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Register a new userDimensional Reduction for Directed Branched Polymers
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John Z. Imbrie
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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: $gamma(D+1)=alpha(D)$ (the exponent describing the singularity of the pressure), and $ u_{perp}(D+1)= u(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $theta(D+1)=1+sigma(D)$, where $sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.
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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.
This article will review recent results on dimensional reduction for branched polymers, and discuss implications for critical phenomena. Parisi and Sourlas argued in 1981 that branched polymers fall into the universality class of the Yang-Lee edge in two fewer dimensions. Brydges and I have proven in [math-ph/0107005] that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions (which is in the Yang-Lee or $i phi^3$ class). I will describe how this equivalence arises from an underlying supersymmetry of the branched polymer model. - I will also use dimensional reduction to analyze the crossover of two-dimensional branched polymers to their mean-field limit, and to show that the scaling is given by an Airy function (the same as in [cond-mat/0107223]).
In [math-ph/0107005] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This result explains why the critical behavior of branched polymers should be the same as that of the $i phi^3$ (or Yang-Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). - In this article we review and generalize the results of [math-ph/0107005]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.
A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.
We explore the effect of an attractive interaction between parallel-aligned polymers, which are perpendicularly grafted on a substrate. Such an attractive interaction could be due to, e.g., reversible cross-links. The competition between permanent grafting favoring a homogeneous state of the polymer brush and the attraction, which tends to induce in-plane collapse of the aligned polymers, gives rise to an instability of the homogeneous phase to a bundled state. In this latter state the in-plane translational symmetry is spontaneously broken and the density is modulated with a finite wavelength, which is set by the length scale of transverse fluctuations of the grafted polymers. We analyze the instability for two models of aligned polymers: directed polymers with a line tension and weakly bending chains with a bending stiffness.
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