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If L(h) denotes the law of h, it would probably be more appropriate to write pN (β, L(h)) rather than pN (β, h). The simpler notation pN (β, h) is motivated by the fact that the most important case (at least in the sense that it is as hard as the general case) is the case where h is constant. If this notation disturbs you, please assume everywhere that h is constant and you will not lose much. Thus with these notations we have pN (β, h) = ϕ(1) . s z and h. 69). 7. (Guerra’s replica-symmetric bound).

8. 68), so that for ν0 there is no correlation between sites, so √ this is a (nice) exercise in Calculus. 68) means that −H0 (σ) = i≤N σi Yi . 74) we have ν0 (σi1 σi2 ) = E σi1 σi2 0 = E σi 2 0 = Eth2 Yi = q . d. s of expectation q, for which all kinds of estimates are classical. Nonetheless we give a simple self-contained proof. The main step of this proof is to show that for every u we have ν0 exp N u(R1,2 − q) ≤ exp N u2 . v. with Eu2 = 2λ/N , independent of all the other sources of randomness.

92), we get, by deﬁnition of · t , U (σ, τ )C(σ, τ , u(t)) = n(n + 1) U (σ n+1 , σ n+2 )f (σ 1 , . . 90). 91). ,σ n and where 1{σ · · · wσn f (σ 1 , . . , σ n ) exp =σ} = 1 if σ = σ and is 0 otherwise. 93) σ,τ = Z(u(t))−n U (σ , σ )wσ1 · · · wσn f (σ 1 , . . ,σ n = U (σ , σ )f (σ 1 , . . 90). The case of the other terms is similar. 4. 91). 2 until she fully understands it. 90) will occur on several occasions (since Gibbs’ measures are intrinsically given by a ratio of two quantities). More generally, calculations of a similar nature will be needed again and again.