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Register a new userDimensional Reduction Formulas for Branched Polymer Correlation Functions
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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.
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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.
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]).
We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.
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