The Effective Radius of Self Repelling Elastic Manifolds

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 $[-N,N]^d\cap \mathbb{Z}^d$, that takes values in $\mathbb{R}^d$. Our main result states that in two dimensions ($d=2$), the effective radius $R_N$ of the manifold is approximately $N$. This verifies the conjecture of Kantor, Kardar and Nelson [8] up to a logarithmic correction. Our results in $d\geq 3$ give a similar lower bound on $R_N$ and an upper of order $N^{d/2}$. This result implies that self-repelling elastic manifolds undergo a substantial stretching at any dimension.