Dimensional Reduction for Directed Branched Polymers

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 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 or R 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) 1 Figure 1: A directed branched polymer in Z. which relates the DBP generating function to the density of an associated repulsive gas model. In contrast to the undirected case, the reduction in dimension is one. If we define α from the singularity ρHC ∼ (z − zc) of the hard-core gas at the negative activity critical point, and γ from the singularity ZDBP ∼ (z − z̃c) , then we find z̃c = −zc and γ(D + 1) = α(D). Define an exponent θ from the asymptotic behavior dN ∼ z c N. Then θ = 2− γ. Noting that the Yang-Lee edge exponent σ [10] can be identified with 1− α [11, 12], we obtain the relation θ(D + 1) = 1 + σ(D). (2) In one and two dimensions, σ can be determined from the exact values α(1) = 3 2 [11], α(2) = 7 6 (the latter follows from the solution to the hard-hexagon model [13] at the negative activity critical point, see [14, 15]; alternatively from [16], assuming the model is in the Yang-Lee class.) Hence θ(2) = 1 2 and θ(3) = 5 6 . We also obtain identities relating DBP correlations with repulsive gas correlations, which imply that the DBP exponent for the transverse correlation length ν⊥(D + 1) equals the repulsive gas exponent ν(D). The related problem of directed lattice animals (DA) has been studied extensively [17, 18, 19, 20, 21, 22, 14, 15, 23, 24, 25, 26, 27]. It is generally believed that the loop-free condition does not affect the critical behavior, so that both systems should have the same exponents. For DA, there are exact results in two dimensions (see [22, 23, 24], the review [25], and references therein) and in three dimensions [14]. Also, in any dimension, models of DA have been related to dynamical models of hard-core lattice gases [14] (see also [28], which connects Lorentzian semi-random lattices with DA and gives an alternative derivation of Dhar’s equivalence). They have also been related to the critical dynamics of the Ising model in an imaginary field [17, 18]. As a result, the identity (2) is believed to hold for DA [17, 18, 19], and the values in D = 2, 3 should be the same as the ones given above for DBP. We consider a class of DBP models where the vertices yk = (tk, xk) have a time compo2 nent tk ∈ R+ and a space component xk ∈ S, where S is either R or Z. If a lattice model is desired, the time component can be discretized by taking limits (see examples below). Let T be a tree graph on {1, . . . , N}, and let yk be the position of the k vertex. Fix the vertex 1 as the root, with y1 = 0. For each pair (i, j) ≡ ij, define yij := (tij , xij) := (|ti − tj |, xi − xj). (3) Each link of T connects a vertex j to a vertex i, where i is one step closer than j to the root along T . As we are considering directed BP, we require tj ≥ ti. The weight associated with each DBP configuration depends on a linking weight V (y) = V (t, x) and a repulsive weight U(y) = U(t, x) = U(t,−x). The generating function for (rooted) DBP is written as ZDBP(z) = ∞