The effective radius of self repelling elastic manifolds

Abstract

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 ${[-\mathit{N},\mathit{N}]}^{\mathit{d}}\cap {\mathbb{Z}}^{\mathit{d}}$, that takes values in ${\mathbb{R}}^{\mathit{d}}$. Our main result states that in two dimensions ($\mathit{d}=2$), the effective radius ${\mathit{R}}_{\mathit{N}}$ 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 $\mathit{d}\ge 3$ give a similar lower bound on ${\mathit{R}}_{\mathit{N}}$ and an upper of order ${\mathit{N}}^{\mathit{d}/2}$. This result implies that self-repelling elastic manifolds undergo a substantial stretching at any dimension.