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By M. Aizenman (Chief Editor)

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32) Passive Advection and the Degenerate Elliptic Operators Mn 35 Since (1 + |yi |)2−ξ −d ≤ 1 for any i ∈ U , we can conclude that (∗3 ) ≤ C3 (1 + |Lyi |)2−ξ −d := (∗4 ). 5 we have (1 + |Lxi |)2−ξ −d =: (∗5 ). 5 we have |Lxi | ≤ C for i ∈ U1 we can finally conclude that n (∗5 ) ≤ C5 (1 + |Lx2i−1 |)2−ξ −d . 5 and thus in this case we have n−1 y∈Bjx d (n−1)d y EC2n−1 (x, y) (1 + |Ly2i−1 |)2−ξ −d χ (|Ly2n−1 |) = 0. 37) i=1 So we may assume 2n − 1 ∈ U1 . 38) for i ∈ U2 . Therefore n y∈Bjx \Bjx−1 d (2n−1)d y EC2n−1 (x, y) i=1 (1 + |Lxi |)2−ξ −d ≤ C7 (1 + |Ly2i−1 |)2−ξ −d i∈U2 y∈Bjx \Bjx−1 d (2n−1)d y ECn (x − y) (1 + |Lyi |)2−ξ −d χ (Ly2n−1 ) = (∗6 ).

Such a graph masquerades under various names in combinatorics: bipartite graph, or a complete matching (cf. [LW], for example), and is nothing more than another way to think about fixed-point-free involutions. Pictorially the complete matching looks like r1 • —– • s1 r2 • —– • s2 .. rb • —– • sb Fig. 1 We denote the complete matching determined by ϕ by the symbol µϕ . Any complete matching on a set labelled by positive integers corresponds to a fixed-point-free involution and a state v. Let us agree that two complete matchings µ1 , µ2 are isomorphic if there is a bijection from the node set of µ1 to the node set of µ2 that preserves labels.

Such a graph masquerades under various names in combinatorics: bipartite graph, or a complete matching (cf. [LW], for example), and is nothing more than another way to think about fixed-point-free involutions. Pictorially the complete matching looks like r1 • —– • s1 r2 • —– • s2 .. rb • —– • sb Fig. 1 We denote the complete matching determined by ϕ by the symbol µϕ . Any complete matching on a set labelled by positive integers corresponds to a fixed-point-free involution and a state v. Let us agree that two complete matchings µ1 , µ2 are isomorphic if there is a bijection from the node set of µ1 to the node set of µ2 that preserves labels.

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Communications in Mathematical Physics - Volume 235 by M. Aizenman (Chief Editor)


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