2004/02/29 by John Imbrie, John Z. Imbrie
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #msc:82B20 #msc:82B21 #msc:82B27 #msc:82B41
paper · pdf · doi:10.1088/0305-4470/37/12/l03
published as J. Phys. A: Math. Gen. 37, L137--L142 (2004) · 7 pages, 1 eps figure, ref 24 corrected
openalex publication_date 2004/03/09 · arxiv created 2004/03/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D +1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: γ(D + 1) = α(D) (the exponent describing the singularity of the pressure), and ν⊥(D + 1) = ν(D) (the correlation length exponent of the repulsive gas). It also leads to the relation θ(D + 1) = 1 + σ(D), where σ(D) is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions. PACS numbers: 64.60.Fr, 04.20.Jb, 04.60.Nc, 05.20.Jj The phenomenon of dimensional reduction has attracted considerable attention over the years. The first example was the controversial random field Ising model (RFIM), whose critical behavior was conjectured to be the same as the pure Ising model in two fewer dimensions [1]. A proof of long-range order for the RFIM in three dimensions [2, 3] showed that dimensional reduction fails there, and recent work [4, 5, 6] has elucidated what goes wrong. A second example is the Parisi-Sourlas reduction of branched polymers (BP) in D + 2 dimensions to the Yang-Lee edge or iϕ 3 field theory in D dimensions [7]. This was recently confirmed with the discovery of an exact relationship between BP models and repulsive gases at negative activity in two fewer dimensions [8, 9]. The failure of the heuristic arguments for dimensional reduction in the RFIM underscores the importance of having an exact result. In this letter we give a third example, in which directed branched polymers (DBP) reduce to the repulsive gas at negative activity in one fewer dimension. We consider directed branched polymers as self-avoiding tree graphs embedded in Z D+1 or R D+1 so that every vertex can be reached from the root at 0 by a sequence of links which move forward with respect to a preferred direction. See Figure 1. Let dN denote the number of DBP with N vertices, and let ZDBP(z) = ∑ N dNz N. We prove the identity ρHC(z) = −ZDBP(−z), (1)