On the critical percolation probabilities

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Lucio Russo. Istituto Matematico, Universitfi di Modena, 1-41100 Modena (Italy). Summary. We prove that the critical probabilities of site percolation on the.
Zeitschrift for

Wahrscheinlichkeitstheorie

Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 229 237 (1981)

und verwandte O e b i e t e 9 Springer-Verlag 198I

On the Critical Percolation Probabilities* Lucio Russo Istituto Matematico, Universitfi di Modena, 1-41100 Modena (Italy)

Summary. We prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c + p * = l . Furthermore we prove the continuity of the function "percolation probability".

1. Introduction It was conjectured in [1] that in any pair of dual graphs the critical probabilities of percolation, Pc and Pc*, satisfy the relation Pc + P* = 1.

(1.1)

If, in particular, as in the bond percolation in the square lattice, the graph is self-dual, so that Pc-Pc,* (1.1) becomes pc=1/2.

(1.2)

In the case of bond percolation in the square lattice (1.2) has been recently proved by H. Kesten [2]. Here we extend his result by proving (1.1). The present paper deals only with site percolation in the square lattice, but it seems possible to extend his results to other regular planar graphs. We call #x the Bernoulli probability measure according to which each element of the graph is equal to +1 ("open" in the bond terminology) with probability x. In his paper Kesten determines the #x-probability of suitable events, whose #~-probability is known, by a sequence of modifications of the measure #~. In particular he uses the fact that, by self-duality, the #~-probability of the crucial events A L, + 1 (defined in Sect. 2) for any L equals 1/2. The main new A + for x~[1 tool in our proof of (1.1) is an uniform bound on the functions #x(L,K), --Pc, Pc]. This bound, proved in Sect. 3, allows us to prove also the continuity of the function "percolation probability" (we remark that in the self-dual case, conversely, this last statement is a simple consequence of (1.2)). Section4 contains a remark which allows us to simplify thc main proof. The main result is proved in Sect. 5. * Work supported in part by U.S. National Science Foundation Grant PHY78-25390

0044- 3719/81/0056/0229/$01.80

230

L. Russo

2. Definitions and Some Preliminary Results We shall employ the following terminology and notations. Two elements i, j of Z 2 are adjacent if I i l - j l [ + [ i 2 - j 2 l =1, they are *adjacent if Max([il-jl[, [i2 - j 2 [ ) = l . A finite sequence (i 1.... ,i,) of distinct elements of Z 2 is a (selfavoiding) chain [*chain] if ir and is are adjacent [*adjacent] if and only if [r-s] = 1 (throughout this paper chains and *chains will always be understood to be self-avoiding, in the above specified sense). (iz, ..., in) is a circuit [*circuit] if for any r~(1 .... ,n) (ir, i~+D...,i,,i 1.... ,i~_z) is a chain [*chain]. A set X c Z 2 is connected [*connected] if for any pair i, j of points in X there is a chain [*chain] made up of points in X having i, j as terminal points. We consider the configuration space ~2={-1,1} z2. We define in f2 the partial order __< by putting co1 R [ , l(X) [ 1 - ( 1 --

,

(2.2)

R/2 ~ (x) > R L,* (x) [ 1 - (1 - Rs (x))~] 6,

(2.2 a)

--

R L, § 1 ~,c x]!~ 136

R[,3(x)> [ R [ l(X)]3 [ 1 -t(1 - R +L,1~,txn~q 1 2a, , ,

(2.3)

R L,*3(X)> [ RL,- ,I(X)] 3 [1--(1-- RE, - ,l(x)) ~] 12.

(2.3a)

Proof We consider, besides AL,,, the other square in Z2: A'L, 1 ={ieZ210=#x(F~). Hence, by using once more the same argument, we get, for any s~S(AL,1):

#x(VslE)__>1_(1_R-~L, 1(X))~"

(2.6)

Since G and H~ are positive events Harris' inequality implies: #~(G c~H.) > #~(H.) #~(G) = #~(H.) ~ #x(F~l E) #x(Es) s~Sl

>- [1 - (1 - RE, I(X))~-] 3. By observing that G ~ H . c A[, 3/2 we get R L, + 3/2 (X) => [1

--

(1 -

RLI(x))~]

3.

(2.7)

If we consider the rectangle A'L, 3/2 = { i ~ Z 2 1 - 2 L < i a 1 - 5 -4 Besides AL,3 we consider the other rectangle

A'L, 3 --{i~Z2liil[k; on the other hand we have

#x(M~,t[L~,t)#x(L~,t)

s~A1 t~A2

#x(/~,,)#x(L~,,)>__y~ #x(nf)#~(Ls,,)

scA1 tcA2

s~A1 tcA2

=#~(Df) ~ ~ #~(Ns,,IE.)#~(e.)>_--#.(Df)Y, ~ #x(Ns,~)#~(E.) s~A1 t~A2

s~AI

t~A 2

L >=#~(DE) ~ #x(E~)R~*2(x)>=R[,2(x)R-* L,2(x )#x(Dk)" s ~ AI

Lemma 5.

There exists c~>O such that, VL, Vxe[1-P*,Pc]

RF,~(x)>~, RL*(x)>=~.

236

L. R u s s o

Proof If x~[1-p*,p~], by proposition l, P~(x)=P~*(x)=O. Hence, by lemma 2, R L. + 3 (x)_< _ 1 - 5 - 4, R~ .~ (x) _< 1 - 5 - 4; by using lemma 1 we get R/., l(x) < p,

R[,~(x)l- ~, R ~ ( x ) > l - f l . If we choose e = ( 1 - f l ) ( 1 - ill)6, the statement of the lemma easily follows. Lemma 6. For any positive integer k, there exists Lo(k) such that if x ~ [ 1 - p * , Pc], L~Lo(k), then ~x(DD_>_ L 1/2. _

Proof Since D~ is a negative event, for any x~[1 -p~,p~], * we have /~(D~)~#vr on the other hand since, by proposition 1, P+(p~)=0, #v-a.s. there are infinitely many disjoint (-,)circuits surroinding the origin; hence, for any k, lira #v~(Ok)-L _ 1. L~oo

Now we can easily prove our main result. Theorem 1. Pc + P~* --- 1 . P r o o f Suppose p~> l - p * ;

then we can choose an integer k such that k>2[(p~ + p * - - 1) c~2]-1

(5.1)

(where ~ is the number defined in the proof of Lemma 5). Furthermore we choose an integer L > L o ( k ) (where Lo(k) is the function defined in Lemma6). Then Lemmas 4, 5, 6 imply:

Vxe[1-p~,p~]

~x(o~eafn(A;~,~)(o)_->E)>~V2.

Hence (5.1) implies: Vx6[1 -p~,p~]

u >kcd/2>(p~+p*-l)-l.

By using Lemma 3 we get p pc(A2+L, 1)--]~l_p2(A2+L,1):>(pc+p * - 1)

min

d : - #x(A~L.1)> 1.

x~[1 -p~, p~l d x

The last inequality gives a contradiction, since Vx~[0, 1] #x(A2+L,1)~[0, I]. This proves Theorem I. We call S+(x) [-S-*(x)] the mean size, with respect to the measure #x, of the finite (+)clusters [ ( - .)clusters]. Theorem 1, together with results of Refs. 4 and 5 implies the following theorem Theorem 2. There exists pc~(O,1) such that:

a) If x 0, P•*(x)=0,

S+(x)