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].
Categories: math.PR math-ph math.MP