TAILIEUCHUNG - Đề tài " Branched polymers and dimensional reduction "

We establish an exact relation between self-avoiding branched polymers in D + 2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions. We review conjectures and results on critical exponents for D + 2 = 2, 3, 4 and show that they are corollaries of our result. We explain the connection (first proposed by Parisi and Sourlas) between branched polymers in D + 2 dimensions and the Yang-Lee edge singularity in D dimensions. | Annals of Mathematics Branched polymers and dimensional reduction By David C. Brydges and John Z. Imbrie Annals of Mathematics 158 2003 1019 1039 Branched polymers and dimensional reduction By David C. BRyDGES and John Z. Imbrie Abstract We establish an exact relation between self-avoiding branched polymers in D 2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions. We review conjectures and results on critical exponents for D 2 2 3 4 and show that they are corollaries of our result. We explain the connection first proposed by Parisi and Sourlas between branched polymers in D 2 dimensions and the Yang-Lee edge singularity in D dimensions. 1. Introduction A branched polymer is usually defined Sla99 to be a finite subset y1 . yN of the lattice ZD 2 together with a tree graph whose vertices are y1 . yN and whose edges yi yj are such that yi yj 1 so that points in an edge of the tree graph are necessarily nearest neighbors. A tree graph is a connected graph without loops. Since the points yi are distinct branched polymers are self-avoiding. Figure 1 shows a branched polymer with N 9 vertices on a two-dimensional lattice. Critical exponents may be defined by considering statistical ensembles of branched polymers. Define two branched polymers to be equivalent when one is a lattice translate of the other and let CN be the number of equivalence classes of branched polymers with N vertices. For example c1 c2 c3 1 2 6 respectively in Z2. Some authors prefer to consider the number of branched polymers that contain the origin. This is Ncn since there are N representatives of each class which contain the origin. Research supported by NSF Grant DMS-9706166 to David Brydges and Natural Sciences and Engineering Research Council of Canada. 1020 DAVID C. BRYDGES AND JOHN Z. IMBRIE Figure 1. One expects that cN has an asymptotic law of the form cN - N--z N in the sense that limN inN ln cNzN 0. The critical exponent 0 is conjectured to be .

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