By M. Aizenman (Chief Editor)

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V N . This done, use φv to denote the homomorphism from V to φ(Wv ) × V3 × . . × V N that is defined as follows: Let G = g1 g2 . . g N , where each gk ∈ Vk , and let (g1 g2 )v denote the component of g1 g2 in Wv . Now set φ(G) = φ((g1 g2 ))g3 . . g N . With the stage now set, here is the promised 32 C. H. 2. With the point z = (z 0 , z 3 , . . 13) as z 1 and z 2 converge to z 0 . 14) here the notation uses Eˆ 0 to denote the smallest of the numbers in the set Eˆ v , and φ0 to denote the corresponding version of φv .

24) is identical to the (1, 0) portion of the differential at z 1 of (z,F) when the latter is considered as a function of the first component of z with the remaining components fixed. 4. The appearance of the required singular behavior near the diagonal comes from the second derivatives of a τ at τ = 0 that appear via the chain τ rule when computing those of (z,F) . 18). 17) can be used to compute the latter derivative. 7). 1). The detailed manipulations are reasonably straightforward in both cases and left to the reader.

6) is used to write exp( Aˆ (∗z,z ) )V as an absolutely convergent sum that has the schematic form 1 M×M +σ 0 1≤a≤d α 1 +σ2 1≤a,b≤d α,β 0 s1 0 M1−s tα,a Ms × M1−s tα,a Ms ds M1−s1 tα,a Ms1 −s2 tβ,b Ms2 × M1−s1 tα,a Ms1 −s2 tβ,b Ms2 ds2 ds1 + . . 7) Here, the unprimed terms are defined by z and operator only on F¯ , while the primed terms are defined by z and operate only on F . Also, M = exp( Aˆ z ) and M = exp( Aˆ z ), each viewed as an endomorphism of a particular finite dimensional subspace of C ∞ (× N P).

### Communications In Mathematical Physics - Volume 267 by M. Aizenman (Chief Editor)

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