The Effective Radius of Self Repelling Elastic Manifolds

TL;DR

This paper investigates the size and shape of self-repelling elastic manifolds modeled by Gaussian free fields, establishing bounds on their effective radius and confirming a conjecture in two dimensions.

Contribution

It provides rigorous bounds on the effective radius of self-repelling elastic manifolds, confirming a conjecture for 2D and extending understanding to higher dimensions.

Findings

Effective radius in 2D is approximately N, confirming the conjecture.

In dimensions d ≥ 3, the radius is at least of order N and at most N^{d/2}.

Self-repelling elastic manifolds exhibit significant stretching across dimensions.

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 $[-N,N]^d\cap \mathbb{Z}^d$, 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.