Summary of the expansion for A Proof of the identity 2.35 b;{o} e;{o} in ˜C

4.3 Summary of the expansion for A

~x J Recall 4.28 and 4.54, and define, for N ≥ 0, a N ~x J \I , ~x I = a N ~x J \I , ~x I ; 2 + X y 1 a N y 1 , ~x I ; 3 + a N y 1 , ~x I ; 4 τ~x J \I − y 1 , 4.55 let a N ~x J be given by 2.26 and define a ~x J = ∞ X N =0 −1 N a N ~x J , φy 1 , y 2 ± = ∞ X N =0 −1 N φ N y 1 , y 2 ± . 4.56 Now, we can summarize the expansion in the previous two subsections as follows: Proposition 4.5 Expansion for A ~x J . For any λ ≥ 0, J 6= ∅ and ~x J ∈ Λ |J| , A ~x J = a ~x J + X ∅6=IJ 1 X y 1 ,y 2 Cy 1 , y 2 τ~x J \I − y 1 τ~x I − y 2 , 4.57 where Cy 1 , y 2 = φy 1 , y 2 + + φy 2 , y 1 + − φy 2 , y 1 − . 4.58 Proof. We substitute 4.54 into 4.28. Note that, by 4.7, j I 1 precisely when 1 ∈ I. Thus, also taking notice of the difference in J \ I, which contains 1 in 2.18, but may not in 4.28, we split the sum over I arising from in 4.28 as X y 1 ,y 2 X ∅6=I⊂J 1 φy 1 , y 2 + τ~x J \I − y 1 τ~x I − y 2 + X 1 ∈IJ φy 1 , y 2 + − φy 1 , y 2 − τ~x J \I − y 1 τ~x I − y 2 = X y 1 ,y 2 X ∅6=I⊂J 1 φy 1 , y 2 + τ~x J \I − y 1 τ~x I − y 2 + X y ′ 1 ,y ′ 2 X ∅6=I ′ ⊂J 1 φy ′ 2 , y ′ 1 + − φy ′ 2 , y ′ 1 + τ~x J \I ′ − y ′ 1 τ~x I ′ − y ′ 2 = X y 1 ,y 2 X ∅6=I⊂J 1 φy 1 , y 2 + + φy 2 , y 1 + − φy 2 , y 1 − τ~x J \I − y 1 τ~x I − y 2 , 4.59 where y ′ 1 , y ′ 2 and I ′ in the middle expression correspond to y ′ 1 = y 2 , y ′ 2 = y 1 and I ′ = J \ I on the left hand side of 4.59. Therefore, we arrive at 4.57–4.58. This completes the derivation of the lace expansion for the r-point function.

4.4 Proof of the identity 2.35

Note that, by 2.24, 2.35 is equivalent to C ǫ,ǫ y 1 , y 2 ≡ C y 1 , ǫ, y 2 , ǫ = p ǫ y 1 p ǫ y 2 1 − δ y 1 , y 2 . 4.60 839 By 4.58, 4.60 follows when we show that φ ǫ,ǫ y 1 , y 2 ± = p ǫ y 1 p ǫ y 2 1 − δ y 1 , y 2 . 4.61 According to 4.50, φ N ǫ,ǫ y 1 , y 2 ± = 0 unless N = 0. Also, by 4.27, we see that φ ǫ,ǫ y 1 , y 2 + = φ ǫ,ǫ y 1 , y 2 − . Therefore, since p ǫ y 1 p ǫ y 2 1 − δ y 1 , y 2 is symmetric in y 1 , y 2 , it suffices to show that φ ǫ,ǫ y 1 , y 2 + ≡ X b,e b6=e p b p e ˜ E b h 1 E ′

o,b;{o}

1 {H ǫ

o,e;{o} in ˜C

e o} B δ

b, y

1 , ǫ; Co B δ

e, y

2 , ǫ; ˜ C e o i = p ǫ y 1 p ǫ y 2 1 − δ y 1 , y 2 . 4.62 However, this immediately follows from the fact that the product of the two indicators in ˜ E b is 1 {b=e=o} cf., 3.4 and 4.35 and that, by 4.21, B δ

b, y

1 , ǫ; Co = δ

b, y

1 , ǫ and B δ

e, y

2 , ǫ; ˜ C e o = δ

e, y

2 , ǫ . This completes the proof of 2.35. 5 Bounds on Bx and A ~x J In this section, we prove the following proposition, in which we denote the second-largest element of {t j } j ∈J by ¯t = ¯t J : Proposition 5.1 Bounds on the coefficients of the linear expansion. i Let d 4 and L ≫ 1. For λ ≤ λ ǫ c , N ≥ 0, t ∈ ǫN, ~t J ∈ ǫZ + |J| and q = 0, 2, X x |x| q B N t x ≤ 1 − ǫδ q,0 + λǫσ q δ t, ǫ δ N ,0 + ǫ 2 O β 1 ∨N σ q 1 + t d−q2 , 5.1 X ~x J A N ~t J ~x J ≤ ǫOβ N O 1 + ¯t r −3 , 5.2 where the constant in the O β term is independent of ǫ, L, N and t or ¯t in 5.2. ii Let d ≤ 4 with α ≡ bd − 4 −d 2 0, ˆ β T = β 1 T −α with α ∈ 0, α, and L 1 ≫ 1. For λ ≤ λ ǫ c , N ≥ 0, t ∈ ǫN ∩ [0, T log T ], ~t J ∈ ǫZ + |J| with max j ∈J t j ≤ T log T and q = 0, 2, X x |x| q B N t x ≤ 1 − ǫδ q,0 + λǫσ q T δ t, ǫ δ N ,0 + ǫ 2 O β T O ˆ β T ∨N−1 σ q T 1 + t d−q2 , 5.3 X ~x J A N ~t J ~x J ≤ ǫO ˆ β T N O 1 + ¯t r −3 , 5.4 where the constants in the O β T and O ˆ β T terms are independent of ǫ, L 1 , T, N and t or ¯t in 5.4. In Section 5.1, we define several constructions that will be used later to define bounding diagrams for Bx , A ~x , Cy 1 , y 2 and a ~x . There, we also summarize effects of these constructions. Then, we prove the above bounds on Bx in Section 5.2, and the bounds on A ~x J in Section 5.3. Throughout Sections 5–7, we shall frequently assume that λ ≤ 2, which follows from 2.5 for d 4 and L ≫ 1, and from the restriction on λ T in Theorem 1.1 for d ≤ 4 and L 1 ≫ 1. 840 a v x u + v u x b u=v x + u=v x Figure 9: Schematic representation of Lu, v ; x for a u 6= v and b u = v. Here, the tilted arrows denote spatial bonds, while the short double line segments at u in Case a denote unspecified bonds that could be spatial or temporal.

5.1 Constructions: I