Communications in Mathematical Physics - Volume 218 - download pdf or read online

By M. Aizenman (Chief Editor)

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For i ≤ n, an interval J ⊂ ˜ n and aˆ ∈ J , we say i,n (a) ˆ has a smooth continuation to J if there is a map g : i,n (a) ˆ × J → R0 such that – g(·, a) = i,n (a) for all a and ˆ a → g(z, a) is smooth. – for each z ∈ i,n (a), Likewise one has the notion of the critical regions C (i) deforming continuously as a ranges over J . 4. Let aˆ ∈ n and J = [aˆ − ρ 2n , aˆ + ρ 2n ]. Then J ⊂ ˜ n ; moreover, has a smooth continuation to J , and C (i) , i ≤ n, deform continuously on J . ˆ n,n (a) The structual stability of the critical regions comes from the fact that the components of C (i) are stacked together in a very rigid way, and their relations to the components of C (i−1) are equally rigid.

They must lie on the two horizontal boundaries of Q(k−i0 ) (zk ) because γk−i0 passes through zk and intersects no other point of ∂Rk−i0 . 3 is < ( DT 200 b)k−i0 < (Kb) 100 k . 1. By the Borel–Cantelli Lemma, it suffices to show that −k Z (k) ) < ∞. We estimate m(T −k Z (k) ) by k m(T m(T −k Z (k) ) = m(T −k (Q(k) ∩ Z (k) )) ≤ max m(T −k (Q(k) ∩ Z (k) )) m(T −k Q(k) ) m(T −k Q(k) ), where the summations and maximum are taken over all components Q(k) of C (k) . Note also that m(T −k Q(k) ) < 1. 2 and the regularity of det(DT ) in (**), we obtain (k) ∩ Z (k) ) m(T −k (Q(k) ∩ Z (k) )) 2k m(Q · ≤ K m(T −k Q(k) ) m(Q(k) ) (Kb) 100 k · b 20 k 99 ≤ K 2k · ( Kb )k+1 · ρ k 1 b 25 k · b ρk 1 1 ≤ K 4k which decreases geometrically in k as desired.

2) With the properties of θN in (1) having been established, we observe that continuing to use θN as the source of control, the material in Sects. 3 are now valid for times up to min(m, 3N ). (3) We are now ready to argue that (IA2)–(IA6) hold for all z0 ∈ 3θN . For each z0 ∈ 3θN , whether it is in θN or of generation > θ N, there exists z0 ∈ θN such that |z0 −z0 | = O(b e−β3N θN 4 ). This implies, for i ≤ 3N , that |zi −zi | < b θ 4 θN 4 DT 3N << 1 −β . 2e provided θ is chosen so that b DT < (IA2) follows immediately from the corresponding condition for z0 .

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Communications in Mathematical Physics - Volume 218 by M. Aizenman (Chief Editor)


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