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holds. Let A be an increasing subset ofwith A₆₌∅, . For p_∈(0,1), d

d pφp(A)≥cφp(A)(1−φp(A))log[1/(2 maxe IA(e))], where IA(e)is the influence of e on A with respect to the measureφp.

Theorem 4.81 takes an especially simple form when A has a certain property of symmetry. In such a case, the following sharp-threshold theorem implies that f(p) ₌ φp(A)increases from (near) 0 to (near) 1 over an

interval of p-values with length of order not exceeding 1/log N .

Let5be the group of permutations of E. Any π _∈ 5 acts onby π ω₌(ω(πe): e∈ E). We say that a subgroupAof5acts transitively

on E if, for all pairs j,k_∈E, there existsα_∈A_{withαj =k.}

LetA_{be a subgroup of5. A probability measureφon(,}F_{)is called} A_{-invariant ifφ(ω)=φ(αω)for all α ∈} A_{. An event A ∈} F _{is called} A_{-invariant if A=αA for allα∈}A_{. It is easily seen that, for any subgroup} A,φpisA-invariant.

4.82 Theorem (Sharp threshold) [93]. There exists a constant c satisfying c_∈(0,_∞)such that the following holds. Let N _{= |}E_{| ≥}1. Let A _∈F

be an increasing event, and suppose there exists a subgroupA_of5acting

transitively on E such that A isA_{-invariant. Then}

(4.83) d

4.7 Russo’s formula and sharp thresholds 77

Proof. We show first that the influences IA(e)are constant for e∈ E. Let e,f _∈E, and findα _∈A_{such thatαe= f . Under the given conditions,}

φp(A,1f =1)= X ω∈A φp(ω)1f(ω)= X ω∈A φp(αω)1e(αω) = X ω′∈A φp(ω′)1e(ω′)=φp(A,1e=1),

where 1g is the indicator function that ω(g) = 1. On setting A = ,

we deduce thatφp(1f = 1)= φp(1e =1). On dividing, we obtain that

φp(A | 1f = 1) = φp(A | 1e = 1). A similar equality holds with 1

replaced by 0, and therefore IA(e)=IA(f).

It follows that _X

f∈E

IA(f)=N IA(e).

By Theorem 4.38 applied to the product space(,F_{, φp), the right side is}

at least cφp(A)(1−φp(A))log N , and (4.83) is a consequence of Theorem

4.79.

Letǫ_∈(0,1₂)and let A be increasing and non-trivial. Under the condi- tions of Theorem 4.82,φp(A)increases fromǫto 1−ǫover an interval of

values of p having length of order not exceeding 1/log N . This amounts to a quantification of the so-called S-shape results described and cited in [106, Sect. 2.5]. An early step in the direction of sharp thresholds was taken by Russo [213] (see also [231]), but without the quantification of log N .

Essentially the same conclusions hold for a family_{µp : p ∈ (0,1)}

of probability measures given as follows in terms of a positive measureµ satisfying the FKG lattice condition. For p_∈(0,1), letµpbe given by

(4.84) µp(ω)= 1 Zp Y e∈E pω(e)(1₋p)1−ω(e) µ(ω), ω_∈, where Zp is chosen in such a way thatµpis a probability measure. It is

easy to check that eachµpsatisfies the FKG lattice condition. It turns out

that, for an increasing event A₆₌∅, , (4.85) d d pµp(A)≥ cξp p(1₋p)µp(A)(1−µp(A))log[1/(2 maxe JA(e))], where ξp=min e∈E µp(ω(e)=1)µp(ω(e)=0) .

The proof uses inequality (4.36), see [100, 101]. This extension of Theorem 4.81 does not appear to have been noted before. It may be used in the study

of the random-cluster model, and of the Ising model with external field (see [101]).

A slight variant of Theorem 4.82 is valid for measuresφpgiven by (4.84),

with the positive probability measureµ satisfying: µsatisfies the FKG lattice condition, andµisA_{-invariant. See (4.85) and [100, 109].}

From amongst the issues arising from the sharp-threshold Theorem 4.82, we identify two. First, to what degree is information about the groupA

relevant to the sharpness of the threshold? Secondly, what can be said when

p ₌ pN tends to 0 as N → ∞. The reader is referred to [147] for some

answers to these questions.

4.8 Exercises

4.1 Let Xn,Yn∈L2(,F,P)be such that Xn→X , Yn→Y in L2. Show

that XnYn → X Y in L1. [Reminder: Lpis the set of random variables Z with

E(_|Z_|p) < _∞, and Zn → Z in LpifE(|Zn−Z|p) → 0. You may use any

standard fact such as the Cauchy–Schwarz inequality.]

4.2 [135] LetP_pbe the product measure on the space_{0,1_}nwith density p. Show by induction on n thatP_psatisfies the Harris–FKG inequality, which is to say thatPp(A∩B)≥Pp(A)Pp(B)for any pair A, B of increasing events.

4.3 (continuation) Consider bond percolation on the square latticeZ2. Let X and Y be increasing functions on the sample space, such thatEp(X2),Ep(Y2) <

∞. Show that X and Y are positively correlated in thatE(X Y)_≥E(X)E(Y).

4.4 Coupling.

(a) Take=[0,1], with the Borelσ-field and Lebesgue measureP. For any distribution function F, define a random variable ZFonby

Z_F(ω)₌inf_{z :ω_≤ F(z)_}, ω_∈. Prove that

P(ZF ≤z)=P [0,F(z)]=F(z),

whence Z_Fhas distribution function F.

(b) For real-valued random variables X , Y , we write X ≤stY ifP(X≤u)≥ P(Y_≤u)for all u. Show that X_≤stY if and only if there exist random variables X′, Y′on, with the same respective distributions as X and Y , such thatP(X′_≤Y′)₌1.

4.5 [109] Letµbe a positive probability measure on the finite product space _{= {}0,1_}E.

(a) Show thatµsatisfies the FKG lattice condition

µ(ω1∨ω2)µ(ω1∧ω2)≥µ(ω1)µ(ω2), ω1, ω2∈, if and only if this inequality holds for all pairsω1,ω2that differ on exactly two elements of E.

4.8 Exercises 79 (b) Show that the FKG lattice condition is equivalent to the statement thatµ

is monotone, in that, for e∈E,

f(e, ξ ):₌µ ω(e)₌1ω(f)₌ξ(f)for f ₆₌e is non-decreasing inξ∈ {0,1}E\{e}_.

4.6 [109] Letµ1,µ2be positive probability measures on the finite product = {0,1}E_{. Assume that they satisfy}

µ2(ω1∨ω2)µ1(ω1∧ω2)≥µ1(ω1)µ2(ω2),

for all pairsω₁, ω_2∈that differ on exactly one element of E, and in addition that eitherµ1orµ2satisfies the FKG lattice condition. Show thatµ2≥stµ1.

4.7 Let X₁,X₂, . . .be independent Bernoulli random variables with parameter p, and Sn=X1+X2+ · · · +Xn. Show by Hoeffding’s inequality or otherwise

that

P _|Sn−np| ≥x√n≤2 exp(−1₂x2/m2), x>0,

where m=max{p,1−p}.

4.8 Let Gn,pbe the random graph with vertex set V= {1,2, . . . ,n}obtained

by joining each pair of distinct vertices by an edge with probability p (different pairs are joined independently). Show that the chromatic numberχn,psatisfies

P _|χn,p−Eχn,p| ≥x≤2 exp(−1₂x2/n), x>0.

4.9 Russo’s formula. Let X be a random variable on the finite sample space