d =, 2 Z d (i.e., E 0 [T 0 ]= ). 2.3 (i) α> α s Γ(α +) ( s) α+ (s ). (ii) α = s = log s. α> log(/s) s (s ) : α = log. 0 x α s x dx = ( log s ) α Γ(α +). 2

15 F 00 (s) =Q 00 (s)/q 00(s) 2, d =, q (2) 0,0 / π ( ), s Q 00 (s) =+ Q 00 (s) = s 2 q (2) 0,0 + 2s 2 q (2) 0,0 s 2 Γ(/2) π π ( s 2 ), /2 2s 2 2 Γ(3/2) π π ( s 2 ). 3/2 F 00(s) = Q 00(s) Q 00 (s) 2 2 πγ(3/2) Γ(/2) 2 s (s ). s 2 E 0 [T 0 ] = lim F s 00(s) =. d =2 q (2) 0,0 /(π) ( ), s Q 00(s) E 0 [T 0 ] = lim s 2 Q 00 (s) π log s 2, 2 πs( s 2 ) 2 π( s 2 ) [ π( s 2 ) ( log ) ] 2 s 2 =. 2.3 (Oe-dimesioal ati-symmetric Radom Walks) Z {X } p (0 <p<), p. p /2, {X = X (p) }. d, d, ( 0 ). q j,j+ = q 0, = p, q j,j = q 0, = p, j,k = ( ) +j k p ( j+k)/2 ( p) (+j k)/2 ( + j k 2Z) 2 0 ( + j k 2Z +) [ + l, m, = l + m. k j =?.] 3

16 q (2) 0,0 = ( ) 2 (p( p)) (4p( p)) π ( ) 2.4. p /2 4p( p) < : 2.4 {X = X (p) } (0 <p<,p /2)., :. ( P lim ) X =2p =. 2.5 p>/2 j, u j (s) :=F j0 (s) = m sm P j (T 0 = m) (0<s<) lim j u j(s) =0,,. u (s) = psu 2 (s)+( p)s u j (s) = psu j+ (s)+( p)su j (s) (j 2) ( ) j 4p( p)s F j0 (s) = 2 (0 <s<) 2ps ( ) j p P j (T 0 < ) = (j ) p. [ {X = j +}, {X = j }., P j (T 0 = m) =P j (T 0 = m X = j +)P j (X = j +)+P j (T 0 = m X = j )P j (X = j ), P j (T 0 = m X = j ) j 2 P j (T 0 = m ), j = P (T 0 = m X =0)=. ] 2.6 u j := P j (T 0 < ) j Z. p>/2 j, u j = P j (T 0 < ) =, j = 0 u 0 = pu +( p), P 0 (T 0 < ) =u 0 = 2( p) <. 4

17 3 (Percolatios) 3. Z 2, B 2 = {{x, y}; x, y Z 2, x y =} 2, bod.,, p (0 p ) ope, p closed. X b = X (p) b = X (p) b (ω) b B 2, X = {X b ; b B 2 }. P p. S = {b B 2 ; X b =}, O C O, (ope cluster). C O C O. p H : Hammersley (critical probability) θ(p) =P p ( C O = ), : p H = if{p [0, ]; θ(p) > 0}. p T : Temperley χ(p) =E p [ C O ]=, : P p ( C O = )+ P p ( C O = ) p T = if{p [0, ]; χ(p) = }. p H p T, p H = p T.. 3. Z 2, p H, p T, p H = p T = /2, p c. p>p c,, p p c. 3.. Z d (d 3), p c =/2., /3 p H 2/3. θ(p) =P p ( C O = ) p., p,. 3. p H p T.,,,,.,. Ω=Ξ:={0, } B2 ω = ω(b); B 2 {0, }, (0 ) Ω ; (cylider set) A i,...,i b,...,b = {ω; ω(b )=i,...,ω(b )=i } (b k B 2,i k = 0 or,k =,...,) 5

