Christian von Ferber

Applied Mathematics Research Centre, Coventry University
In this lecture, we discuss the scaling properties of long flexible polymer chains in two dimensions. We compare perturbative expansions of the Edwards model, lattice Monte Carlo simulations, and exact results using conformal invariance and 2D quantum gravity for the (scaling) properties of random walks with self and/or mutual avoidance interactions. We are especially interested hi the question of the universality of the problem of self and mutually avoiding walks hi two dimensions (2D), as well as hi validating multifraetality found hi these situations by field theoretic methods based on the Edwards model and by a conformal theory.
We focus on model star copolymers hi two dimensions: walks or polymers of different species with a common starting point; the species avoid each other mutually.
In our field theoretic approach we mapped the problem of finding the scaling properties of the copolymer star to that of determining the anomalous dimensions of appropriate local field operator products. Re-summation of the perturbation series for these dimensions provides reliable numeric values for a family of exponents that displays multifractal behavior.
A recent extension of the conformal theory for 2D polymers to random graphs using methods of 2D gravity has revealed an exact derivation of this multifractal spectrum which is in remarkable coincidence with the perturbative results for a number of situations.
To further investigate this coincidence with respect to universality and moreover to uncover the reasons for deviations, we have undertaken a series of MC simulations on the lattice where the implementation of avoidance and topological restrictions of 2D polymers is most natural. While we confirm the universality of the 2D star copolymer problem of walks with topological avoidance it appeal's to constitute a class separate from the 2D Edwards and O(n) models with repelling interactions.
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