By Andre Mercier

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Let A be an increasing event, and write θ ( p, q) = φ p,q (A). 1 we shall construct a Markov chain Z t = (X t , Yt ) taking values in the product space 2 . 25). Let De (ω) be the indicator function of the event that the endvertices of e are connected by no open path of E \ {e}. We deﬁne the functions H, H A : 2 → R as follows. 31) for ω ∈ and e ∈ E. Secondly, H (ω, ω ) = 0 for other pairs ω, ω with ω = ω . 32) H A (ω, ω ) = H (ω, ω )1 A (ω ∧ ω ) if ω = ω . The diagonal terms H (ω, ω) and H A (ω, ω) are chosen in such a way that H A (ω, ω ) = 0, H (ω, ω ) = ω∈ ω∈ .

K − 1 , Je denotes both the event {ω ∈ : ω(e) = 1} and its indicator function. interesting aspect of the proof of this theorem is the use of discrete Fourier transforms and hypercontractivity. 31) f k (x) = 1 if x k > 1 − ak , 0 otherwise. It may be shown as follows that the function f : [0, 1] E → {0, 1} E is nondecreasing. 30) corresponding to the vectors x and x . Clearly a1 = a1 , so that f1 (x) ≤ f 1 (x ). Since µ is monotonic, a2 ≤ a2 , implying that f 2 (x) ≤ f2 (x ). Continuing inductively, we ﬁnd that fk (x) ≤ f k (x ) for all k, which is to say that f (x) ≤ f (x ).

We write = {0, 1} E as usual, and F for the set of all subsets of . Let µ be a probability measure on ( , F ), and A an increasing event. 27) where J = (Je : e ∈ E) denotes8 the identity function on . There has been an extensive study of the largest inﬂuence, maxe I A (e), when µ is a product measure, and this has been used to obtain concentration theorems for φ p (A) viewed as a function of p, where φ p denotes product measure with density p on . Such results have applications to several topics including random graphs, random walks, and percolation.