Surface geometry and local critical behaviour : the self-avoiding-walk

(Received 21 October 1992, accepted 27 November 1992)

Abstract
The statistics of a polymer chain confined inside a system which is limited by a parabolic-like surface is studied through Monte-Carlo simulations in two dimensions. In agreement with scaling considerations, the surface geometry is found to be a relevant perturbation to the flat surface behaviour when the shape exponent k is smaller than one. In this case the system becomes anisotropic with a radius exponent along the parabola greater than the exponent in the transverse direction. When k < 1 the anisotropy ratio z adjusts itself to the value k^-1 for which the surface geometry is a marginal perturbation. The exponents obtained analytically, using either the blob picture approach or a Flory approximation, are in good agreement with the 2d simulation results.