By M. Aizenman (Chief Editor)

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5. 94) d−2 2 (1+(n−1)δ) , then χ S1c ≤ e−L d−2 (1+(n−1)δ) 4 . 95) Proof. For α ∈ R, b > 0, we have eαb χ(|φz − φz | > b) ≤ 2 cosh(α(φz − φz )) − ≤2 L 20 f cosh(αφz )ex p(− n Q n (φ)) 2 ≤ 2ex p( α 2 2n L 20 L d−2(1+(n−1)δ) dφ − f ex p(− L220 n Q n (φ))dφ ). 1]. Letting α= bL d−2(1+(n−1)δ) 22n L 20 we have χ (|φz − φz | > b) ≤ e For each y ∈ − b2 n 42 L 20 L d−2(1+(n−1)δ) . 97) , define B y = {φ ∈ R| | ||φ y | > |x−y| 1 L d−2 4 (1+(n−1)δ) B y = {φ ∈ R| | ||φ y − φ y | > e L 2(1+(n−1)δ+(n−1) ) }, 1 2L d−2 4 (1+(n−1)δ) The inequality in these definitions holds at one point y ∈ By ⊂ By .

111) 40 H. 111) becomes d {( f n,ψx − f n,ψx dψw | w ( f n,ψz − f n,ψz w ( f n,ψz ≤ w {ψ1 +tψ2 } {ψ1 +tψ2 } ) + f n,ψx ; f n,ψw : 1 c {ψ1 +tψ2 } ) ( f n,ψ y − f n,ψ y {ψ1 +tψ2 } ) − f n,ψz {ψ1 +tψ2 } ) {ψ1 +tψ2 } ψ2,w | + · · · maxφ (| f n,ψx ψw − f n,ψx ψw · ( f n,ψ y − f n,ψ y − f n,ψ y {ψ1 +tψ2 } )} {ψ1 +tψ2 } ψ2,w | | ( f n,ψx ψw − f n,ψx ψw ≤ {ψ1 +tψ2 } )( f n,ψ y {ψ1 +tψ2 } ) 2 1 2 {ψ1 +tψ2 } {ψ1 +tψ2 } + f n,ψx ; f n,ψw : 1 ( f n,ψz − f n,ψz ≤e −L n {ψ1 +tψ2 } ) 2 c {ψ1 +tψ2 } |) 1 2 {ψ1 +tψ2 } |ψ2,w | + · · · .

129) is bounded by e−L . 6. nδ 44 H. 135) e L 2(1+(n−1)δ+(n−1) ) , |φw | |w−z| d−2 4 (1+(n−1)δ) e L 2(1+(n−1)δ+(n−1) ) for all w ∈ }. 6, | f n,ψx ; f n,ψ y ; f n,ψz : χ S1c c {ψ1 } | d−2 (1+(n−1)δ) 4 ≤ e−L . 139) where I3 = {x1 , x2 , y1 , y2 , z 1 , z 2 ∈ |max(|x −x1 |, |x −x2 |, |x1 − x2 |) ≤ L 1+(n−1)δ+(n−1)2 , max(|y − y1 |, |y − y2 |, |y1 − y2 |) ≤ L 1+(n−1)δ+(n−1)2 , max(|z − z 1 |, |z − z 2 |, |z 1 − z 2 |) ≤ L 1+(n−1)δ+(n−1)2 }. 1. Towards a Nonperturbative Renormalization Group Analysis 45 5.

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Communications In Mathematical Physics - Volume 282 by M. Aizenman (Chief Editor)


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