18 C. F = B(Ξ) := σ(c) (C σ-field, C σ-field). B(Ξ) = {G 2 Ω ; G C σ-field}. P = P p cylider set, P p (A i,,i b,...,b )=p i+ +i ( p) ( i)+ +( i). (.), b B 2 (Ω b, F b,p b,p ) Ω b = {ω(b) =,ω(b) =0}, F b =2 Ω b, P b,p (ω(b) = ) = p,. X b = X (p) b X b (ω) =ω(b). X b P p (X b =)= P p (ω(b) = ) = p, X = {X b ; b B 2 } P p. X(ω) =ω. (.) θ(p) =P p ( C O = ) p. θ (p) :=P p ( C O ) θ(p) = lim θ (p) θ (p). b B 2, Z b Z b (ω) [0, ], {Z b }. Q Q(Z b p) =p = P p (X b =). S(p) ={b B 2 ; Z b p}, O C O (p), p θ (p) =P p ( C O ) =Q( C O (p) ) θ (p). 3.2 /3 p H 2/3 p H,. Peierls. 3.2 Hammersley p H /3 p H 2/3. = if{p [0, ]; θ(p) = P p ( C O = ) > 0},. γ = {x 0,b,x,b 2,...,b,x } (, path) (i) b = {x i,x i }, (ii) i j b i b j., γ = {b,...,b }. 3.2 [p H /3 ] p</3,θ(p) =0., γ ( ), P p (γ C O )=P p (X b =, b γ )=p. γ 4 3, P p ( γ C O ) 4 3 p (< if p</3). 6

19 C O = N, N; γ C O, p</3, Borel- Catelli θ(p) =P p ( C O = ) P p { γ C O } =0. N N p H /3. [p H 2/3 ] p>2/3,θ(p) > 0. Z 2 (Z 2 ) := {(m +/2,+/2); m, Z}. (Z 2 ) b (B 2 ) b B 2. X b := X b {X b ; b (B 2 ) } = {X b ; b B 2 }. N, V N := {(m, ) Z 2 ; m := max( m, ) N}, V N. p>2/3 N = N(p) S = {b B 2 ; X b =}, P p (S V N ) 2 (3). C O < C O (B 2 ) (closed path) γ. O V N, P p (S V N ) = P p (V N (B 2 ) γ ) P p (X b =0, b γ ) γ ; V N, γ = k V N k 4(2N +) = 8N +4 ( 8N), γ [0,k] {0} (, {(j +/2, /2); j k} ) γ, k 4 3 k. P p (S V N ) 4k 3 k ( p) k. p>2/3 N, 0 ( ). N = N(p) (3). (3) {X b ; b V N }, {X b ; b V N } ( P p {Xb =, b V N } {S V N } ) = P p (X b =, b V N )P p (S V N ) 2 P p(x b =, b V N )= p4n 2 > 0. 2 P p ( C O = ) ( ), p>2/3 θ(p) > 0,, p H 2/ [0 p</3,θ(p) =0 p H /3] [2/3 < p,θ(p) > 0 p H 2/3]. 3.3 p>2/3 k3 k ( p) k <. k k 8N 3.4 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), i.e., S(p + h) =S(p) ( ), θ(p + h) θ(p) (h 0). h>0, θ h (p) (p ), θ (p) θ(p) ( ). 3.5 f (x) [0, ] f f ( ) f(x) [0, ]. 7

20 4.,. 4. ( 2.2 (iii)) 4.. π =(π j ) π j =/E j [T j ] > 0,. i, j S, 2. Q ij (s) =δ ij + F ij (s)q jj (s) i j s lim( s)q jj (s) = lim s s F jj (s) = F jj ( ) = E j [T j ]. lim( s)q ij (s) = F ij() s E j [T j ]., 2.2 F ij () = P i (T j < ) = i, j S, lim( s)q ij (s) = s E j [T j ] (=: π j ). E j [T j ] < 0 <π j. j S ( s)q ij (s) = Fatou j S π j, k S, j S π j q j,k lim if s ( s)q ij (s)q j,k j S lim( s) s =0 s q (+) i,k = lim s ( s)s (Q ik (s) δ ik ) = π k, k π j q j,k = π k (k S) j S. π j ( s)q jk (s) =( s) j S =0 s j S π j j,k = π k. (4) s Lebesgue j π j π k = π k j π j =. π =(π j ). π =(π j ), (4) s π k = F jk () π j E k [T k ] + π k E k [T k ] E k [T k ] j k. k; π k > 0 E k [T k ] <,. k S E k [T k ] <, 2.2 F jk () = P j (T k < ) =( j, k S) 8

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