We study elastic manifolds with self-repelling terms and estimate their
effective radius. This class of manifolds is modelled by a self-repelling
vector-valued Gaussian free field with Neumann boundary conditions over
the domain [−N,N]
d ∩ Zd , that takes values in Rd . Our main result states
that in two dimensions (d = 2), the effective radius RN of the manifold is
approximately N. This verifies the conjecture of Kantor, Kardar and Nelson
(Phys. Rev. Lett. 58 (1987) 1289–1292) up to a logarithmic correction. Our results in d ≥ 3 give a similar lower bound on RN and an upper of order Nd/2.
This result implies that self-repelling elastic manifolds undergo a substantial
stretching at any dimension.