Renormalization Group Approach to Interacting Polymerised Manifolds

Abstract:

We propose to study the infrared behaviour of polymerised (or tethered) random manifolds of dimension D interacting via an exclusion condition with a fixed impurity in d-dimensional Euclidean space in which the manifold is embedded. In this paper we take D=1, but modify the underlying free Gaussian covariance (thereby changing the canonical scaling dimension of the Gaussian random field) so as to simulate a polymerised manifold with fractional dimension . The canonical dimension of the coupling constant is , where −β/2 is the canonical scaling dimension of the Gaussian embedding field. β is held strictly positive and sufficiently small. For ɛ>0, sufficiently small, we prove for this model that the iterations of Wilson's renormalisation group transformations converge to a non-Gaussian fixed point. Although ɛ is small, our analysis is non-perturbative in ɛ. A similar model was studied earlier [CM] in the hierarchical approximation.