论文标题
较低范围的自我抑制弹性歧管
Self-Repelling Elastic Manifolds with Low Dimensional Range
论文作者
论文摘要
我们认为具有域$ [-n，n]^d \ cap \ mathbb {z}^d $的自我撤销弹性歧管，该弹性歧管在$ \ mathbb {r}^d $中取值。我们的主要结果指出，当域的尺寸为$ d = 2 $并且范围的尺寸为$ d = 1 $时，歧管的有效半径$ r_n $约为$ n^{4/3} $。这验证了Kantor，Kardar和Nelson [7]的猜想。我们在$ d \ geq 3 $和$ d <d $的情况下给出了$ n^{\ frac {\ frac {1} {d} {d} \ left（d- \ frac {2（d-d）} {d+2} {d+2} \ right）} $的下限，并获得上限的优势， $ n^{\ frac {d} {2}+\ frac {d-d} {d+2}} $。这些结果表明，比作者在[10]中研究的$ d = d $的情况下，具有较低维度范围的自我撤销弹性歧管经历了明显的伸展。
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].
