Vincent Rivasseau (Chief Editor)'s Annales Henri Poincaré - Volume 1 PDF

By Vincent Rivasseau (Chief Editor)

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Correlation Function, Lattice Spin Systems, Exponential Decay, Witten Laplacian. 1) Rm in the limit m → ∞ and for large β > 0. 3) is the expectation of xj . In [10], the authors studied exponential decay of the correlations under assumptions on the function H containing that of uniform strict convexity. They exhibited a certain matrix Schr¨ odinger operator for gradients and studied it by means of a maximum principle. The global convexity was quite crucial for the maximum principle to apply. 60 V.

This subgraph gj must have j > i hence have t(j) > i. Therefore for each half-line in one of these sets there is always at least one corresponding half-line in Si0 (C) (Si1 (C)). Lemma 10 For any connected component in Tik we have xki ≥ 1/2. Proof. We distinguish three situations. 50 M. Disertori and V. Rivasseau Ann. Henri Poincar´ e • If |Eik (C)| ≥ 5, in fact, by parity of the number of external half-lines of any subgraph, |Eik (C)| ≥ 6 and then xki ≥ (1/4)(|Eik (C)| − 4) ≥ 1/2. 106) • If |Eik (C)| = 4, then there must be a subgraph gj ∈ DC0c (P ) with j ≥ i (j = i only if li belongs to the connected component Tik (J, P )) and A(j) < i.

N ! n ¯ −1 0≤w1 ≤···≤wn−1 ≤1 q=1 ¯ n ¯ −1 RGki Gk i ∈Dµ q =1 Λ ,wq DΛ0 dwq v (T , Ω) d2 x1 . . d2 xn¯ o−T E,µ Col,Ω λw(v) N δmw(v ) v δζw(v ) v ∂ Λ0 ,wq (¯ xlq , xlq ) D ∂Λ Λ (¯ xlq , xlq ) det M(µ) φ1 (xi1 ) . . 126) q=q a) b) Figure 23 0 The second term LΓΛΛ 2p (φ1 , . . , φ2p ) is the series obtained when the derivative falls on a loop line in the determinant (see Figure 23b): ∞ 0 LΓΛΛ 2p (φ1 , . . n ! n ¯ −1 0≤w1 ≤···≤wn−1 ≤1 q=1 ¯ n ¯ −1 v Λ ,wq DΛ0 RGki Gk i ∈Dµ dwq (T , Ω) d2 x1 . .

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