Self-Repelling Elastic Manifolds with Low Dimensional Range

TL;DR

This paper analyzes self-repelling elastic manifolds with low-dimensional range, establishing the effective radius scaling for specific dimensions and confirming a conjecture for the case where the domain is two-dimensional and the range is one-dimensional.

Contribution

The paper provides rigorous bounds on the effective radius of self-repelling elastic manifolds, verifying a conjecture and extending understanding to cases with low-dimensional range.

Findings

Effective radius scales as N^{4/3} for d=2, D=1.

Lower and upper bounds for R_N when d≥3 and D<d.

Manifolds with low-dimensional range stretch more than when D=d.

Abstract

We consider self-repelling elastic manifolds with a domain $[-N,N]^d \cap \mathbb{Z}^d$, that take values in $\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for the case where $d \geq 3$ and $D <d$ give a lower bound on $R_N$ of order $N^{\frac{1}{D} \left(d-\frac{2(d-D)}{D+2} \right)}$ and an upper bound proportional to $N^{\frac{d}{2}+\frac{d-D}{D+2}}$. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where $d=D$, which was studied by the authors in [10].